qu.1.topic=Section 1.2: Polynomials, Powers and Logs@

qu.1.1.mode=Multipart@
qu.1.1.name=1.2.14 Polynomials@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=int(rint(2)+2); $b=int(rint(3)+1); $x=int(rint(3)+1); $ansb=int($a*$x*($x)+ $b); $xc=int($x+1); $xcm=int(-$xc); $c1=int($xc*$xc); $ansc=int($a*($xc)*($xc) + $b);$qd=int($b-1); $xmax=int(2*$xc); $ymax=int($a*$xmax*$xmax + $b + 2);@
qu.1.1.uid=94edfbe9-40e9-40d0-9097-76c61acc2149@
qu.1.1.question=

          <p class="noindent">Let 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">y</mi>

            <mo class="MathClass-rel">=</mo>

            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>
          </math>

          where 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <msup>
              <mrow>
                <mi class="MathClass-ord">x</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">2</mn>
              </mrow>
            </msup>

            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>
          </math>

          .</p>
        @
qu.1.1.weighting=1,1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=

            <p class="noindent">Find the value of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">y</mi>
            </math>

            when 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            is zero.</p>
          @
qu.1.1.part.1.answer=${b}? 0.05@
qu.1.1.part.1.mode=Plain Number@
qu.1.1.part.1.comment=

            <p class="noindent">We are asked for the value of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">y</mi>
            </math>

            when 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            is zero. That is, we are asked for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>
            </math>

            . Plugging in we get 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mo class="MathClass-open">(</mo>

                  <mn class="MathClass-ord">0</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn>${b}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">
            </p>
          @
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.question=

            <p class="noindent">What is 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mn>${x}</mn>

              <mo class="MathClass-close">)</mo>
            </math>

            ?</p>
          @
qu.1.1.part.2.answer=${ansb} ? 0.05@
qu.1.1.part.2.mode=Plain Number@
qu.1.1.part.2.comment=

            <p class="noindent">Substituting we get 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mn>${x}</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mo class="MathClass-open">(</mo>

                  <mn>${x}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn>${ansb}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">
            </p>
          @
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.question=

            <p class="noindent">Which values of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            give 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">y</mi>
            </math>

            a value of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn>${ansc}</mn>
            </math>

            ?</p>
          @
qu.1.1.part.3.answer=${xc};${xcm}@
qu.1.1.part.3.mode=Multi Formula@
qu.1.1.part.3.comment=

          <p class="noindent">Asking what values of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>
          </math>

          give a 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">y</mi>
          </math>

          -value of 11 is the same as solving 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mtable columnspacing="0" class="eqnarray-star">
              <mtr>
                <mtd columnalign="right">
                  <mi class="MathClass-ord">y</mi>

                  <mo class="MathClass-rel">=</mo>

                  <mn>${ansc}</mn>

                  <mspace width=".167em"/>
                </mtd>

                <mtd columnalign="center">
                  <mo class="MathClass-rel">=</mo>
                </mtd>

                <mtd columnalign="left">
                  <mn>${a}</mn>

                  <msup>
                    <mrow>
                      <mi class="MathClass-ord">x</mi>
                    </mrow>

                    <mrow>
                      <mn class="MathClass-ord">2</mn>
                    </mrow>
                  </msup>

                  <mo class="MathClass-bin">+</mo>

                  <mn>${b}</mn>
                </mtd>

                <mtd columnalign="left">
                  <mtext class="eqnarray">
                  </mtext>
                </mtd>
              </mtr>

              <mtr>
                <mtd columnalign="right">
                  <msup>
                    <mrow>
                      <mi class="MathClass-ord">x</mi>
                    </mrow>

                    <mrow>
                      <mn class="MathClass-ord">2</mn>
                    </mrow>
                  </msup>

                  <mspace width=".167em"/>
                </mtd>

                <mtd columnalign="center">
                  <mo class="MathClass-rel">=</mo>
                </mtd>

                <mtd columnalign="left">
                  <mn>${c1}</mn>
                </mtd>

                <mtd columnalign="left">
                  <mtext class="eqnarray">
                  </mtext>
                </mtd>
              </mtr>

              <mtr>
                <mtd columnalign="right">
                  <mi class="MathClass-ord">x</mi>

                  <mspace width=".167em"/>
                </mtd>

                <mtd columnalign="center">
                  <mo class="MathClass-rel">=</mo>
                </mtd>

                <mtd columnalign="left">
                  <mo class="MathClass-bin">&#177;</mo>

                  <mn>${xc}</mn>

                  <mo class="MathClass-punc">.</mo>
                </mtd>

                <mtd columnalign="left">
                  <mtext class="eqnarray">
                  </mtext>
                </mtd>
              </mtr>
            </mtable>
          </math>


          <p class="nopar">We can also solve this problem graphically. Looking at the graph of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <msup>
              <mrow>
                <mi class="MathClass-ord">x</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">2</mn>
              </mrow>
            </msup>

            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>
          </math>

          below 
          <br class="newline"/>
          </p>

          <applet code="applets.grapher.Graph" width="250" height="250" archive="graphing.jar">
            <param name="y1" value="${a}*x*x+${b}">
            </param>

            <param name="gridLines" value="1">
            </param>

            <param name="xMin" value="0">
            </param>

            <param name="xMax" value="${xmax}">
            </param>

            <param name="yMin" value="0">
            </param>

            <param name="yMax" value="${ymax}">
            </param>
          </applet>

          <br class="newline"/>

          We see that the graph of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>
          </math>

          intersects the line 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">y</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${ansc}</mn>
          </math>

          at 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${xc}</mn>
          </math>

          and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mo class="MathClass-bin">-</mo>

            <mn>${xc}</mn>
          </math>

          . Thus, when 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>
          </math>

          equals 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mn>${xc}</mn>
          </math>

          or 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>
          </math>

          equals 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mo class="MathClass-bin">-</mo>

            <mn>${xc}</mn>
          </math>

          we have 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${ansc}</mn>
          </math>

          .@
qu.1.1.part.4.extra=@
qu.1.1.part.4.editing=useHTML@
qu.1.1.part.4.question=

            <p class="noindent">Are there any values of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            that give 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">y</mi>
            </math>

            a value of ${qd}? 
            <br class="newline"/>

            <1>
            </p>
          @
qu.1.1.part.4.blank.1=No, yes@
qu.1.1.part.4.grader.1=menu@
qu.1.1.part.4.mode=Blanks@
qu.1.1.part.4.comment=

            <p class="noindent">No. No matter what, 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>
            </math>

            is greater than or equal to 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn class="MathClass-ord">0</mn>
            </math>

            , so 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">y</mi>

              <mo class="MathClass-rel">=</mo>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn class="MathClass-ord">2</mn>
            </math>

            is greater than or equal to 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn class="MathClass-ord">2</mn>
            </math>

            .</p>
          @

qu.1.2.mode=Multipart@
qu.1.2.name=1.2.18 Polynomials@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$x=int(rint(4)+2); $ansb=decimal(0,8+10*($x)/3-65*($x)^2/36+7*($x)^3/36); $x1=int(rint(3)+1);@
qu.1.2.uid=9d2a65bb-db31-4a1c-8853-d873314bd175@
qu.1.2.question=

        <p class="noindent">Consider the function 

        <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
          <mi class="MathClass-ord">v</mi>

          <mo class="MathClass-rel">=</mo>

          <mi class="MathClass-ord">f</mi>

          <mo class="MathClass-open">(</mo>

          <mi class="MathClass-ord">t</mi>

          <mo class="MathClass-close">)</mo>
        </math>

        show below. 
        <br class="newline"/>

        <br class="newline"/>
        </p>

        <applet code="applets.grapher.Graph" width="250" height="250" archive="graphing.jar">
          <param name="y1" value="8+10*x/3-65*x^2/36+7*x^3/36">
          </param>

          <param name="gridLines" value="6">
          </param>

          <param name="xMin" value="0">
          </param>

          <param name="xMax" value="6">
          </param>

          <param name="yMin" value="0">
          </param>

          <param name="yMax" value="12">
          </param>
        </applet>

        <br class="newline"/>

        <br class="newline"/>


        <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
          <mi class="MathClass-ord">t</mi>
        </math>

        is plotted on the horizontal axis, 

        <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
          <mi class="MathClass-ord">v</mi>
        </math>

        on the vertical axis.@
qu.1.2.weighting=1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.extra=@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.question=

            <p class="noindent">What is the value of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">v</mi>
            </math>

            when 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            ? 
            <1>
            </p>
          @
qu.1.2.part.1.blank.1=8%3f0.25@
qu.1.2.part.1.grader.1=formula@
qu.1.2.part.1.mode=Blanks@
qu.1.2.part.1.comment=

            <p class="noindent">At 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            , we see that 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">v</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">8</mn>
            </math>

            .</p>
          @
qu.1.2.part.2.extra=@
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.question=

            <p class="noindent">

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">v</mi>

              <mo class="MathClass-open">(</mo>

              <mn>${x}</mn>

              <mo class="MathClass-close">)</mo>
            </math>

            =
            <1>

            to the nearest integer.</p>
          @
qu.1.2.part.2.blank.1=%24%7bansb%7d %3f 0.25@
qu.1.2.part.2.grader.1=formula@
qu.1.2.part.2.mode=Blanks@
qu.1.2.part.2.comment=

            <p class="noindent">When 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mn>${x}</mn>
            </math>

            , we see that 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">v</mi>

              <mo class="MathClass-rel">=</mo>

              <mn>${ansb}</mn>
            </math>

            .</p>
          @

qu.1.3.mode=Blanks@
qu.1.3.name=1.2.9 Using logs@
qu.1.3.comment=

          <p class="noindent">Dividing both sides by 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">P</mi>
          </math>

          we get 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mn>${a}</mn>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mn>${c}</mn>

                <mo class="MathClass-punc">&#183;</mo>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">Taking the natural log of both sides gives 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mn>${c}</mn>

                <mo class="MathClass-punc">&#183;</mo>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${a}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">This gives 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mn>${c}</mn>

            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${a}</mn>

            <mo class="MathClass-close">)</mo>
          </math>


          <p class="nopar">or in other words 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>

            <mstyle displaystyle="true">
              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>ln</mi>

                  <mo>&#8289;</mo>


                  <mo class="MathClass-open">(</mo>

                  <mn>${a}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mn>${c}</mn>
                </mrow>
              </mfrac>
            </mstyle>

            <mo class="MathClass-rel">&#8776;</mo>

            <mn>${ans}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          </p>
        @
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=int(rint(5)+2); $c=(rint(10)+1)/10; $ans=decimal(4,(1/$c)*ln($a));@
qu.1.3.uid=abcf41db-de05-4797-ae88-63760a5bac8e@
qu.1.3.question=

          <p class="noindent">Solve for 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>
          </math>

          using natural logarithms 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mn>${a}</mn>

            <mi class="MathClass-ord">P</mi>

            <mo class="MathClass-rel">=</mo>

            <mi class="MathClass-ord">P</mi>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mn>${c}</mn>

                <mo class="MathClass-punc">&#183;</mo>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>

          .</p>


          <p class="noindent">

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>
          </math>

          <1>

          correct to 2 decimal places.</p>
        @
qu.1.3.blank.1=%24%7bans%7d %3f 0.005@
qu.1.3.grader.1=formula@
qu.1.3.extra=@

qu.1.4.mode=Equation@
qu.1.4.name=1.2.11 Exponential functions@
qu.1.4.comment=

          <p class="noindent">Since we want 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <msup>
              <mrow>
                <mo class="MathClass-open">(</mo>

                <mn>${b}</mn>

                <mo class="MathClass-close">)</mo>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">k</mi>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mo class="MathClass-open">(</mo>

                <msup>
                  <mrow>
                    <mi class="MathClass-ord">e</mi>
                  </mrow>

                  <mrow>
                    <mi class="MathClass-ord">k</mi>
                  </mrow>
                </msup>

                <mo class="MathClass-close">)</mo>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>

          , so 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mn>${b}</mn>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">k</mi>
              </mrow>
            </msup>
          </math>

          , and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">k</mi>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${b}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${c0}</mn>
          </math>

          . Thus, 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">P</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mn>${c}</mn>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>

          .</p>
        @
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=int(100*rint(5)+2); $b=(5+rint(10))/10; $c0=decimal(4,ln($b)); $c=decimal(2,$c0);@
qu.1.4.uid=d4a7f234-24fb-4508-8233-ad31653ac233@
qu.1.4.question=

          <p class="noindent">Write the function 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">P</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <msup>
              <mrow>
                <mo class="MathClass-open">(</mo>

                <mn>${b}</mn>

                <mo class="MathClass-close">)</mo>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>

          in the form 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">P</mi>

            <mo class="MathClass-rel">=</mo>

            <msub>
              <mrow>
                <mi class="MathClass-ord">P</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">0</mn>
              </mrow>
            </msub>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">k</mi>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          <br class="newline"/>

          <br class="newline"/>

          (Give 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">k</mi>
          </math>

          correct to 2 decimal places.)</p>
        @
qu.1.4.answer=P=${a}*e^(${c}*t)@

qu.1.5.mode=Plain Number@
qu.1.5.name=1.2.7 Using logs@
qu.1.5.comment=

          <p class="noindent">Taking natural logs of both sides we get 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${a}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${b}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-bin">+</mo>

            <mi class="MathClass-ord">t</mi>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn class="MathClass-ord">3</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">This gives 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn class="MathClass-ord">3</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${a}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-bin">-</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${b}</mn>

            <mo class="MathClass-close">)</mo>
          </math>


          <p class="nopar">or in other words 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>

            <mstyle displaystyle="true">
              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>ln</mi>

                  <mo>&#8289;</mo>


                  <mo class="MathClass-open">(</mo>

                  <mn>${a}</mn>

                  <mo class="MathClass-bin">-</mo>

                  <mo>&#8290;</mo>

                  <mi>ln</mi>

                  <mo>&#8289;</mo>


                  <mo class="MathClass-open">(</mo>

                  <mn>${b}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mo>&#8290;</mo>

                  <mi>ln</mi>

                  <mo>&#8289;</mo>


                  <mo class="MathClass-open">(</mo>

                  <mn class="MathClass-ord">3</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>
              </mfrac>
            </mstyle>

            <mo class="MathClass-rel">&#8776;</mo>

            <mn>${ans}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          </p>
        @
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$a1=int(rint(5)+2); $a=int(if(eq($a1,3),4,$a1)); $b=int(rint(5)+7); $ans=decimal(4,(log($a)-log($b))/log(3) );@
qu.1.5.uid=848d4b9c-b34d-4e60-b93b-a4c4dcaf7c95@
qu.1.5.question=

          <p class="noindent">Solve 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mn>${a}</mn>

            <mo class="MathClass-rel">=</mo>

            <mn>${b}</mn>

            <mo class="MathClass-bin">&#215;</mo>

            <msup>
              <mrow>
                <mn class="MathClass-ord">3</mn>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>


          <p class="nopar">for 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>
          </math>

          taking natural logarithms. 
          <br class="newline"/>

          Give your answer correct to 2 decimal places.</p>
        @
qu.1.5.answer=${ans} ? 0.005@

qu.1.6.mode=Formula@
qu.1.6.name=1.2.1 Functions@
qu.1.6.comment=

          <p class="noindent">First we re-arrange the equation to give 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mn>${a}</mn>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${c}</mn>

            <mo class="MathClass-bin">-</mo>

            <mn>${b}</mn>

            <mo class="MathClass-rel">=</mo>

            <mn>${cmb}</mn>
          </math>

          and then divide by 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mn>${a}</mn>
          </math>

          to give 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${ans}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>
          </p>
        @
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$a=int(rint(11)+2); $b=int(rint(12)+1); $c=int(rint(25)-12); $cmb=int($c-$b); $ansint=int($cmb/$a); $ans=if(eq($cmb/$a,$ansint),"$ansint","$cmb/$a");@
qu.1.6.uid=af4d7cbe-d9c8-4cf7-8def-fb72ab8082fe@
qu.1.6.question=

          <p class="noindent">Find the value of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>
          </math>

          that is the solution of the linear equation 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mn>${a}</mn>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>

            <mo class="MathClass-rel">=</mo>

            <mn>${c}</mn>
          </math>

          .</p>
        @
qu.1.6.answer=${ans}@

qu.1.7.mode=Blanks@
qu.1.7.name=1.2.15 Power functions@
qu.1.7.comment=

          <p class="noindent">

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">y</mi>

            <mo class="MathClass-rel">=</mo>

            <mfrac>
              <mrow>
                <mn>${k}</mn>
              </mrow>

              <mrow>
                <mn>${b}</mn>
              </mrow>
            </mfrac>

            <msup>
              <mrow>
                <mi class="MathClass-ord">x</mi>
              </mrow>

              <mrow>
                <mn>${p}</mn>
              </mrow>
            </msup>
          </math>

          ; 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">k</mi>

            <mo class="MathClass-rel">=</mo>

            <mfrac>
              <mrow>
                <mn>${k}</mn>
              </mrow>

              <mrow>
                <mn>${b}</mn>
              </mrow>
            </mfrac>
          </math>

          , 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">p</mi>

            <mo class="MathClass-rel">=</mo>

            <mo class="MathClass-bin">-</mo>

            <mn>${p}</mn>
          </math>

          .</p>
        @
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$k=int(rint(3)+2); $c=rint(1); $p=if($c,int(rint(5)+2),(2*rint(2)+1)/2); $b=int($k+int(3));@
qu.1.7.uid=b394f83a-adc6-4576-8a9d-c29ed7ae0466@
qu.1.7.question=

          <p class="noindent">Write the function</p>


          <p class="noindent">

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">y</mi>

            <mo class="MathClass-rel">=</mo>

            <mstyle displaystyle="true">
              <mfrac>
                <mrow>
                  <mn>${k}</mn>
                </mrow>

                <mrow>
                  <mn>${b}</mn>

                  <msup>
                    <mrow>
                      <mi class="MathClass-ord">x</mi>
                    </mrow>

                    <mrow>
                      <mn>${p}</mn>
                    </mrow>
                  </msup>
                </mrow>
              </mfrac>
            </mstyle>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          </p>


          <p class="noindent">in the form 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">y</mi>

            <mo class="MathClass-rel">=</mo>

            <mi class="MathClass-ord">k</mi>

            <msup>
              <mrow>
                <mi class="MathClass-ord">x</mi>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">p</mi>
              </mrow>
            </msup>
          </math>

          and give the values of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">k</mi>

            <mo class="MathClass-rel">=</mo>
          </math>

          <1>

          and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">p</mi>
          </math>

          =
          <2>

          .</p>
        @
qu.1.7.blank.1=%24%7bk%7d%2f%24%7bb%7d@
qu.1.7.blank.2=-%24%7bp%7d@
qu.1.7.grader.1=formula@
qu.1.7.grader.2=formula@
qu.1.7.extra=@

qu.1.8.mode=Multipart@
qu.1.8.name=1.2.13 Polynomials@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=@
qu.1.8.uid=1a38abe3-3bef-4741-88ee-ee9b49ebd73d@
qu.1.8.question=

          <p class="noindent">For each part find all polynomials 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">p</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>
          </math>

          of degree 2 or less that satisfy the given conditions.</p>
        @
qu.1.8.weighting=1,1,1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.extra=@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.question=

            <p class="noindent">

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">1</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">2</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>

            <br class="newline"/>

            <br class="newline"/>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>
            </math>

            <1>

            .</p>
          @
qu.1.8.part.1.algorithm=$a=int(rint(3)+1);@
qu.1.8.part.1.blank.1=%24%7ba%7d@
qu.1.8.part.1.grader.1=formula@
qu.1.8.part.1.mode=Blanks@
qu.1.8.part.1.comment=

            <p class="noindent">If 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">a</mi>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">b</mi>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">c</mi>
            </math>

            , then 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>
            </math>

            gives 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">c</mi>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>
            </math>

            , 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">1</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">c</mi>
            </math>

            gives 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${a}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>
            </math>

            and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">2</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">4</mn>

              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mn class="MathClass-ord">2</mn>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${a}</mn>
            </math>

            gives 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn class="MathClass-ord">4</mn>

              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mn class="MathClass-ord">2</mn>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${a}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>
            </math>

            . Solving these equations simultaneously gives 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            . So 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">
            </p>
          @
qu.1.8.part.2.extra=@
qu.1.8.part.2.editing=useHTML@
qu.1.8.part.2.question=

            <p class="noindent">

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">1</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">2</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${c}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>

            <br class="newline"/>

            <br class="newline"/>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>
            </math>

            <1>

            .</p>
          @
qu.1.8.part.2.algorithm=$a=int(-(rint(3)+2)); $b=int(-$a); $c=int(4*$a+2*$b); $ansb=$a*x^2+ $b*x;@
qu.1.8.part.2.blank.1=%24%7bansb%7d@
qu.1.8.part.2.grader.1=formula@
qu.1.8.part.2.mode=Blanks@
qu.1.8.part.2.comment=

            <p class="noindent">As in part (a) we have 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">c</mi>
            </math>

            so 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">c</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            . Similarly, 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">1</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">c</mi>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">2</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">4</mn>

              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mn class="MathClass-ord">2</mn>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-rel">=</mo>

              <mn>${c}</mn>
            </math>

            . This gives the simultaneous equations 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mtable columnspacing="0" class="eqnarray-star">
                <mtr>
                  <mtd columnalign="right">
                    <mi class="MathClass-ord">a</mi>

                    <mo class="MathClass-bin">+</mo>

                    <mi class="MathClass-ord">b</mi>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mn class="MathClass-ord">0</mn>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mn class="MathClass-ord">4</mn>

                    <mi class="MathClass-ord">a</mi>

                    <mo class="MathClass-bin">+</mo>

                    <mn class="MathClass-ord">2</mn>

                    <mi class="MathClass-ord">b</mi>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mn>${c}</mn>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                  </mtd>

                  <mtd columnalign="left">
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>
              </mtable>
            </math>


            <p class="nopar">So 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">
            </p>
          @
qu.1.8.part.3.editing=useHTML@
qu.1.8.part.3.fixed=@
qu.1.8.part.3.question=

            <p class="noindent">There a unique quadratic which satisfies 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">1</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${c}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>
            </p>
          @
qu.1.8.part.3.choice.2=

            <p class="noindent">False</p>
          @
qu.1.8.part.3.choice.1=

            <p class="noindent">True</p>
          @
qu.1.8.part.3.comment=

            <p class="noindent">Since 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">c</mi>
            </math>

            we have 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">c</mi>

              <mo class="MathClass-rel">=</mo>

              <mn>${c}</mn>
            </math>

            . Similarly, when 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">1</mn>
            </math>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">a</mi>

              <mo class="MathClass-bin">+</mo>

              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${c}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn>${c}</mn>
            </math>

            or 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">b</mi>

              <mo class="MathClass-rel">=</mo>

              <mo class="MathClass-bin">-</mo>

              <mi class="MathClass-ord">a</mi>
            </math>

            . Thus 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">p</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">a</mi>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">-</mo>

              <mi class="MathClass-ord">a</mi>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${c}</mn>
            </math>


            <p class="nopar">for any value of a, so there is not a unique quadratic through the given points.</p>
          @
qu.1.8.part.3.mode=True False@
qu.1.8.part.3.algorithm=$c=int(rint(3)+1);@
qu.1.8.part.3.answer=2@

qu.1.9.mode=Blanks@
qu.1.9.name=1.2.16 Power functions@
qu.1.9.comment=

          <p class="noindent">This is not a power function because of the 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>
          </math>

          .</p>
        @
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$k=int(rint(3)+2); $p=int(rint(5)+2); $b=int(rint(3)+2); $ans=($k)^($p); $power=$p*$b;@
qu.1.9.uid=36c718f8-d0e0-4a77-97bd-b358b6a0bbd3@
qu.1.9.question=

          <p class="noindent">Is the function given below a power function? 
          <br class="newline"/>

          <br class="newline"/>


          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">y</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${k}</mn>

            <msup>
              <mrow>
                <mi class="MathClass-ord">x</mi>
              </mrow>

              <mrow>
                <mn>${p}</mn>
              </mrow>
            </msup>

            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
            <br class="newline"/>

            <br class="newline"/>

            <1>
          </p>
        @
qu.1.9.blank.1=no, yes@
qu.1.9.grader.1=menu@
qu.1.9.extra=@

qu.1.10.mode=Multipart@
qu.1.10.name=1.2.17 Polynomials@
qu.1.10.comment=@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$deg=int(rint(5)+3); $a=int(rint(4)+2); $tps=int($deg-1); $b=int(rint(5)+2); $c=int(rint(5)+2);@
qu.1.10.uid=a99bd442-8743-4110-8c4e-d7fbf0d6c3a9@
qu.1.10.question=
        @
qu.1.10.weighting=1,1,1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.blank.2=%24%7ba%7d@
qu.1.10.part.1.blank.1=%24%7bdeg%7d@
qu.1.10.part.1.extra=@
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.question=

            <p class="noindent">The degree of the polynomial 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn>${deg}</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">-</mo>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${c}</mn>
            </math>

            is 
            <1>

            . 
            <br class="newline"/>

            The leading coefficient is 
            <2>

            .</p>
          @
qu.1.10.part.1.comment=

            <p class="noindent">The degree of the polynomial 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn>${deg}</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">-</mo>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">2</mn>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${c}</mn>
            </math>

            is ${deg}. 
            <br class="newline"/>

            The leading coefficient is ${a}.</p>
          @
qu.1.10.part.1.mode=Blanks@
qu.1.10.part.1.grader.2=formula@
qu.1.10.part.1.grader.1=formula@
qu.1.10.part.2.editing=useHTML@
qu.1.10.part.2.question=

            <p class="noindent">What power of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            approximates 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>
            </math>

            for large 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            ?.</p>
          @
qu.1.10.part.2.answer=${deg}@
qu.1.10.part.2.mode=Plain Number@
qu.1.10.part.2.comment=

            <p class="noindent">For large values of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">x</mi>
            </math>

            the function looks like 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">x</mi>
                </mrow>

                <mrow>
                  <mn>${deg}</mn>
                </mrow>
              </msup>
            </math>

            .</p>
          @
qu.1.10.part.3.blank.2=one less than the degree, the same as the degree, one more than the degree, unrelated to the degree@
qu.1.10.part.3.blank.1=%24%7btps%7d@
qu.1.10.part.3.extra=@
qu.1.10.part.3.editing=useHTML@
qu.1.10.part.3.question=

            <p class="noindent">Sketch the function 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>
            </math>

            , for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-rel">&#8804;</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-rel">&#8804;</mo>

              <mn class="MathClass-ord">1</mn>

              <mn class="MathClass-ord">0</mn>
            </math>

            . 
            <br class="newline"/>

            This function has at most 
            <1>

            turning points. 
            <br class="newline"/>

            The number of turning points is 
            <span class="cmbx-12">at most</span>

            <2>

            of the polynomial.</p>
          @
qu.1.10.part.3.comment=

            <p class="noindent">A polynomial of degree ${deg} has at most ${tps} turning points.</p>
          @
qu.1.10.part.3.mode=Blanks@
qu.1.10.part.3.grader.2=menu@
qu.1.10.part.3.grader.1=formula@

qu.1.11.mode=Multipart@
qu.1.11.name=1.2.3 Doubling Time@
qu.1.11.comment=@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$r1=0.1+rint(7)*0.1; $r2=$r1+1; $r3=$r2+1; $p1=decimal(3,1+$r1/100); $p2=decimal(3,1+$r2/100); $p3=decimal(3,1+$r3/100);$t1=log(2)/log($p1);$t2=log(2)/log($p2); $t3=log(2)/log($p3);$t1d=decimal(2,$t1); $t2d=decimal(2,$t2); $t3d=decimal(2,$t3);@
qu.1.11.uid=2f637ed3-e0cb-4ef9-861f-795f8444c2c8@
qu.1.11.question=
        @
qu.1.11.weighting=1,1@
qu.1.11.numbering=alpha@
qu.1.11.part.1.blank.3=%24%7bt3%7d %3f 0.05@
qu.1.11.part.1.blank.2=%24%7bt2%7d %3f 0.05@
qu.1.11.part.1.blank.1=%24%7bt1%7d %3f 0.05@
qu.1.11.part.1.extra=@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.question=

            <p class="noindent">Find the doubling time for annual growth rates of ${r1}%, ${r2}% and ${r3}%. 
            <br class="newline"/>

            <br class="newline"/>

            Doubling time for ${r1}% is 
            <1>

            <br class="newline"/>

            <br class="newline"/>

            Doubling time for ${r2}% is 
            <2>

            <br class="newline"/>

            <br class="newline"/>

            Doubling time for ${r3}% is 
            <3>
            </p>
          @
qu.1.11.part.1.comment=

            <p class="noindent">For interest of ${r1}%, we have 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msup>
                <mrow>
                  <mo class="MathClass-open">(</mo>

                  <mn>${p1}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>
              </msup>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">2</mn>
            </math>

            , so 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn class="MathClass-ord">2</mn>
                </mrow>

                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn>${p1}</mn>
                </mrow>
              </mfrac>

              <mo class="MathClass-rel">&#8776;</mo>

              <mn>${t1d}</mn>
            </math>

            years.</p>


            <p class="noindent">For interest of ${r2}%, we have 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msup>
                <mrow>
                  <mo class="MathClass-open">(</mo>

                  <mn>${p2}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>
              </msup>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">2</mn>
            </math>

            , so 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn class="MathClass-ord">2</mn>
                </mrow>

                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn>${p2}</mn>
                </mrow>
              </mfrac>

              <mo class="MathClass-rel">&#8776;</mo>

              <mn>${t2d}</mn>
            </math>

            years.</p>


            <p class="noindent">For interest of ${r3}%, we have 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msup>
                <mrow>
                  <mo class="MathClass-open">(</mo>

                  <mn>${p3}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>
              </msup>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">2</mn>
            </math>

            , so 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn class="MathClass-ord">2</mn>
                </mrow>

                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn>${p3}</mn>
                </mrow>
              </mfrac>

              <mo class="MathClass-rel">&#8776;</mo>

              <mn>${t3d}</mn>
            </math>

            years.</p>


            <p class="noindent">Give your answers to two decimal places.</p>
          @
qu.1.11.part.1.mode=Blanks@
qu.1.11.part.1.grader.3=formula@
qu.1.11.part.1.grader.2=formula@
qu.1.11.part.1.grader.1=formula@
qu.1.11.part.2.editing=useHTML@
qu.1.11.part.2.question=

            <p class="noindent">Does the change in doubling time suggest that the doubling time, 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">D</mi>
            </math>

            , is a linear function of the growth rate, 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">r</mi>
            </math>

            ?</p>
          @
qu.1.11.part.2.fixed=@
qu.1.11.part.2.choice.2=

            <p class="noindent">False</p>
          @
qu.1.11.part.2.answer=2@
qu.1.11.part.2.choice.1=

            <p class="noindent">True</p>
          @
qu.1.11.part.2.mode=True False@
qu.1.11.part.2.comment=

          <p class="noindent">This is definitely not a linear relation; in fact, from the above calculations we can see that for a growth rate of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">r</mi>

            <mi class="MathClass-ord">%</mi>
          </math>

          , 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>

            <mstyle displaystyle="true">
              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mn class="MathClass-ord">2</mn>
                </mrow>

                <mrow>
                  <mo>&#8290;</mo>

                  <mi>log</mi>

                  <mo>&#8289;</mo>


                  <mo class="MathClass-open">(</mo>

                  <mn class="MathClass-ord">1</mn>

                  <mo class="MathClass-bin">+</mo>

                  <mfrac>
                    <mrow>
                      <mi class="MathClass-ord">r</mi>
                    </mrow>

                    <mrow>
                      <mn class="MathClass-ord">1</mn>

                      <mn class="MathClass-ord">0</mn>

                      <mn class="MathClass-ord">0</mn>
                    </mrow>
                  </mfrac>

                  <mo class="MathClass-close">)</mo>
                </mrow>
              </mfrac>
            </mstyle>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
            <br class="newline"/>

            <br class="newline"/>
          </p>

          <applet code="applets.grapher.Graph" width="250" height="250" archive="graphing.jar">
            <param name="y1" value="log(2)/(log(1+x/100))">
            </param>

            <param name="gridLines" value="10">
            </param>

            <param name="xMin" value="0">
            </param>

            <param name="xMax" value="10">
            </param>

            <param name="yMin" value="0">
            </param>

            <param name="yMax" value="100">
            </param>
          </applet>

          <br class="newline"/>

          <br class="newline"/>

          The doubling time, 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>
          </math>

          , is plotted on the vertical axis, the interest rate, 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">r</mi>
          </math>

          % is plotted on the horizontal axis.@

qu.1.12.mode=Plain Number@
qu.1.12.name=1.2.6 Using logs@
qu.1.12.comment=

          <p class="noindent">Taking natural logs of both sides we get 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mn>${a}</mn>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <msup>
              <mrow>
                <mn>${b}</mn>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">This gives 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-open">(</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mn>${b}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mn>${a}</mn>
          </math>


          <p class="nopar">or in other words 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>

            <mstyle displaystyle="true">
              <mfrac>
                <mrow>
                  <mo>&#8290;</mo>

                  <mi>ln</mi>

                  <mo>&#8289;</mo>


                  <mn>${a}</mn>
                </mrow>

                <mrow>
                  <mo>&#8290;</mo>

                  <mi>ln</mi>

                  <mo>&#8289;</mo>


                  <mn>${b}</mn>
                </mrow>
              </mfrac>
            </mstyle>

            <mo class="MathClass-rel">=</mo>

            <mn>${ans}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          </p>
        @
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$a=int(rint(5)+2); $b1=int(rint(5)+2); $b=int(if(eq($a,$b1),8,$b1)); $ans=log($a)/log($b);@
qu.1.12.uid=a6f4d368-9563-4f0a-b88f-20851a8fd7e3@
qu.1.12.question=

          <p class="noindent">Solve 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mn>${a}</mn>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mn>${b}</mn>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>


          <p class="nopar">for 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>
          </math>

          taking natural logarithms. 
          <br class="newline"/>

          (Give your answer correct to 2 decimal places.)</p>
        @
qu.1.12.answer=${ans} ? 0.005@

qu.1.13.mode=Blanks@
qu.1.13.name=1.2.2 Average value@
qu.1.13.comment=

          <p class="noindent">The average rate of change, 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">R</mi>
          </math>

          , between 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${x0}</mn>
          </math>

          and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${x1}</mn>
          </math>

          is 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">R</mi>

            <mo class="MathClass-rel">=</mo>

            <mstyle displaystyle="true">
              <mfrac>
                <mrow>
                  <mi class="MathClass-ord">f</mi>

                  <mo class="MathClass-open">(</mo>

                  <mn>${x1}</mn>

                  <mo class="MathClass-close">)</mo>

                  <mo class="MathClass-bin">-</mo>

                  <mi class="MathClass-ord">f</mi>

                  <mo class="MathClass-open">(</mo>

                  <mn>${x0}</mn>

                  <mo class="MathClass-close">)</mo>
                </mrow>

                <mrow>
                  <mn>${x1}</mn>

                  <mo class="MathClass-bin">-</mo>

                  <mn>${x0}</mn>
                </mrow>
              </mfrac>
            </mstyle>

            <mo class="MathClass-rel">=</mo>

            <mn>${ans}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          </p>
        @
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$a=int(rint(3)+2); $b=int(rint(3)+1); $x0=int(rint(3)+1); $x1=int($x0+rint(3)+2); $f0=$a*($x0)^2+$b; $f1=$a*($x1)^2+$b; $ans=($f1-$f0)/(($x1)-($x0));@
qu.1.13.uid=8e036e7b-0266-4593-9b48-57ce0a53afd0@
qu.1.13.info=  difficulty=medium;
@
qu.1.13.question=

          <p class="noindent">Find the average rate of change of 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <msup>
              <mrow>
                <mi class="MathClass-ord">x</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">2</mn>
              </mrow>
            </msup>

            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>
          </math>

          between 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${x0}</mn>
          </math>

          and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-rel">=</mo>

            <mn>${x1}</mn>
          </math>

          . 
          <br class="newline"/>

          <br class="newline"/>

          Average rate of change =
          <1>

          to 3 decimal places.</p>
        @
qu.1.13.blank.1=%24%7bans%7d%3f0.05@
qu.1.13.grader.1=formula@
qu.1.13.extra=@

qu.1.14.mode=Blanks@
qu.1.14.name=1.2.10 Power functions@
qu.1.14.comment=

          <p class="noindent">The initial quantity is 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>
          </math>

          ; growth rate 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mo class="MathClass-rel">=</mo>

            <mn>${r}</mn>

            <mo class="MathClass-rel">=</mo>

            <mn>${c}</mn>
          </math>

          %</p>
        @
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$a=int(3*rint(5)+2)/10; $b=int(-rint(5)+10)/10; $r=$b-1;$c=int(100*$r);@
qu.1.14.uid=f8c8b9c1-1970-4b83-86a8-a3c198454957@
qu.1.14.question=

          <p class="noindent">What is the initial quantity for the function 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">Q</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <msup>
              <mrow>
                <mo class="MathClass-open">(</mo>

                <mn>${b}</mn>

                <mo class="MathClass-close">)</mo>
              </mrow>

              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>

          ? 
          <br class="newline"/>
          </p>


          <p class="noindent">
            <1>

            <br class="newline"/>
          </p>


          <p class="noindent">What is the growth rate? (Expressed as a percentage.) 
          <br class="newline"/>

          <2>

          .</p>
        @
qu.1.14.blank.1=%24%7ba%7d@
qu.1.14.blank.2=%24%7bc%7d@
qu.1.14.grader.1=formula@
qu.1.14.grader.2=formula@
qu.1.14.extra=@

qu.1.15.mode=Multipart@
qu.1.15.name=1.2.5 Functions@
qu.1.15.comment=@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$a=int(rint(3)+2); $b=int(rint(4)+2); $c=int(rint(3)+2); $ca=int($a*$c); $cb=int($b*$c);$aa=int($a*$a); $aba=int($a*$b+$b);@
qu.1.15.uid=434b5e58-758b-49cc-b0a7-af74ab8f1809@
qu.1.15.question=

          <p class="noindent">Let 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">f</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${a}</mn>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-bin">+</mo>

            <mn>${b}</mn>
          </math>

          and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">g</mi>

            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">x</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mn>${c}</mn>

                <mi class="MathClass-ord">x</mi>
              </mrow>
            </msup>
          </math>

          . Find formulas for each of the following functions. 
          <br class="newline"/>

          <br class="newline"/>

          Use exp to denote the exponential function.</p>
        @
qu.1.15.weighting=1,1,1@
qu.1.15.numbering=alpha@
qu.1.15.part.1.editing=useHTML@
qu.1.15.part.1.question=

            <p class="noindent">

              <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
                <mi class="MathClass-ord">g</mi>

                <mo class="MathClass-open">(</mo>

                <mi class="MathClass-ord">f</mi>

                <mo class="MathClass-open">(</mo>

                <mi class="MathClass-ord">x</mi>

                <mo class="MathClass-close">)</mo>

                <mo class="MathClass-close">)</mo>
              </math>
            </p>
          @
qu.1.15.part.1.answer=exp(${ca}*x + ${cb})@
qu.1.15.part.1.mode=Formula@
qu.1.15.part.1.comment=

            <p class="noindent">We know that 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>
            </math>

            and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">g</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mn>${c}</mn>

                  <mi class="MathClass-ord">x</mi>
                </mrow>
              </msup>
            </math>

            . Thus, 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mtable columnspacing="0" class="eqnarray-star">
                <mtr>
                  <mtd columnalign="right">
                    <mi class="MathClass-ord">g</mi>

                    <mo class="MathClass-open">(</mo>

                    <mi class="MathClass-ord">f</mi>

                    <mo class="MathClass-open">(</mo>

                    <mi class="MathClass-ord">x</mi>

                    <mo class="MathClass-close">)</mo>

                    <mo class="MathClass-close">)</mo>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mi class="MathClass-ord">g</mi>

                    <mo class="MathClass-open">(</mo>

                    <mn>${a}</mn>

                    <mi class="MathClass-ord">x</mi>

                    <mo class="MathClass-bin">+</mo>

                    <mn>${b}</mn>

                    <mo class="MathClass-close">)</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <msup>
                      <mrow>
                        <mi class="MathClass-ord">e</mi>
                      </mrow>

                      <mrow>
                        <mn>${c}</mn>

                        <mo class="MathClass-open">(</mo>

                        <mn>${a}</mn>

                        <mi class="MathClass-ord">x</mi>

                        <mo class="MathClass-bin">+</mo>

                        <mn>${b}</mn>

                        <mo class="MathClass-close">)</mo>
                      </mrow>
                    </msup>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <msup>
                      <mrow>
                        <mi class="MathClass-ord">e</mi>
                      </mrow>

                      <mrow>
                        <mn>${ca}</mn>

                        <mi class="MathClass-ord">x</mi>

                        <mo class="MathClass-bin">+</mo>

                        <mn>${cb}</mn>
                      </mrow>
                    </msup>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mi class="MathClass-ord">e</mi>

                    <mi class="MathClass-ord">x</mi>

                    <mi class="MathClass-ord">p</mi>

                    <mo class="MathClass-open">(</mo>

                    <mn>${ca}</mn>

                    <mi class="MathClass-ord">x</mi>

                    <mo class="MathClass-bin">+</mo>

                    <mn>${cb}</mn>

                    <mo class="MathClass-close">)</mo>

                    <mo class="MathClass-punc">.</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>
              </mtable>
            </math>


            <p class="nopar">
            </p>
          @
qu.1.15.part.2.editing=useHTML@
qu.1.15.part.2.question=

            <p class="noindent">

              <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
                <mi class="MathClass-ord">f</mi>

                <mo class="MathClass-open">(</mo>

                <mi class="MathClass-ord">g</mi>

                <mo class="MathClass-open">(</mo>

                <mi class="MathClass-ord">x</mi>

                <mo class="MathClass-close">)</mo>

                <mo class="MathClass-close">)</mo>
              </math>
            </p>
          @
qu.1.15.part.2.answer=${a}*exp(${c}*x) + ${b}@
qu.1.15.part.2.mode=Formula@
qu.1.15.part.2.comment=

            <p class="noindent">We know that 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">g</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mn>${c}</mn>

                  <mi class="MathClass-ord">x</mi>
                </mrow>
              </msup>
            </math>

            and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>
            </math>

            . Thus, 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">g</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mn>${c}</mn>

                  <mi class="MathClass-ord">x</mi>
                </mrow>
              </msup>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mn>${c}</mn>

                  <mi class="MathClass-ord">x</mi>
                </mrow>
              </msup>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">
            </p>
          @
qu.1.15.part.3.editing=useHTML@
qu.1.15.part.3.question=

            <p class="noindent">

              <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
                <mi class="MathClass-ord">f</mi>

                <mo class="MathClass-open">(</mo>

                <mi class="MathClass-ord">f</mi>

                <mo class="MathClass-open">(</mo>

                <mi class="MathClass-ord">x</mi>

                <mo class="MathClass-close">)</mo>

                <mo class="MathClass-close">)</mo>
              </math>
            </p>
          @
qu.1.15.part.3.answer=${aa}x +${aba}@
qu.1.15.part.3.mode=Formula@
qu.1.15.part.3.comment=

            <p class="noindent">We know that 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>
            </math>

            . Thus, 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mi class="MathClass-ord">f</mi>

              <mo class="MathClass-open">(</mo>

              <mn>${a}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn>${a}</mn>

              <mo class="MathClass-open">(</mo>

              <mn>${a}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-bin">+</mo>

              <mn>${b}</mn>

              <mo class="MathClass-rel">=</mo>

              <mn>${aa}</mn>

              <mi class="MathClass-ord">x</mi>

              <mo class="MathClass-bin">+</mo>

              <mn>${aba}</mn>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">
            </p>
          @

qu.1.16.mode=Multipart@
qu.1.16.name=1.2.12 Application@
qu.1.16.comment=@
qu.1.16.editing=useHTML@
qu.1.16.solution=@
qu.1.16.algorithm=$r=int(rint(5)+3); $cns=$r/100; $ecns8=exp($cns*8);$ecns8d=decimal(4,$ecns8); $ans1=decimal(3,5000*$ecns8); $t=int(rint(4)+5); $r2=decimal(3,ln(1.6)/$t); $ans2=decimal(2,100*$r2);@
qu.1.16.uid=accd6965-65b4-4be3-b3c3-917c35e47c9a@
qu.1.16.question=

          <p class="noindent">You invest 5000 dollars in an account which pays interest compounded continuously.</p>
        @
qu.1.16.weighting=1,1@
qu.1.16.numbering=alpha@
qu.1.16.part.1.editing=useHTML@
qu.1.16.part.1.question=

            <p class="noindent">How much money is in the account after 8 years, if the annual interest rate is ${r}%? 
            <br class="newline"/>

            (Give your answer to the nearest cent.)</p>
          @
qu.1.16.part.1.answer=${ans1} ? 0.05@
qu.1.16.part.1.mode=Plain Number@
qu.1.16.part.1.comment=

            <p class="noindent">We know that the formula for the account balance at time 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>
            </math>

            in an account compounded continuously is given by the formula 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">P</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <msub>
                <mrow>
                  <mi class="MathClass-ord">P</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mi class="MathClass-ord">r</mi>

                  <mi class="MathClass-ord">t</mi>
                </mrow>
              </msup>
            </math>


            <p class="nopar">where 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">P</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            is the initial deposit and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">r</mi>
            </math>

            is the annual rate. Thus, in our case the formula would be 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mi class="MathClass-ord">P</mi>

              <mo class="MathClass-open">(</mo>

              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-close">)</mo>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">5</mn>

              <mn class="MathClass-ord">0</mn>

              <mn class="MathClass-ord">0</mn>

              <mn class="MathClass-ord">0</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mn>${cns}</mn>

                  <mi class="MathClass-ord">t</mi>
                </mrow>
              </msup>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">Substituting the value 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">8</mn>
            </math>

            we get 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mtable columnspacing="0" class="eqnarray-star">
                <mtr>
                  <mtd columnalign="right">
                    <mi class="MathClass-ord">P</mi>

                    <mo class="MathClass-open">(</mo>

                    <mn class="MathClass-ord">8</mn>

                    <mo class="MathClass-close">)</mo>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mn class="MathClass-ord">5</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <msup>
                      <mrow>
                        <mi class="MathClass-ord">e</mi>
                      </mrow>

                      <mrow>
                        <mn>${cns}</mn>

                        <mo class="MathClass-open">(</mo>

                        <mn class="MathClass-ord">8</mn>

                        <mo class="MathClass-close">)</mo>
                      </mrow>
                    </msup>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">&#8776;</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mn class="MathClass-ord">5</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mo class="MathClass-open">(</mo>

                    <mn>${ecns8d}</mn>

                    <mo class="MathClass-close">)</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">&#8776;</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mn>${ans1}</mn>

                    <mo class="MathClass-punc">.</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>
              </mtable>
            </math>


            <p class="nopar">Thus, the balance at the end of eight years would be about \\$${ans1}.</p>
          @
qu.1.16.part.2.editing=useHTML@
qu.1.16.part.2.question=

            <p class="noindent">If you want the account to contain \\$8000 after ${t} years, what yearly interest rate is needed?</p>


            <p class="noindent">Give your answer to one decimal place.</p>
          @
qu.1.16.part.2.answer=${ans2} ? 0.1@
qu.1.16.part.2.mode=Plain Number@
qu.1.16.part.2.comment=

            <p class="noindent">We are asked to solve for the rate, 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">r</mi>
            </math>

            , that would give us an \\$8000 balance at the end of eight years. In other words we are asked to solve for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">r</mi>
            </math>

            in the equation 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mn class="MathClass-ord">8</mn>

              <mn class="MathClass-ord">0</mn>

              <mn class="MathClass-ord">0</mn>

              <mn class="MathClass-ord">0</mn>

              <mo class="MathClass-rel">=</mo>

              <mn class="MathClass-ord">5</mn>

              <mn class="MathClass-ord">0</mn>

              <mn class="MathClass-ord">0</mn>

              <mn class="MathClass-ord">0</mn>

              <msup>
                <mrow>
                  <mi class="MathClass-ord">e</mi>
                </mrow>

                <mrow>
                  <mn>${t}</mn>

                  <mi class="MathClass-ord">r</mi>
                </mrow>
              </msup>

              <mo class="MathClass-punc">.</mo>
            </math>


            <p class="nopar">Solving we get 

            </p>


            <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
              <mtable columnspacing="0" class="eqnarray-star">
                <mtr>
                  <mtd columnalign="right">
                    <mn class="MathClass-ord">8</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mn class="MathClass-ord">5</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <mn class="MathClass-ord">0</mn>

                    <msup>
                      <mrow>
                        <mi class="MathClass-ord">e</mi>
                      </mrow>

                      <mrow>
                        <mn>${t}</mn>

                        <mi class="MathClass-ord">r</mi>
                      </mrow>
                    </msup>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <msup>
                      <mrow>
                        <mi class="MathClass-ord">e</mi>
                      </mrow>

                      <mrow>
                        <mn>${t}</mn>

                        <mi class="MathClass-ord">r</mi>
                      </mrow>
                    </msup>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mstyle displaystyle="true">
                      <mfrac>
                        <mrow>
                          <mn class="MathClass-ord">8</mn>

                          <mn class="MathClass-ord">0</mn>

                          <mn class="MathClass-ord">0</mn>

                          <mn class="MathClass-ord">0</mn>
                        </mrow>

                        <mrow>
                          <mn class="MathClass-ord">5</mn>

                          <mn class="MathClass-ord">0</mn>

                          <mn class="MathClass-ord">0</mn>

                          <mn class="MathClass-ord">0</mn>
                        </mrow>
                      </mfrac>
                    </mstyle>

                    <mo class="MathClass-rel">=</mo>

                    <mn class="MathClass-ord">1</mn>

                    <mo class="MathClass-punc">.</mo>

                    <mn class="MathClass-ord">6</mn>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mo>&#8290;</mo>

                    <mi>ln</mi>

                    <mo>&#8289;</mo>


                    <msup>
                      <mrow>
                        <mi class="MathClass-ord">e</mi>
                      </mrow>

                      <mrow>
                        <mn>${t}</mn>

                        <mi class="MathClass-ord">r</mi>
                      </mrow>
                    </msup>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mo>&#8290;</mo>

                    <mi>ln</mi>

                    <mo>&#8289;</mo>


                    <mn class="MathClass-ord">1</mn>

                    <mo class="MathClass-punc">.</mo>

                    <mn class="MathClass-ord">6</mn>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mn>${t}</mn>

                    <mi class="MathClass-ord">r</mi>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mo>&#8290;</mo>

                    <mi>ln</mi>

                    <mo>&#8289;</mo>


                    <mn class="MathClass-ord">1</mn>

                    <mo class="MathClass-punc">.</mo>

                    <mn class="MathClass-ord">6</mn>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>

                <mtr>
                  <mtd columnalign="right">
                    <mi class="MathClass-ord">r</mi>

                    <mspace width=".167em"/>
                  </mtd>

                  <mtd columnalign="center">
                    <mo class="MathClass-rel">=</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mstyle displaystyle="true">
                      <mfrac>
                        <mrow>
                          <mo>&#8290;</mo>

                          <mi>ln</mi>

                          <mo>&#8289;</mo>


                          <mn class="MathClass-ord">1</mn>

                          <mo class="MathClass-punc">.</mo>

                          <mn class="MathClass-ord">6</mn>
                        </mrow>

                        <mrow>
                          <mn>${t}</mn>
                        </mrow>
                      </mfrac>
                    </mstyle>

                    <mo class="MathClass-rel">&#8776;</mo>

                    <mn>${r2}</mn>

                    <mo class="MathClass-punc">.</mo>
                  </mtd>

                  <mtd columnalign="left">
                    <mtext class="eqnarray">
                    </mtext>
                  </mtd>
                </mtr>
              </mtable>
            </math>


            <p class="nopar">Thus, the rate we would need in order to have a balance of \\$8000 at the end of eight years is about 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mn>${ans2}</mn>

              <mi class="MathClass-ord">%</mi>
            </math>

            .</p>
          @

qu.1.17.mode=Blanks@
qu.1.17.name=1.2.8 Using logs@
qu.1.17.comment=

          <p class="noindent">Taking natural logs of both sides we get 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${a}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${b}</mn>

            <mi class="MathClass-ord">t</mi>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mi class="MathClass-ord">e</mi>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-rel">=</mo>

            <mn>${b}</mn>

            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">This gives 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>

            <mo>&#8290;</mo>

            <mi>ln</mi>

            <mo>&#8289;</mo>


            <mo class="MathClass-open">(</mo>

            <mn>${a}</mn>

            <mo class="MathClass-close">)</mo>

            <mo class="MathClass-bin">/</mo>

            <mn>${b}</mn>

            <mo class="MathClass-rel">&#8776;</mo>

            <mn>${ans}</mn>

            <mo class="MathClass-punc">.</mo>
          </math>


          <p class="nopar">
          </p>
        @
qu.1.17.editing=useHTML@
qu.1.17.solution=@
qu.1.17.algorithm=$a=int(rint(5)+2); $b=int(10*(rint(5)+1)); $ans=decimal(3,(1/$b)*ln($a));@
qu.1.17.uid=ea6acdaa-b070-4d31-86d9-8adb0d2399e1@
qu.1.17.question=

          <p class="noindent">Solve 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mn>${a}</mn>

            <mo class="MathClass-rel">=</mo>

            <msup>
              <mrow>
                <mi class="MathClass-ord">e</mi>
              </mrow>

              <mrow>
                <mn>${b}</mn>

                <mi class="MathClass-ord">t</mi>
              </mrow>
            </msup>
          </math>


          <p class="nopar">for 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>
          </math>

          taking natural logarithms. 
          <br class="newline"/>
          </p>


          <p class="noindent">

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>

            <mo class="MathClass-rel">=</mo>
          </math>

          <1>

          correct to 2 decimal places.</p>
        @
qu.1.17.blank.1=%24%7bans%7d %3f 0.005@
qu.1.17.grader.1=formula@
qu.1.17.extra=@

qu.1.18.mode=Multipart@
qu.1.18.name=1.2.4 Exponential, Log functions@
qu.1.18.comment=@
qu.1.18.editing=useHTML@
qu.1.18.solution=@
qu.1.18.algorithm=$v=int(10*(5+rint(4)));$t=int(2+rint(3)); $f1=decimal(4,$v*(1-exp(-1/$t))); $f3=decimal(4,$v*(1-exp(-3/$t)));$f5=decimal(4,$v*(1-exp(-5/$t)));@
qu.1.18.uid=3edf43d0-b1de-44f9-bf50-5d54bd20ff8f@
qu.1.18.question=

          <p class="noindent">Under certain circumstances, the velocity, 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">v</mi>
          </math>

          , of a falling raindrop is given by 

          </p>


          <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
            <mi class="MathClass-ord">v</mi>

            <mo class="MathClass-rel">=</mo>

            <msub>
              <mrow>
                <mi class="MathClass-ord">v</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">0</mn>
              </mrow>
            </msub>

            <mfenced open="(" close=")">
              <mrow>
                <mn class="MathClass-ord">1</mn>

                <mo class="MathClass-bin">-</mo>

                <msup>
                  <mrow>
                    <mi class="MathClass-ord">e</mi>
                  </mrow>

                  <mrow>
                    <mo class="MathClass-bin">-</mo>

                    <mi class="MathClass-ord">t</mi>

                    <mo class="MathClass-bin">/</mo>

                    <msub>
                      <mrow>
                        <mi class="MathClass-ord">t</mi>
                      </mrow>

                      <mrow>
                        <mn class="MathClass-ord">0</mn>
                      </mrow>
                    </msub>
                  </mrow>
                </msup>
              </mrow>
            </mfenced>
          </math>


          <p class="nopar">where 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <mi class="MathClass-ord">t</mi>
          </math>

          is time, 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <msub>
              <mrow>
                <mi class="MathClass-ord">v</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">0</mn>
              </mrow>
            </msub>
          </math>

          and 

          <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
            <msub>
              <mrow>
                <mi class="MathClass-ord">t</mi>
              </mrow>

              <mrow>
                <mn class="MathClass-ord">0</mn>
              </mrow>
            </msub>
          </math>

          are positive constants.</p>
        @
qu.1.18.weighting=1,1@
qu.1.18.numbering=alpha@
qu.1.18.part.1.example=0,0 1,${f1} 3,${f3} 5,${f5}@
qu.1.18.part.1.editing=useHTML@
qu.1.18.part.1.question=

            <p class="noindent">Sketch a rough graph of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">v</mi>
            </math>

            as a function of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>
            </math>

            , for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <mi class="MathClass-ord">t</mi>

              <mo class="MathClass-rel">&#8805;</mo>

              <mn class="MathClass-ord">0</mn>
            </math>

            , when 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">v</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>

              <mo class="MathClass-rel">=</mo>

              <mn>${v}</mn>
            </math>

            and 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>

              <mo class="MathClass-rel">=</mo>

              <mn>${t}</mn>
            </math>

            .</p>
          @
qu.1.18.part.1.axes.labeled=true@
qu.1.18.part.1.answer=check(( increasing($1) ) && ( concave_down($1) ) && ( value($1,4)<${v} ))@
qu.1.18.part.1.axes=0,5,0,100@
qu.1.18.part.1.gridlines=5@
qu.1.18.part.1.mode=Sketch@
qu.1.18.part.2.extra=@
qu.1.18.part.2.editing=useHTML@
qu.1.18.part.2.question=

            <p class="noindent">How does increasing 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            change the shape of the graph? 
            <br class="newline"/>

            <br class="newline"/>

            a raindrop falling for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            large 
            <1>

            speed up as quickly as a raindrop falling for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            small.</p>
          @
qu.1.18.part.2.blank.1=does not, does@
qu.1.18.part.2.grader.1=menu@
qu.1.18.part.2.mode=Blanks@
qu.1.18.part.2.comment=

            <p class="noindent">As is plain from (a), increasing the value of 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            stretches the curve to the right. In other words, a raindrop falling for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            large does not speed up as quickly as a raindrop falling for 

            <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
              <msub>
                <mrow>
                  <mi class="MathClass-ord">t</mi>
                </mrow>

                <mrow>
                  <mn class="MathClass-ord">0</mn>
                </mrow>
              </msub>
            </math>

            small.</p>
          @

