Question: Calculus

2.   Limaçons: Polar graphs of the form r(t)=1+ksint are called limaçons, French for snails, possibly because of their vague resemblance to the shape of these animals for certain values of k. 

      a)   See what you think of the resemblance by sketching the polar graph of r(t)=1+2sint, for t in the interval [0, 2p]. 

      b)   Your sketch for k = 2 should show a smaller loop inside a larger loop. Plot the rectangular graph of r(t)=1+2sint

      c)   Note where this rectangular curve crosses the x-axis. For what values of x is the curve below the x-axis?                                

      d)   What portion of the polar graph corresponds to the part of the rectangular graph that is below the x-axis?

      e)   Sketch the limaçons for k = 0.75 and k = 0.25 to see other possible shapes. One should be dented, the other egg-like. Sketch the limaçons for several additional values of k. Besides being looped, dented, or egg-like, did you find any other shapes occurring as limaçons? Describe what happens to the shape of the limaçons as k changes. 

 

3.   Spirals: If r(t) is positive and is strictly increasing as a function of t, the polar graph of r will be shaped like a spiral. For the spirals considered here, we will be mainly interested in their behavior as they cross the positive x-axis, for positive values of t. 

      a)   Plot both the polar and rectangular graphs of r(t)=lnt . For which values of t does the polar curve cross the positive x-axis?

      b)   What are the values of r at those points?

      c)   Find a function r(t) whose polar graph is a spiral which meets the positive x-axis at precisely the integer values: 1, 2, 3, ... . Begin your search by listing the values of t for which the spiral will cross the positive x-axis, and then consider what the value of r will need to be at these points. Only when you think you’ve got the right function, sketch the graph for values of t in the interval [0, 8p] to be sure. What was your function r(t)?

      d)   Find a function r(t)whose polar graph meets the positive x-axis at precisely the powers of 2: 1, 2, 4, 8, ... . Proceed as you did with arithmetic spirals, listing the values of t for which the spiral crosses the positive x-axis, and then considering what the value for r will need to be at those points. Sketch your graph for values of t in the interval [0, 8p] to confirm you have the right function. What was your function r(t)?

 

4.   a)   In problem 1 above, you conjectured the number of petals on various flowers. Which numbers of petals never occur in a polar graph of r(t)=sinkt?

      b)   Create polar graphs that have the number of petals that can never occur in a graph of r(t)=sinkt . How did you do this?

 

5.   Make up your own polar function using trigonometric functions and other basic mathematical tools. See if you can find something that looks pretty or interesting! When we're done, we'll have an "art gallery" of polar curves. Explore the nature of the function by changing it in a small way: for instance, place a coefficient in front of one of the terms [as in r=1+cos-alpha, which can be changed to r=2+3cos-alpha or r=-1+0.4cos3alpha ], etc. or change slightly the argument of one of the trigonometric quantities. Explain what happens to your curve as you make this change. Use different colors, Animation, etc. (This project is given in an assignment at the U.S. Naval Academy each year. The most spectacular example one of the students there found was r = 1+ sin(t / 24) + cos(2t). This is a neat example to plot!