jmurphy

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16 years, 121 days

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First we need to find the slope of the tangent line at (-4,-3) A line that is tangent to a curve has two properties: 1) It shares a common point with the curve           In our case that point is (-4,-3) 2) At the shared point, the derivate of the of the curve is equal to the slope of the tangent line           To determine the slope of the tangent line solve for y in terms of x to get y=+/-sqrt(25-x^2). Next,           Differentiate the equation of the bottom half of the circle y=-sqrt(25-x^2) w.r.t y to get           x/(sqrt(25-x^2))           Finally, substitute x=-4 into the derivative x/(sqrt(25-x^2)) and get m=-4/3 If two lines are perpendicular then their slopes are negative reciprocals meaning m_1=-1/m_2           So the slope of the line perpendicular to the tangent line is m=3/4           Now, we create the equation of our line using the slope m=3/4 and the point (-4,3) *recall the           formula m=(y_2-y_1)/(x_2-x_1)* to get y-(-3)=3/4(x-(-4)) after simplification our equation           becomes y=(3/4)x Therefore, the equation of the line perpendicular to the curve x^2+y^2=25 going through the point (-4,3) is y=3/4x. Below the problem is solved using Maple 11. To view the plot, it is best to download and view it in Maple 11. View 111_Equation of a line.mw on MapleNet or Download 111_Equation of a line.mw
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Hi, I thought I would try and give a little more insight on how to arrive at the correct answer. Hopefully this will help you with future assignments and tests :) Q1. To calculate the correct answer for this question you should use the Unit Circle - very useful in trig calculations. From this we can determine that: tan(15)=(1/4)/(sqrt(3)/4) After some simplification on the right hand side we arrive at: tan(15)=1/sqrt(3) We can then substitute 1/sqrt(3) for tan(15) into (2*tan(15))/(1-(tan(15))^2) And arrive at the final answer: 3/sqrt(3) Q2. To solve this problem we need to use the trig identities. Most people memorize these however they can be derived - this way you don't have to remember as much :) We will derivation method so you can see where the answer comes from. Some things we need to know: tan(x)=sin(x)/cos(x) cos(x)^2+sin(x)^2=1 sec(x)=1/cos(x) Ok so we know cos(x)^2+sin(x)^2=1 but our question is in terms of tan(x)^2 Fist we figure out how we can rewrite cos(x)^2+sin(x)^2=1 in terms of tan(x)^2 We know that tan(x)=sin(x)/cos(x) and tan(x)^2=sin(x)^2/cos(x)^2 so a good start would be to divide the equation cos(x)^2+sin(x)^2=1 by cos(x)^2 To get 1+sin(x)^2/cos(x)^2=1/cos(x)^2 Now we can simplify 1+tan(x)^2=sec(x)^2 From this we can conclude our answer of sec(x)^2 Q3. Again, we use trig identities to solve the problem. cot(A)*sin(A) =(1/tan(A))*sin(A) =(1/(sin(A)/cos(A)))*sin(A) =(cos(A)/sin(A))*sin(A) =cos(A) I will leave it at this. It seems like you were able to get the last 2 on your own. I know this is a bit hard on the eyes but hopefully you will find it useful. Jenna
Hi, I thought I would try and give a little more insight on how to arrive at the correct answer. Hopefully this will help you with future assignments and tests :) Q1. To calculate the correct answer for this question you should use the Unit Circle - very useful in trig calculations. From this we can determine that: tan(15)=(1/4)/(sqrt(3)/4) After some simplification on the right hand side we arrive at: tan(15)=1/sqrt(3) We can then substitute 1/sqrt(3) for tan(15) into (2*tan(15))/(1-(tan(15))^2) And arrive at the final answer: 3/sqrt(3) Q2. To solve this problem we need to use the trig identities. Most people memorize these however they can be derived - this way you don't have to remember as much :) We will derivation method so you can see where the answer comes from. Some things we need to know: tan(x)=sin(x)/cos(x) cos(x)^2+sin(x)^2=1 sec(x)=1/cos(x) Ok so we know cos(x)^2+sin(x)^2=1 but our question is in terms of tan(x)^2 Fist we figure out how we can rewrite cos(x)^2+sin(x)^2=1 in terms of tan(x)^2 We know that tan(x)=sin(x)/cos(x) and tan(x)^2=sin(x)^2/cos(x)^2 so a good start would be to divide the equation cos(x)^2+sin(x)^2=1 by cos(x)^2 To get 1+sin(x)^2/cos(x)^2=1/cos(x)^2 Now we can simplify 1+tan(x)^2=sec(x)^2 From this we can conclude our answer of sec(x)^2 Q3. Again, we use trig identities to solve the problem. cot(A)*sin(A) =(1/tan(A))*sin(A) =(1/(sin(A)/cos(A)))*sin(A) =(cos(A)/sin(A))*sin(A) =cos(A) I will leave it at this. It seems like you were able to get the last 2 on your own. I know this is a bit hard on the eyes but hopefully you will find it useful. Jenna
I think Maple is a great tool. I love that it has the capabilities of being used at all levels - High School, College, University and that you can continue to use it in your Professional or Academic career. As an employee of Maplesoft, I continue to learn about new and existing features and functionalities which amaze me and make me realize how powerful Maple can be. Please keep posting any questions you have and in no time you will be up to speed! Jenna
I think Maple is a great tool. I love that it has the capabilities of being used at all levels - High School, College, University and that you can continue to use it in your Professional or Academic career. As an employee of Maplesoft, I continue to learn about new and existing features and functionalities which amaze me and make me realize how powerful Maple can be. Please keep posting any questions you have and in no time you will be up to speed! Jenna
Maple did not calculate the integral because it did not interpret the integrand (x-2)(2x-3) correctly. Try entering a space or a multiplication symbol ' * ' between the braces, for example: (x-2) (2x-3) or (x-2)*(2x-3). Once this is done Maple should calculate the integral. In Maple 10, implicit multiplication was introduced; this is where you can enter a space in replace of the multiplication symbol. For best practices always enter a space or the actual multiplication symbol when multiplying. Jenna
Maple did not calculate the integral because it did not interpret the integrand (x-2)(2x-3) correctly. Try entering a space or a multiplication symbol ' * ' between the braces, for example: (x-2) (2x-3) or (x-2)*(2x-3). Once this is done Maple should calculate the integral. In Maple 10, implicit multiplication was introduced; this is where you can enter a space in replace of the multiplication symbol. For best practices always enter a space or the actual multiplication symbol when multiplying. Jenna
Another alternative would be to use the derivative button in the Expressions Palette. On the left hand side of your Maple worksheet you should see a series of palettes. Open the Expressions palette by clicking on it. Once it is open you should see a variety of buttons. Click on the derivative button to insert it into your worksheet. Once it is inserted you can enter the required information by using the Tab key to navigate through the fields. Again, remember the difference in Pi being used. Pi is used in calculations where as pi is a symbolic representation. Jenna
Another alternative would be to use the derivative button in the Expressions Palette. On the left hand side of your Maple worksheet you should see a series of palettes. Open the Expressions palette by clicking on it. Once it is open you should see a variety of buttons. Click on the derivative button to insert it into your worksheet. Once it is inserted you can enter the required information by using the Tab key to navigate through the fields. Again, remember the difference in Pi being used. Pi is used in calculations where as pi is a symbolic representation. Jenna
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