The Chain Rule and Parametric Equations Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Given and find

(2) Given and find

(3) Given and find

(4) Write the function in the form and ; and then find

(5) Write the function in the form and ; and then find

(6) Write the function in the form and ; and then find

(7) Write the function in the form and ; and then find

(8) Find the derivative of

(9) Find the derivative of   

(10) Find the derivative of    

(11) Find the derivative of   

(12) Find the derivative of   

(13) Find the derivative of   

(14) Find the derivative of   

(15) Find the derivative of    

(16) Find the derivative of   

(17) Find given

(18) Find given

(19) Find given

(20) Find given

(21) Find given

(22) Find given

(23) Suppose that the functions , , and their derivatives with respect to have the following values at and



Find the derivatives with respect to of the following combinations at a given value of

    (a)
    (b)
    (c)
    (d)
    (e)
    (f)
    (g)

(24) Find when if and

(25) (a) Find the tangent to the curve at (b) What is the smallest value the slope of the curve can ever have on the interval Give reasons for you answer.   

(26) Sketch the graph of the parametric equations and , then indicate the direction of increasing

(27) Sketch the graph of the parametric equations and , then indicate the direction of increasing

(28) Sketch the graph of the parametric equations and , then indicate the direction of increasing

(29) Find parametric equations and a parameter interval for the motion of a particle that starts at and traces the ellipse (a) once clockwise (b) once counterclockwise (c) twice clockwise (d) twice counterclockwise.  

(30) Find a parametrization for the curve whose graph is the lower half of the parabola

(31) Find a parametrization for the curve whose graph is the ray (half-line) with initial point that passes through the point

(32) Find an equation for the line tangent to the curve and at Also, find the value of at this point.  

(33) Find an equation for the line tangent to the curve and at Also, find the value of at this point.  

(34) Suppose that is differentiable at is differentiable at and is negative. What, if anything, can be said about the values of and

(35) Differentiate the function given by

(36) Differentiate the function given by

(37) Differentiate the function given by

(38) If where is a differentiable function, find

(39) Suppose is a differentiable function on Let and Find expressions for and

(40) Determine if the following statement is true or false. Then justify your claim.
If is a differentiable function of is a differentiable function of and is a differentiable function of then

(41) Let be a differentiable function of Use to prove that



when Use the formula to find given

(42) Show that What do you think is the importance of the exercise?

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The Chain Rule And Parametric Equations Homework
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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