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compute using the parametric equations I= cost, y = sin t, te (-x, 0) Write as a function (i) of t. (ii) of r and y. (Express the derivative as a rational function, not a piecewise-defined function.)

compute using the parametric equations I= cost, y = sin t, te (-x, 0) Write as a function (i) of t. (ii) of r and y. (Express the derivative as a rational function, not a piecewise-defined function.)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Problem 2: a. Using the chain rule dy dy dr dr dt dt compute 4 using the parametric equations x = cost, y = sin t, te (-0, 00) Write as a function (i) of t. (ii) of æ and y. (Express the derivative as a rational function, not a piecewise-defined function.) b. Use the chain rule to express in terms of d (dy dt dr dr and dt c. Use the result in part (a) and the chain rule again to compute 4 as a function (i) of t. (ii) of r and y. (Express the second derivative as a rational function, not a piecewise-defined function.) d. Find the derivatives and 4, as in the previous parts, for the parametric equations x = cos 3t, y = sin 3t, te (-0, 0). e. More generally, let f(t) and g(t) be differentiable functions defined over t € (-0, 0). Suppose the curve C1 has parametric equations r = f(t), y = 9(t), te(-∞,00) and the curve C, has parametric equations x = f(2t), y = g(2t), te (-0, a0). Compute both of the derivatives 4 for C1 and for C2, then describe the relationship between these derivatives and give a reason for this relationship.
Transcribed Image Text:Problem 2: a. Using the chain rule dy dy dr dr dt dt compute 4 using the parametric equations x = cost, y = sin t, te (-0, 00) Write as a function (i) of t. (ii) of æ and y. (Express the derivative as a rational function, not a piecewise-defined function.) b. Use the chain rule to express in terms of d (dy dt dr dr and dt c. Use the result in part (a) and the chain rule again to compute 4 as a function (i) of t. (ii) of r and y. (Express the second derivative as a rational function, not a piecewise-defined function.) d. Find the derivatives and 4, as in the previous parts, for the parametric equations x = cos 3t, y = sin 3t, te (-0, 0). e. More generally, let f(t) and g(t) be differentiable functions defined over t € (-0, 0). Suppose the curve C1 has parametric equations r = f(t), y = 9(t), te(-∞,00) and the curve C, has parametric equations x = f(2t), y = g(2t), te (-0, a0). Compute both of the derivatives 4 for C1 and for C2, then describe the relationship between these derivatives and give a reason for this relationship.
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