Explanation - For finding the equation y = f ( x ) , we have to solve both parametric equations and have to eliminate the variable t . (a) Derivative of a function in parametric form can be found by differentiating both functions with respect to the parametric variable and then by dividing them d y d x = d y d s d x d s = y ' ( s ) x ' ( s ) (b) After finding the equation , we have to differentiate the equation with respect to x, and have to find f ' ( x ) . Given:- x = s 3 , y = s 6 + s − 3 Formula Used:- (1) Differentiation of x n is d ( x n ) d x = n x n − 1 (2) d y d x = d y d s d x d s = y ' ( s ) x ' ( s ) Calculation:- For finding f ( x ) , use x = s 3 , y = s 6 + s − 3 . x = s 3 Putting the value of s , in y . y = s 6 + s − 3 y = ( s 3 ) 2 + ( s 3 ) − 1 y = x 2 + x − 1 ð f ( x ) = x 2 + x − 1 Part (a) x = s 3 , y = s 6 + s − 3 x = s 3 d x d s = 3 s 2 y = s 6 + s − 3 d y d s = d ( s 6 ) d s + d ( s − 3 ) d s

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Author: Rogawski

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Question

Chapter 12.1, Problem 61E

To determine

- Find an equation $y = f (x)$ for the parametric curve and compute $\frac{d y}{d x}$ in two ways for

$x = s^{3}, y = s^{6} + s^{- 3}$

(a) Using $\frac{d y}{d x} = \frac{\frac{d y}{d s}}{\frac{d x}{d s}} = \frac{y' (s)}{x' (s)}$ .

(b) By differentiation of $f (x)$ .

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Chapter 12.1, Problem 60E

Chapter 12.1, Problem 62E

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Chapter 12 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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- Find an equation y = f ( x ) for the parametric curve and compute d y d x in two ways for x = s 3 , y = s 6 + s − 3 (a) Using d y d x = d y d s d x d s = y ' ( s ) x ' ( s ) . (b) By differentiation of f ( x ) .

Solution Summary: The author explains how to find the equation y = f ( x ) for the parametric curve and computed it in two ways.

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- Find an equation $y = f (x)$ for the parametric curve and compute $\frac{d y}{d x}$ in two ways for

$x = s^{3}, y = s^{6} + s^{- 3}$

(a) Using $\frac{d y}{d x} = \frac{\frac{d y}{d s}}{\frac{d x}{d s}} = \frac{y' (s)}{x' (s)}$ .

(b) By differentiation of $f (x)$ .

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Chapter 12 Solutions

- Find an equation y = f ( x ) for the parametric curve and compute d y d x in two ways for x = s 3 , y = s 6 + s − 3 (a) Using d y d x = d y d s d x d s = y ' ( s ) x ' ( s ) . (b) By differentiation of f ( x ) .

Solution Summary: The author explains how to find the equation y = f ( x ) for the parametric curve and computed it in two ways.

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- Find an equation y=f(x) for the parametric curve and compute dydx in two ways for x=s3,y=s6+s−3 (a) Using dydx=dydsdxds=y'(s)x'(s). (b) By differentiation of f(x).

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Chapter 12 Solutions

- Find an equation $y = f (x)$ for the parametric curve and compute $\frac{d y}{d x}$ in two ways for

$x = s^{3}, y = s^{6} + s^{- 3}$

(a) Using $\frac{d y}{d x} = \frac{\frac{d y}{d s}}{\frac{d x}{d s}} = \frac{y' (s)}{x' (s)}$ .

(b) By differentiation of $f (x)$ .