Explanation Given: The bug is moving along the right side of the parabola y = x 2 . The distance from the origin is increasing at 1 cm/min . Calculation: Consider A bug is located at a point P ( x , x 2 ) on the parabola and z be a distance from the bug P ( x , x 2 ) to the origin ( 0 , 0 ) . Here, the bug is moving in parabola. That is, the value of x changing with respect to time t . z = ( x − 0 ) 2 + ( x 2 − 0 ) 2 = x 2 + x 4 = x 1 + x 2 The variable z depends on x implies that the variable z is changing with respect to time t . Thus, z ( t ) and x ( t ) both are related variable. z ( t ) = x ( t ) 1 + x ( t ) 2 Differentiate with respect to t , z ′ ( t ) = d d t ( x ( t ) 1 + x ( t ) 2 ) = ( 1 + x ( t ) 2 ) d d t ( x ( t ) ) + ( x ( t ) ) d d t ( 1 + x ( t ) 2 ) = x ′ ( t ) 1 + x ( t ) 2 + ( x ( t ) ) d d t ( 1 + x ( t ) 2 ) Let u = 1 + x ( t ) 2 and apply the chain rule, z ′ ( t ) = x ′ ( t ) 1 + x ( t ) 2 + ( x ( t ) ) ( d d u ( u ) d u d t ) = x ′ ( t ) 1 + x ( t ) 2 + ( x ( t ) ) 1 2 u d u d t Substitute u = 1 + x ( t ) 2 in z ′ ( t ) , z ′ ( t ) = x ′ ( t ) 1 + x ( t ) 2 + ( 1 2 1 + x ( t ) 2 d d t ( 1 + x ( t ) 2 ) ) ( x ( t ) ) = x ′ ( t ) 1 + x ( t ) 2 + x ( t ) 2 1 + x ( t ) 2 ( d d t ( 1 ) + d d t ( x ( t ) 2 ) ) = x ′ ( t )

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 3.11, Problem 16E

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To find: The rate of change of the x- and y-coordinates of the bug increasing when the bug is at the point (2,4) .

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Chapter 3.11, Problem 15E

Chapter 3.11, Problem 17E

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Chapter 3 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 3.1 - Use definition (1) (p. 127) for the slope of a...Ch. 3.1 - Explain why the slope of a secant line can be...Ch. 3.1 - Explain why the slope of the tangent line can be...Ch. 3.1 - Prob. 4E Ch. 3.1 - Given a function f and a point a in its domain,...Ch. 3.1 - Explain the relationships among the slope of a...Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (1) a....

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The rate of change of the x - and y- coordinates of the bug increasing when the bug is at the point ( 2 , 4 ) .

Solution Summary: The author calculates the x- and y-coordinates of the bug increasing at the rate sqrt59cm/min

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To find: The rate of change of the x- and y-coordinates of the bug increasing when the bug is at the point (2,4) .

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Chapter 3 Solutions