Explanation Given: The function is f ( x , y , z ) = z ln ( y x ) Concept Used: ∇ f ( x 0 , y 0 , z 0 ) = f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f z ( x 0 , y 0 , z 0 ) The direction in which a function f ( x , y ) increases most rapidlyfrom point P 0 ( x 0 , y 0 , z 0 ) is given by ∇ f ( x 0 , y 0 , z 0 ) and magnitude of greatest rate of increase is ‖ ∇ f ( x 0 , y 0 , z 0 ) ‖ Calculation: In order to find the direction from P 0 ( 1 , e , − 1 ) in which the function f <

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.6, Problem 32PS

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To find the direction from P0(1,e,−1) in which the given function increases most rapidly also to compute the magnitude of the greatest rate of increase.

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Chapter 11.6, Problem 31PS

Chapter 11.6, Problem 33PS

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5. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.5.AE.003. y y= ex² 0 Video Example x EXAMPLE 3 (a) Use the Midpoint Rule with n = 10 to approximate the integral कर L'ex² dx. (b) Give an upper bound for the error involved in this approximation. SOLUTION 8+2 1 L'ex² d (a) Since a = 0, b = 1, and n = 10, the Midpoint Rule gives the following. (Round your answer to six decimal places.) dx Ax[f(0.05) + f(0.15) + ... + f(0.85) + f(0.95)] 0.1 [0.0025 +0.0225 + + e0.0625 + 0.1225 e0.3025 + e0.4225 + e0.2025 + + e0.5625 €0.7225 +0.9025] The figure illustrates this approximation. (b) Since f(x) = ex², we have f'(x) = 0 ≤ f'(x) = < 6e. ASK YOUR TEACHER and f'(x) = Also, since 0 ≤ x ≤ 1 we have x² ≤ and so Taking K = 6e, a = 0, b = 1, and n = 10 in the error estimate, we see that an upper bound for the error is as follows. (Round your final answer to five decimal places.) 6e(1)3 e 24( = ≈

Chapter 11 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

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To find the direction from P 0 ( 1 , e , − 1 ) in which the given function increases most rapidly also to compute the magnitude of the greatest rate of increase.

Solution Summary: The author calculates the direction from P_0(1,e,-1) in which the given function increases most rapidly, and the magnitude of the greatest rate of increase.

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To find the direction from P0(1,e,−1) in which the given function increases most rapidly also to compute the magnitude of the greatest rate of increase.

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