Student Solutions Manual for Calculus & Its Applications and Calculus & Its Applications, Brief Version

14th Edition

ISBN: 9780134463230

Author: Larry J. Goldstein, David I Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Riemann Integral

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Textbook Question

Chapter 9.5, Problem 1CYU

The integral formula ∫ ( a t + b ) e − r t d t = e − r t [ − 1 r ( a t + b ) − a r 2 ] + C ( r ≠ 0 ) is used in many applications in this section. Derive it using integration by parts.

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Chapter 9.4, Problem 42E

Chapter 9.5, Problem 1E

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Total marks 15 3. (i) Let FRN Rm be a mapping and x = RN is a given point. Which of the following statements are true? Construct counterex- amples for any that are false. (a) If F is continuous at x then F is differentiable at x. (b) If F is differentiable at x then F is continuous at x. If F is differentiable at x then F has all 1st order partial (c) derivatives at x. (d) If all 1st order partial derivatives of F exist and are con- tinuous on RN then F is differentiable at x. [5 Marks] (ii) Let mappings F= (F1, F2) R³ → R² and G=(G1, G2) R² → R² : be defined by F₁ (x1, x2, x3) = x1 + x², G1(1, 2) = 31, F2(x1, x2, x3) = x² + x3, G2(1, 2)=sin(1+ y2). By using the chain rule, calculate the Jacobian matrix of the mapping GoF R3 R², i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)? (iii) [7 Marks] Give reasons why the mapping Go F is differentiable at (0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0). [3 Marks]

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Chapter 9 Solutions

Student Solutions Manual for Calculus & Its Applications and Calculus & Its Applications, Brief Version

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The integral formula ∫ ( a t + b ) e − r t d t = e − r t [ − 1 r ( a t + b ) − a r 2 ] + C ( r ≠ 0 ) is used in many applications in this section. Derive it using integration by parts.

Solution Summary: The author explains how the integral formula displaystyle is the product of two functions.

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Chapter 9 Solutions