Explanation Proof: The left sum defined for n rectangles on [ a , b ] is ∑ k = 1 n f ( x k − 1 ) Δ x . Where, Δ x = b − a n , x k = a + k Δ x . The right sum defined for n rectangles on [ a , b ] is ∑ k = 1 n f ( x k ) Δ x . Where, Δ x = b − a n , x k = a + k Δ x

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.2, Problem 53P

To determine

To prove: that if a function f is positive and concave up on [a,b] , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

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Chapter 4.2, Problem 52P

Chapter 4.2, Problem 1TF

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Use the information in the following table to find h' (a) at the given value for a. x|f(x) g(x) f'(x) g(x) 0 0 0 4 3 1 4 4 3 0 2 7 1 2 7 3 3 1 2 9 4 0 4 5 7 h(x) = f(g(x)); a = 0 h' (0) =

Use the information in the following table to find h' (a) at the given value for a. x f(x) g(x) f'(x) g'(x) 0 0 3 2 1 1 0 0 2 0 2 43 22 4 3 3 2 3 1 1 4 1 2 0 4 2 h(x) = (1/(2) ²; 9(x) h' (3)= = ; a=3

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To prove: that if a function f is positive and concave up on [ a , b ] , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Solution Summary: The author proves that if a function f is positive and concave up, the trapezoids are always an over-approximation for the actual area.

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To prove: that if a function f is positive and concave up on [a,b] , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

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Chapter 4 Solutions