Explanation Given: The derivative d 2 d x 2 ∫ 1 e x f ( t ) g ( t ) d t . Concept Used: Second Fundamental Theorem of Calculus stats that if f is continuous on [ a , b ] and we define F ( x ) = ∫ a x f ( t ) d t for all x ∈ [ a , b ] , then F is continuous on [ a , b ] and differentiable on ( a , b ) . Fis antiderivative of f , that is, F ' ( x ) = f ( x ) . If f is continuous on [ a , b ] and u ( x ) is differentiable on [ a , b ] , then for all x ∈ [ a , b ] , d d x ∫ a u ( x ) f ( t ) d t = f ( u ( x ) ) ⋅ u ' ( x ) Calculation: In order to find the second derivative d 2 d x 2 ∫ 1 e x f ( t ) g ( t ) d t , use the Second Fundamental Theorem of Calculus. So, using the Second Fundamental Theorem of Calculus, the first derivative can be computed as shown below: d 2 d x 2 ∫ 1 e x f ( t ) g ( t ) d t = d d x [ d d x ∫ 1 e x f ( t ) g ( t ) d t ] = d d x [ f ( e x ) g ( e x ) ⋅ ( e x ) ' − f ( 1 ) g ( 1 ) ⋅ ( 1 ) ' ] = d d x [ f ( e x ) g ( e x ) ⋅ ( e x ) − f ( 1 ) g ( 1 ) ⋅ ( 0 ) ] = d d x [ e x f ( e x ) g (<

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.7, Problem 44S

To determine

The derivative d2dx2∫1exf(t)g(t) dt using the Second Fundamental Theorem of Calculus.

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Chapter 4.7, Problem 43S

Chapter 4.7, Problem 45S

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Chapter 4 Solutions

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The derivative d 2 d x 2 ∫ 1 e x f ( t ) g ( t ) d t using the Second Fundamental Theorem of Calculus.

Solution Summary: The author explains the second fundamental theorem of calculus, which states that if f is continuous on left[a,bright], then F is differentiable.

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The derivative d2dx2∫1exf(t)g(t) dt using the Second Fundamental Theorem of Calculus.

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