a) Explanation Consider the given integral. ∫ a b f ′ ( x ) d x Integration of a function can be defined by anti derivative of that function. For any function f ( x ) = a x + C , derivative of the function can be evaluated as f ′ ( x ) = a d d x ( x ) + d d x ( C ) = a ( 1 ) + 0 = a Therefore, antiderivative of a is given by ∫ a d x = a x + C Here, a is any constant and C is integrating constant b) To determine To simplify: the derivative d d x ∫ a b f ( x ) d x .

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 4.7, Problem 19C

To determine

Tosimplify:the integral ∫abf′(x)dx .

To determine

To simplify: the derivative ddx∫abf(x)dx .

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Chapter 4.7, Problem 18C

Chapter 4.7, Problem 20C

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Tosimplify: the integral ∫ a b f ′ ( x ) d x .

Solution Summary: The author explains the fundamental theorem of Calculus I.