Explanation Given: The fundamental theorem of calculus relates definite and indefinite integrals. Formula used: ∫ a b f ( x ) d x = F ( b ) − F ( a ) Proof: The fundamental theorem of calculus tells that integrals and derivatives are each other’s opposites. In that it has two uses that are written below: First, if the indefinite integral or an anti-derivative of a function, and then take the derivative of that result then the answer will be the original function. It gives the connection between the derivative and the integral Suppose f ( x ) > 0 when a ≤ x ≤ b , so that the definite integral represents the area under the curve and above the x − a x i s . Define a new function A ( x ) by this rule: A ( x ) = area between the graph of f ( x ) and the x a x i s from a t o x ,. For instance, if the area under the graph from a to 4 i s 35 , then A ( 4 ) = 35 . We now show that A ( x ) is an antiderivative of f ( x ) We must show that A ′ ( x ) = lim A ( x + h ) − A ( x ) is f ( x ) . Look at the situation geometrically. When h is a small positive number, A ( x + h ) is the area under the graph of f ( x ) from a to x + h, and A ( x ) is the area from a to x . Consequently A ( x + h ) − A ( x ) is the area. This area can be approximated with a rectangle having base h and height f ( x )

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Author: Lial, HUNGERFO

Publisher: PEARSON

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Chapter 13.5, Problem 32E

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To prove: The fundamental theorem of calculus relates definite and indefinite integrals.

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Chapter 13.5, Problem 31E

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Chapter 13 Solutions

MYMATHLAB ACCESS CODE - MATHEMATICS WIT

Ch. 13.1 - Checkpoint 1 Find an antiderivative for each of...Ch. 13.1 - Checkpoint 2 Find each of the...Ch. 13.1 - Prob. 3CP Ch. 13.1 - Prob. 4CP Ch. 13.1 - Prob. 5CP Ch. 13.1 - Prob. 6CP Ch. 13.1 - Prob. 7CP Ch. 13.1 - Checkpoint 8 The marginal cost at a level of...Ch. 13.1 - 1. What must be true of F(x) and G(x) if both are...Ch. 13.1 - 2. How is the antiderivative of a function related...

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