   Chapter 6.5, Problem 4PT

Chapter
Section
Textbook Problem

True or False:The Mean Value Theorem for Integrals is a restatement of the Mean Value Theorem (of Chapter 4) but stated in terms of a definite integral instead of in terms of a derivative.

To determine

Whether the statement “The Mean value Theorem for integrals is a restatement of the Mean value Theorem but stated in terms of a definite integral instead of in terms of a derivative.

Explanation

Definition used:

The mean value theorem for integrals is a direct consequence of the mean value theorem for derivatives.

Theorem used:

Mean Value Theorem for derivatives:

“Let f be a function that satisfies the following hypothesis:

1. f is continuous on the closed interval [a,b].

2. f is differentiable on the open interval (a,b).

Then, there is a number c in (a,b) such that f(c)=f(b)f(a)ba.

Or, equivalently, f(b)f(a)=f(c)(ba)”.

The mean value Theorem of integrals:

If f is continuous on a closed interval [a,b] there exists c[a,b] such that abf(x)dx=f(c)(ba), where abf(x)dx is the area under y=f(x) and f(c)(ba) is the area of the rectangle with width (ba) and the height f(c).

Proof:

The result of mean value theorem of derivatives is f(c)=f(b)f(a)ba by using the theorem mentioned above

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