Mean Value Theorem
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 52E
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![59. Generalizing the Mean Value Theorem for Integrals Suppose f and g are
continuous on [a, b] and let
h(x) = (x − b) ſª f(t) dt + (x -
[
·b
- a) g(t) dt.
a. Use Rolle's Theorem to show that there is a number c in (a, b) such
that
[ f(t) dt + [*g(t) dt = f(c)(b − c) + g(c)(c − a),
which is a generalization of the Mean Value Theorem for Integrals.
b. Show that there is a number c in (a, b) such that
Sf(t) dt = f(c)(b - c).
c. Use a sketch to interpret part (b) geometrically.
d. Use the result of part (a) to give an alternative proof of the Mean
Value Theorem for Integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ea0963c-64ff-46a6-b97c-79944ef211ae%2F39725b8f-08fc-4a9b-b336-d9811de888ca%2Fud6zwpt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:59. Generalizing the Mean Value Theorem for Integrals Suppose f and g are
continuous on [a, b] and let
h(x) = (x − b) ſª f(t) dt + (x -
[
·b
- a) g(t) dt.
a. Use Rolle's Theorem to show that there is a number c in (a, b) such
that
[ f(t) dt + [*g(t) dt = f(c)(b − c) + g(c)(c − a),
which is a generalization of the Mean Value Theorem for Integrals.
b. Show that there is a number c in (a, b) such that
Sf(t) dt = f(c)(b - c).
c. Use a sketch to interpret part (b) geometrically.
d. Use the result of part (a) to give an alternative proof of the Mean
Value Theorem for Integrals.
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