Mean Value Theorem

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
Show all work, thank you!
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.
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.
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,