12. Suppose f € C[0, 1]. Prove that lim | f(a" )da = f(0). n>00
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
number 12
![cx+a
8. Let f : R –→ R be continuous. For a > 0, let g(x)
| f(t) dt, x E R. Show g is differentiable, find g'(x).
9. Let f e Cla, b] and g e R[a, b] with g > 0. Prove that 3c e [a, b] such that
:| $(x)g(x)dw = f(c)|.
g(x)dx.
12. Suppose f € C[0, 1]. Prove that lim
f(2" )dx = f(0).
n 00](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae372ae2-2feb-4e9a-b164-90567cadbb97%2F43d59c58-344a-4cb9-9f7f-0c4d60818339%2Fa1bxwki_processed.png&w=3840&q=75)
Transcribed Image Text:cx+a
8. Let f : R –→ R be continuous. For a > 0, let g(x)
| f(t) dt, x E R. Show g is differentiable, find g'(x).
9. Let f e Cla, b] and g e R[a, b] with g > 0. Prove that 3c e [a, b] such that
:| $(x)g(x)dw = f(c)|.
g(x)dx.
12. Suppose f € C[0, 1]. Prove that lim
f(2" )dx = f(0).
n 00
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