Fundamental theorem of the calculus
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
5.10) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.
![Fundamental theorem of the calculus
5.10.
If F(x) = | f(t)dt where f(x) is continuous in [a, b], prove that F'(x) = f(x).
a
F(x+h)- F(x)
1
x+h
J" f(t)dt - f(1)dt}
h
hl
a
a
1
"f(t) dt = f(5) Š between x and x+h
h
x+h
by the first mean value theorem for integrals (Page 99).
Then if x is any point interior to [a, b],
F(x+ h) - F(x)
F'(x) = lim
h 0
lim fE) = f(x)
%3D
h
h 0
since fis continuous.
If x = a or x = b, we use right-or left-hand limits, respectively, and the result holds in these cases as
well.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad530511-9758-42d9-9994-894bc4eea3f3%2F78557837-27d7-4b3c-a904-7284b20c36ee%2Fr77t3lq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Fundamental theorem of the calculus
5.10.
If F(x) = | f(t)dt where f(x) is continuous in [a, b], prove that F'(x) = f(x).
a
F(x+h)- F(x)
1
x+h
J" f(t)dt - f(1)dt}
h
hl
a
a
1
"f(t) dt = f(5) Š between x and x+h
h
x+h
by the first mean value theorem for integrals (Page 99).
Then if x is any point interior to [a, b],
F(x+ h) - F(x)
F'(x) = lim
h 0
lim fE) = f(x)
%3D
h
h 0
since fis continuous.
If x = a or x = b, we use right-or left-hand limits, respectively, and the result holds in these cases as
well.
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