MAT 21B Lecture Notes - Lecture 5: Mean Value Theorem, Antiderivative
5.4: The fundamental theorem of calculus
Differential Calculus &Integral Calculus
•Differential Calculus (Math21A): This part comes from the study of velocity problem; slope problem; rate
of change etc.
•Integral Calculus (Math21B): This part comes from the study of area problem.
•Today, we will see that they are closely related by a very important theorem: The Fundamental Theorem
of Calculus!
Mean Value Theorem
Theorem. If fis continuous on [a, b], then at some point cin [a, b],
f(c) = 1
b−aZb
a
f(x)dx.
Sometimes, we write it as
Zb
a
f(x)dx =f(c)(b−a)
for some cin [a, b].
Proof:
min f(x)≤f(x)≤max f(x)
Zb
a
min f(x)dx ≤Zb
a
f(x)dx ≤Zb
a
max f(x)dx
min f(x)·(b−a)≤Zb
a
f(x)dx ≤max f(x)·(b−a)
min f(x)≤1
b−aZb
a
f(x)dx ≤max f(x)
Since fis continuous, by the intermediate value theorem, there exists some cin [a, b] so that
f(c) = 1
b−aZb
a
f(x)dx.
Suppose fis continuous on [a, b]. For any xin [a, b], consider the function
F(x) = Zx
a
f(t)dt.
What is F0(x)?
F0(x) = lim
h→0
F(x+h)−F(x)
h
= lim
h→0Rx+h
af(t)dt −Rx
af(t)dt
h
= lim
h→0
1
hZx+h
x
f(t)dt
= lim
h→0f(ck),where ckis some point between xand x+h.
=f(x).
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MAT 21B Full Course Notes
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Document Summary
If f is continuous on [a, b], then at some point c in [a, b], f (c) = Sometimes, we write it as for some c in [a, b]. Z b a f (x)dx = f (c)(b a) min f (x) f (x) max f (x) a a min f (x)dx z b. Z b min f (x) (b a) z b. A z b min f (x) . 1 a a f (x)dx z b a max f (x)dx f (x)dx max f (x) (b a) f (x)dx max f (x) Since f is continuous, by the intermediate value theorem, there exists some c in [a, b] so that f (c) = For any x in [a, b], consider the function. F (x) = z x a f (t)dt. F (x + h) f (x) f (t)dt r x a h a f (t)dt f (t)dt. = lim h 0 r x+h h z x+h.
