MATH 131-503 Fall 2015
c Wen Liu
5.4
5.4 The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus: Suppose f is continuous on [a, b].
Rx
1. If g(x) = a f (t)dt, then g 0 (x) = f (x).
Rb
2. a f (x)dx = F (b) − F (a), where F is any antiderivative of f , that is, F 0 = f .
In general, we have
d
dx
Z
h(x)
!
f (t)dt
= f (h(x))h0 (x) − f (g(x))g 0 (x)
g(x)
Examples:
1. (p. 368) f is the function whose graph is shown below and g(x) =
Rx
0
f (t)dt.
(a) find g(0) and g(12).
(b) On what interval is g increasing?
(c) Where does g have a maximum value?
(d) Sketch a rough graph of g.
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MATH 131-503 Fall 2015
c Wen Liu
5.4
2. The graph of a function f is given and s = 6. Consider g(x) =
Rx
0
f (t)dt.
(a) At what values of x do the local maximum
and minimum values of g occur?
(b) Where does g attain its absolute maximum value?
(c) On what intervals is g concave downward?
3. (p. 370) Find the derivative of g(x) =
Rx√
0
1 + t2 dt.
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MATH 131-503 Fall 2015
4. Find the derivative of y =
5.4
R 4x
2x
c Wen Liu
(1 + t3 )10 dt.
5. If f (6) = 10, f 0 is continuous, and
R7
6
f 0 (x)dx = 16, find f (7).
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