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**Unformatted text preview: **c Kendra Kilmer March 1, 2012 Section 5.4 - The Fundamental Theorem of Calculus
x Example 1: Let g(x) = f (t )dt where f is the function whose graph is shown.
0 2
1
1 2 34 5 −1
−2 a) Evaluate g(0), g(2), and g(4). b) On what interval(s) is g increasing? decreasing? c) Where does g have an absolute maximum value? Absolute minimum value? x Example 2: If g(x) = t 3 dt , 1 a) Find a formula for g(x). b) What does your answer represent? c) Find g′ (x). 9 c Kendra Kilmer March 1, 2012 Fundamental Theorem of Calculus
Suppose f is continuous on [a, b].
x 1. If g(x) =
a f (t ) dt , then g is an antiderivative of f , that is g′ (x) = f (x) for a < x < b. Alternate Notation:
b 2.
a d
dx x f (t ) dt = f (x)
a f (x) dx = F (b) − F (a), where F is any antiderivative of f , that is F ′ = f Example 3: Use Part I of the Fundamental Theorem of Calculus to ﬁnd the derivative of the following functions.
x a) g(x) = et 3 2 −t x2 b) h(x) = dt 1 + r3 dr 0 cos x c) g(x) = (1 + v2 )10 dv sin x 10 c Kendra Kilmer March 1, 2012 x Example 4: Let g(x) = f (t ) dt , where f is the function whose graph is shown.
0 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? Section 5.4 Highly Suggested Homework Problems: 3, 5, 7, 9, 11, 15, 17, 19, 25 11 ...

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