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Unformatted text preview: c Kendra Kilmer February 28, 2012 Section 4.8  Antiderivatives Example 1: What are the possible functions whose derivative is x? Deﬁnition: A function F is called an antiderivative of f on an interval I if F ′ (x) = f (x) for all x in I .
Theorem If F is an antiderivative of f on an interval I , then the most general antiderivative of f on I is
where C is an arbitrary constant.
Table of Antidifferentiation Formulas Function General Antiderivative k
xn (n = −1)
1
x
ex
ax
cos x
sin x
sec2 x
sec x tan x
csc2 x
csc x cot x
k f (x)
f (x) ± g(x) 11 c Kendra Kilmer February 28, 2012 Example 2: Find the most general antiderivative of each function.
a) f (x) = 8 b) f (x) = x5 1
c) f (x) = x4
3 d) f (t ) = 2
t9 e) f (x) = (5x4 + x3 − 2) f) f (x) = 3
+ sec2 x
x g) f (x) = ex + x3 3
√
1
h) f (x) = 3 x − 2 − x 2
x i) f (x) = 4x2 + x3
8x j) f (x) = (x − 2)(x + 3) + cos x 12 c Kendra Kilmer February 28, 2012 Example 3: Find y if y(1) = 1 and dy 3 1
=+
dx x x2 Example 4: Find f (x) if f ′′ (x) = 5 − 9x, f (0) = 3, and f (1) = 10. Example 5: A car is traveling at 70 miles per hour when the brakes are fully applied, producing a constant deceleration of 40 ft/s2 . What is the distance covered before the car comes to a stop? Section 4.8 Highly Suggested Homework Problems: 1, 5, 9, 11, 13, 19, 23, 25, 27, 29, 31, 35, 37, 39, 41, 53 13 ...
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 Fall '08
 Allen
 Antiderivatives, Derivative

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