Section 4.9 Antiderivatives
What is an antiderivative?
If f(x) = x2, then f'(x) = __________
How do the graphs of all antiderivatives compare?
Notation: We don't generally use f(x) and f'(x)
when talking about derivatives and antiderivatives.
A function F is an antiderivative of f on an
interval if F'(x) = f(x).
Two Definitions
A function F is an antiderivative of f on an
interval if F'(x) = f(x).
If F is an antiderivative of f on an interval,
then the most general antiderivative of f
on the interval is F(x) + C, where C is any
constant.
Examples: Find the most general antiderivatives of
each of the following functions.
1.
f(x) = cos x
2.
f(x) = 3x2
3.
f(x) = x4
4.
f(x) = x-2
Examples (cont): Find the most general antiderivatives
of each of the following functions.
5.
f(x) =
1
∛x
6.
f(x) =
20
x
Some general rules for antiderivatives:
* For a more complete list, see p.345 in your text.
What about that "C" in the general antiderivative?
With enough information, you can calculate the value of C.
EXAMPLE: Find f if you know that f'(x) = 8x3 + 12x + 3
and f(1) = 6.
EXAMPLE: Find f if you know that f''(x) = 4 - 6x - 40x3
and f(0) = 2 and f'(0) = 1.
EXAMPLE: A stone thrown upward from a 320-ft cliff at 128 ft/sec eventually
falls to the beach below.
(a)
(b)
(c)
(d)
How long does the stone take to reach its highest point?
What is its maximum height?
How long before the stone hits the beach?
What is the velocity of the stone at impact?