Date post: | 22-Dec-2015 |
Category: |
Documents |
Upload: | alexina-harrell |
View: | 229 times |
Download: | 1 times |
7.1 Antiderivatives
OBJECTIVES*Find an antiderivative of a function.
*Evaluate indefinite integrals using the basic integration formulas.
*Use initial conditions, or boundary conditions, to determine an antiderivative.
Slide 4.2 - 1
Who comes up with the ready-made functions we find derivatives for? Isn’t it hard sometimes to find a function for total cost, profit, etc.?
Sometimes it is easier to calculate the rate of change of something and get the function for the total from it.
This process, the reverse of finding a derivative, is antidifferentiation.
Slide 4.2 - 2Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4.2 - 3Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 1: Can you think of a function that would have x2 as its derivative?
Antiderivatives
Slide 4.2 - 4Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
One antiderivative is x3/3. All other antiderivatives differ from this by a constant. So, we can represent any one of them as follows:
To check this, we differentiate .
x3
3C
d
dx
x3
3C
d
dx
x3
3
d
dxC 3
x2
3 0 x2
x3
3C
Slide 4.2 - 5Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
THEOREM
If two functions F and G have the same derivative over an interval, then
F(x) = G(x) + C, where C is a constant.
Antiderivatives
Slide 4.2 - 6Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Antiderivatives
Integrals and Integration
Antidifferentiating is often called integration.
To indicate the antiderivative of x2 is x3/3 +C, we
x2 dx x3
3C,
write
Slide 4.2 - 7Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Antiderivatives
The notationf x dx
is used to represent the antiderivative of f (x).
f x dx F x C,
More generally,
where
F(x) + C is the general form of the antiderivative of f (x).
Slide 4.2 - 8Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1. k dx kx C (k is a constant)
2. xrdx xr1
r 1C, provided r 1
(To integrate a power of x other than 1, increase the
power by 1 and divide by the increased power.)
THEOREM : Basic Integration Formulas
1
1
13. ln , 0
ln , 0
(
4. or ax
ax ax ax
dxx dx dx x C x
x x
x dx x C x
b ebe dx e C e dx C
a a
We will generally assume that x > 0.)
Slide 4.2 - 9Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 2: Evaluate
4.2 Area, Antiderivatives, and Integrals
x9dx.
Slide 4.2 - 10Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
x9dx x10
10C
Check:
d
dx
x10
10C
10
x9
10 0 x9
Slide 4.2 - 11Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 3: Evaluate
4.2 Area, Antiderivatives, and Integrals
5e4 xdx.
Slide 4.2 - 12Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
5e4 xdx 5
4e4 x C
d
dx
5
4e4 x C
5
4e4 x 4 0 5e4 x
Check:
Slide 4.2 - 13Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
THEOREM 4
(The integral of a constant times a function
is the constant times the integral of the function.)
(The integral of a sum or difference is the sum or difference of the integrals.)
4.2 Area, Antiderivatives, and Integrals
Rule A. kf (x)dx k f (x)dx
Rule B. f (x) g(x) dx f (x)dx g(x)dx
Slide 4.2 - 14Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 4: Evaluate
4.2 Area, Antiderivatives, and Integrals
(5x 4x3 )dx.
Slide 4.2 - 15Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
(5x 4x3 )dx 5x dx 4x3dx 5 x dx 4 x3dx 5
x2
2 4
x4
4C
5
2x2 x4 C
Slide 4.2 - 16Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 5: Evaluate and check by differentiation:
4.2 Area, Antiderivatives, and Integrals
a) 7e6 x x dx; b) 1 3
x 1
x4
dx;
Slide 4.2 - 17Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
a) 7e6 x x dx 7e6 xdx x dx
7e6 xdx x1 2
dx
7
6e6 x
x1 21
1
21
C
7
6e6 x
2
3x3 2 C
Antiderivatives
Example 5 (concluded):
Slide 4.2 - 18Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 5 (continued):Check:
Antiderivatives
d
dx
7
6e6 x
2
3x
3
2 C
7
6e6 x 6
3
22
3x1 2 0
7e6 x x
Slide 4.2 - 19Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 5 (continued):
Antiderivatives
44
4 1
3
3
3 1 1b) 1 1 3
3ln4 1
3ln31
3ln3
dx dx dx x dxx x x
xx x C
xx x C
x x Cx
Slide 4.2 - 20Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 5 (concluded):Check:
Antiderivatives
d
dxx 3ln x
x 3
3C
1 31
x 3 1
3x 4 0
1 3
x
1
x4
Slide 4.2 - 21Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 6: Find the function f such that
First find f (x) by integrating.
Antiderivatives
f (x) x2 and f ( 1) 2.
f (x) x2dxf (x)
x3
3C
Slide 4.2 - 22Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example 6 (concluded): Then, the initial condition allows us to find C.
Thus,
Antiderivatives
f ( 1) ( 1)3
3C 2
1
3C 2
C 7
3
f (x) x3
3
7
3.
Slide 4.2 - 23Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
7
Find a function f whose graph has slope '(x) 6 x 4
and goes through the point (1,1).
Example
f
3( ) 2 4 5f x x x
Slide 4.2 - 24Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
8
Suppose a publishing company has found that the marginal cost at a
level of production of x thousand books is given by
50'( ) and that the fixed cost
(cost before the first book can be prod
Example
C xx
uced) is $25,000.
Find the cost function C(x).
1/2C( ) 100 25,000x x