7.1 antiderivatives objectives * find an antiderivative of a function. *evaluate indefinite...
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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.
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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.
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Example 1: Can you think of a function that would have x2 as its derivative?
Antiderivatives
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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
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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
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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
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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).
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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.)
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Example 2: Evaluate
4.2 Area, Antiderivatives, and Integrals
x9dx.
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x9dx x10
10C
Check:
d
dx
x10
10C
10
x9
10 0 x9
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Example 3: Evaluate
4.2 Area, Antiderivatives, and Integrals
5e4 xdx.
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5e4 xdx 5
4e4 x C
d
dx
5
4e4 x C
5
4e4 x 4 0 5e4 x
Check:
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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
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Example 4: Evaluate
4.2 Area, Antiderivatives, and Integrals
(5x 4x3 )dx.
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(5x 4x3 )dx 5x dx 4x3dx 5 x dx 4 x3dx 5
x2
2 4
x4
4C
5
2x2 x4 C
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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;
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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):
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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
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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
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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
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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
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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.
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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
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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
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0.01
9
Find the cost function for the given marginal cost function.
C'(x)=0.03e
with a fixed cost of $8.
x
Example
0.01C( ) 3 5xx e