antiderivatives. think about it suppose this is the graph of the derivative of a function what do we...
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Antiderivatives
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Think About It
Suppose this isthe graph of thederivative of a function
What do we know aboutthe original function?• Critical numbers• Where it is increasing, decreasing
What do we not know?
2
f '(x)
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The work to this point has involved finding and
applying the first or second derivative of a function. In this chapter we will reverse the process. If we know the derivative of a function how
do we obtain the original function? The process is
called antidifferentiation or integration.
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Anti-DerivativesDerivatives give us the rate of change of a function
What if we know the rate of change …• Can we find the original function?
If F '(x) = f(x) • Then F(x) is an antiderivative of f(x)
Example – let F(x) = 12x2 • Then F '(x) = 24x = f(x) • So F(x) = 12x2 is the antiderivative of f(x) = 24x
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Finding An Antiderivative
Given f(x) = 12x3
• What is the antiderivative, F(x)?
Use the power rule backwards• Recall that for f(x) = xn … f '(x) = n • x n – 1
That is … • Multiply the expression by the exponent • Decrease exponent by 1
Now do opposite (in opposite order)• Increase exponent by 1• Divide expression by new exponent
5
4 412( ) 3
4F x x x
4 412( ) 3
4F x x x
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Family of Antiderivatives
Consider a family of parabolas• f(x) = x2 + n
which differ only by value of n
Note that f '(x) is the same foreach version of f
Now go the other way …• The antiderivative of 2x must be different for each of
the original functions
So when we take an antiderivative • We specify F(x) + C• Where C is an arbitrary constant 6
This indicates that multiple
antiderivatives could exist from one derivative
This indicates that multiple
antiderivatives could exist from one derivative
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Indefinite Integral
The family of antiderivatives of a function f indicated by
The symbol is a stylized S to indicate summation
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( )f x dx
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Indefinite Integral
The indefinite integral is a family of functions
The + C represents an arbitrary constant• The constant of integration
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3 41
4x dx x C
2 13 4 3 4x dx x x C
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Properties of Indefinite Integrals
The power rule
The integral of a sum (difference) is the sum (difference) of the integrals
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( ) ( ) ( ) ( )f x g x dx f x dx g x dx
11, 1
1n nx dx x C n
n
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Properties of Indefinite Integrals
The derivative of the indefinite integral is the original function
A constant can be factored out of the integral
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( ) ( )d
f x dx f xdx
( ) ( )f x dx xk k f x d
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• Example : Evaluate (5x 4x3 )dx.
(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 : Find the function f such that
• First find f (x) by integrating.
f (x) x2 and f ( 1) 2.
f (x) x2dxf (x)
x3
3C
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• Example : Evaluate and check by differentiation:
2 4
3 11 dxx x
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Examples
Determine the indefinite integrals as specified below
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5x dx 4 2x x dx 7 dx12x dx
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Integrate
dttt 13
dxx33 dxxx 872 2
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Find each antiderivative
dx
xxx
23
5
4
3 25
dxx
1 dxx2
1
dx
xx 43
54
dxx
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Find each antiderivative
dyyy 43 42
dxxx
321 dxx dxx 232
dx
x
xx4
2 13
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Find each antiderivative
dyx
xx )23)(52( dxxsin4
dxx
3
cos2 dxx2cos
5 dxxx )sin9cos4(
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Find each antiderivative
dxx
x2cos
sin d22 csc2
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Solve the differential equation
3)2(,23)(' fxxf
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Solve the differential equation
1)3(,22)(' 2 fxxxf
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Solve the differential equation
2)1(,1)4(',2)('' ffxf
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Solve the differential equation
1)2(,30)5(',2)('' ffxxf
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Given that the graph of f(x) passes through the point (1,6)
and that the slope of its tangent line at (x.f(x) is 2x+1, find f(6)