4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington.
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Transcript of 4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington.
4.1 : Anti-derivatives
Greg Kelly, Hanford High School, Richland, Washington
First, a little review:
Consider:2 3y x
then: 2y x 2y x
2 5y x or
It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.
However, when we try to reverse the operation:
Given: 2y x find y
2y x C
We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.
If we have some more information we can find C.
Given: and when , find the equation for .2y x y4y 1x
2y x C 24 1 C
3 C2 3y x
This is called an initial value problem. We need the initial values to find the constant.
An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.
4.1 Antiderivatives and Indefinite Integration
Exploration
Finding Antiderivatives For each of the following derivatives, describe the original
function .F
(a) '( ) 2F x x
(b) '( )F x x
2(c) '( )F x x
2
1(d) '( )F x
x
3
1(e) '( )F x
x
(f) '( ) cosF x x
2( )F x x
2( ) / 2F x x
3( ) / 3F x x
( ) 1/F x x
2( ) 1/ 2F x x
( ) sinF x x
4.1 Antiderivatives and Indefinite Integration
2( ) 3f x x
Find the function that is the antiderivative of .F f
3( ) + C becauseF x x3
2[ ]3
d x Cx
dx
2
Here are some members of the
family of antiderivatives of ( ) 3 :f x x
31( ) 5F x x 3
2 ( ) 36F x x ...etc
33( )
3
xF x C
4.1 Antiderivatives and Indefinite Integration
2[ ] 2xD x x
The family of all antiderivatives of ( ) 2 is
represented by
f x x
2( )F x x C
4.1 Antiderivatives and Indefinite Integration
Find the general solution of the differential equation
' 2y 2y x C
4.1 Antiderivatives and Indefinite Integration
:Notation
( )y f x dx ( )F x C
Integrand
Variable of
Integration
Constant of
Integration
is the antiderivative (indefinite integral) of ( )
with respect to .
y f x
x
4.1 Antiderivatives and Indefinite Integration
3xdx 3 xdx 2
32
xC
23
2x C
RewriteIntegrate
Simplify
4.1 Antiderivatives and Indefinite Integration
3
1dx
x 3x dx 2
1
2C
x
2
2
xC
xdx 1/ 2x dx 3 / 2
3/ 2
xC
3 / 22
3x C
2sin xdx 2 sin xdx 2 ( cos )x C
2cos x C
4.1 Antiderivatives and Indefinite Integration
( 2)x dx 2
1 222
xC x C
2xdx dx
2
22
xx C
4.1 Antiderivatives and Indefinite Integration
4 2(3 5 )x x x dx May we skip step 2?????
5 3 23 5 1
5 3 2x x x C
5 3 2
3 55 3 2
x x xC
4.1 Antiderivatives and Indefinite Integration
1xdx
x
1/ 2 1/ 2( )x x dx 3 / 2 1/ 2
3/ 2 1/ 2
x xC 3 / 2 1/ 22
23x x C
Never integrate the numerator and denominator seperately!
Rewrite the Integrand
4.1 Antiderivatives and Indefinite Integration
2 5 3 2x xdx
x
5/2 3/2 1/212 2220
5 3x x x C
4.1 Antiderivatives and Indefinite Integration
2
sin
cos
xdxx
1 sin
cos cos
xdx
x x
sec tanx xdx sec x C
Front Cover#15?
????
4.1 Antiderivatives and Indefinite Integration
Find the general solution of
2
1'( ) , 0f x F x x
x
2( )F x x dx1
1
xC
1C
x
4.1 Antiderivatives and Indefinite Integration
Find the particular solution that
satisfies the initial condition
(1) 0F 1
(1)1
F C 0
1C
1( ) 1, 0F x x
x
1F x C
x
4.1 Antiderivatives and Indefinite Integration
Solve the differential equation given the initial condition.
3 1, 2 3f x x f
231
2f x x x
4.1 Antiderivatives and Indefinite Integration
A ball is thrown upward with an initial velocity of
64 feet per second from an initial height of 80 feet.
(a) Find the position function giving the height as a function of the time .s t
0 initial timet
Given Initial Conditions:Acceleration due to gravity: 32 feet per second per second
''( ) 32s t
'(0) 64s (0) &80s
80 ft
4.1 Antiderivatives and Indefinite Integration
(a) Find the position function giving the height as a function of the time .s t
0 initial timet Given Conditions:Acceleration due to gravity: 32 feet per second per second
''( ) 32s t
'(0) 64s (0) &80s
'( )s t ''( ) 32s t dt dt 132t C
164 32(0) C 1 64C
'( ) 32 64s t t
4.1 Antiderivatives and Indefinite Integration
(a) Find the position function giving the height as a function of the time .s t
( ) '( ) ( 32 64)s t s t dt t dt 2
232 642
tt C
2216 64t t C Using (0) 80:s
2
280 16 0 64 0 C 2 80C 2( ) 16 64 80s t t t
(b) When does the ball hit the ground?
2( ) 64 816 0s t t t 0216( 4 5) 0t t
16( 5)( 1) 0t t
5 secondst
80 ft
5t
Basic Integration Rules
These two equations allow you to obtain integration formulas directly from differentiation formulas, as shown in the following summary.
Basic Integration Rulescont’d
4.1 Antiderivatives and Indefinite Integration
Before you begin the exercise set, be sure you realize
that one of the most important steps in integration is
rewriting the integrand
in a form that fits the basic integration rules.
Practice Exercises
Original Rewrite Integrate Simplify
2dxx
22 1t dt
3
2
3xdx
x
3 4x x dx
4.1 Homework
HW 4.1 Wed: pg. 255, 5-14 all, 15-47 odd,55-63 odd 73,77, 79Thurs:MMM pg. 124-125Fri:More practice problems (4.1)