Antidifferentiation.ppt

1. Antidifferentiation
TS: Making decisions after
reflection and review

2. Objectives
 To define antidifferentiation.
 To investigate antiderivatives, indefinite
integrals, and all of their parts.
 To use basic integration rules to find
antiderivatives.

3. CALCULUS

4. $200
Its derivative is
2x
What is
2
( )
f x x
 ?

5. $400
Its derivative is
2
3x
What is
3
( )
f x x
 ?

6. $600
Its derivative is
4x
What is
2
( ) 2
f x x
 ?

7. $800
Its derivative is
2
1
x
What is
1
( )
f x
x
  ?

8. $1000
Its derivative is
x
What is
3
2
2
3
( )
f x x
 ?

9. Antidifferentiation
 Up to this point in calculus, you have been
concerned primarily with this problem:
given a function, find its derivative
 Many important applications of calculus
involve the inverse problem: given the
derivative of a function, find the function

10. Antidifferentiation
 This operation of determining the original
function from its derivative is the inverse
operation of differentiation and is called
antidifferentiation.
 Antidifferentiation is a process or operation
that reverses differentiation.

11. Antidifferentiation
  7
F x x

 
G x 
8
1
8 x

7 1
x 
1
8
What is the antiderivative of ?
Notice:    
G x F x
 
G is an antiderivative of F.

12. Antiderivatives & Indefinite Integrals
( )
f x dx

Differential
(variable of
integration)
Integral
sign
Integrand
Antiderivative
 The antidifferentiation process is also called integration.
Indefinite
Integral
( )
F x

The derivative of F is f.
   
F x f x
 

13. Antiderivatives & Indefinite Integrals
N
x dx 

1
N
x 
1
N 
, 1
N  
1
N  
1 1
When 1, N
x
N x x
   
Q: What function has the derivative ?
1
x
A: ln x
ln ,
x
This absolute value
prevents you from
having to find the
natural log of a
negative number.
The Power Rule for Integration

14. Antiderivatives & Indefinite Integrals
What if we were to shift the graph up 1 unit?
2x dx 

2
x

15. Antiderivatives & Indefinite Integrals
Do the slopes change?

16. Antiderivatives & Indefinite Integrals
The slopes stay the same.
2
2 1
x dx x
 


17. Antiderivatives & Indefinite Integrals
If a function has an antiderivative,
then it has an infinite number of antiderivates.
2
2x dx x C
 


18. Antiderivatives & Indefinite Integrals
2
2x dx x C
 

To capture the fact that there are infinitely
many antiderivatives we add a constant.
Constant of
Integration

19. Basic Integration Rules
(Number) dx

(Number) x C
 
Evaluate 2 dx 

2x C


20. Basic Integration Rules
c dx cx C
 

Constant Rule for Integration
Evaluate 5 dx
 

5x C
 

21. Basic Integration Rules
 
(Number) f x dx


 
(Number) f x dx
 
The integral of a function times a
constant is equal to the constant
times the integral of the function.

22. Basic Integration Rules
3
Evaluate 5x dx 

3
5 x dx 

5
4
4
x
 C
 
4
5
4 x C

Q: How do you know if you have found the correct antiderivative?
A: Take the derivative of your answer to check.

23. Basic Integration Rules
   
c f x dx c f x dx
 
 
Constant Multiple Rule for Integration
Sum & Difference Rules for Integration
       
f x g x dx f x dx g x dx
 
  
 
  
       
f x g x dx f x dx g x dx
 
  
 
  

24. Basic Integration Rules
 
2 1
Evaluate 6 4 +1
x
x x dx
  

2 1
6 4 1
x
x dx x dx dx dx
   
   
6
3
3
x
C
 
3
2x
4

2
2
x
ln x
 x

2
2x
 ln x x C
  

25. You Try these three.
2
Evaluate 2x dx



Evaluate 5du 

2
3 t dt 

2
Evaluate 3t dt 

Evaluate (3 4)
x x dx
 

3
t
3
3
c
  3
t c

5u c

2
2 x dx



-1
x
2
1
c
 

-2
x
C

2
(3 4 )
x x dx
 

2
3 4
x dx xdx
 
 
3 2
x 2x c
 

26. Conclusion
 Antidifferentiation is a process or operation
that reverses differentiation.
 The antidifferentiation process is also
called integration.
 Similar to differentiation, integration has a
variety of rules that we must remember,
recall, and be able to use.