1. 4001 ANTIDERIVATIVES AND INDEFINITE INTEGRATION
BC Calculus

2. ANTIDERIVATIVES AND INDEFINITE INTEGRATION
Rem:
DEFN: A function F is called an Antiderivative of the function f, if for
every x in f: F /(x) = f(x)
If f (x) = then F(x) =
or since
If f / (x) = then f (x) =

3. Differential Form (REM: A Quantity of change)
Notation:
Differential Equation
Differential Form (REM: A Quantity of change)
Integral symbol =
Integrand =
Variable of Integration =  

4. The Variable of Integration
Newton’s Law of gravitational attraction
NOW: dr tells which variable is being integrated r
Will have more meanings later!

5. The Family of Functions whose derivative is given.
ANTIDERIVATIVES
Layman’s Idea:
A) What is the function that has f (x) as its derivative? .
-Power Rule:
-Trig:
B) The antiderivative is never unique, all answers must include a
+ C (constant of integration)
The Family of Functions whose derivative is given.

6. Verify the statement by showing the derivative of the right side equals the integral of the left side.

7. The Family of Functions whose derivative is given.
Family of Graphs C
The Family of Functions whose derivative is given.

8. ( REM: A Quantity of change) Increment of change
Notation:
Differential Equation
Differential Form
( REM: A Quantity of change)
Increment of change
  Antiderivative or Indefinite Integral
Total (Net) change

9. General Solution
A) Indefinite Integration and the Antiderivative are the same thing.
General Solution
_________________________________________________________
  ILL:

10. General Solution: EX 1.
General Solution: The Family of Functions
EX 1:

11. General Solution: EX 2.
General Solution: The Family of Functions
EX 2:

12. General Solution: EX 3.
General Solution: The Family of Functions
EX 3:
Careful !!!!!

13. Special Considerations

14. Initial Condition Problems:
B) Initial Condition Problems:
Particular solution < the single graph of the Family –
through a given point>
ILL: through the point (1,1)
-Find General solution
-Plug in Point < Initial Condition > and solve for C

15. through the point (1,1)

16. Initial Condition Problems: EX 4.
B) Initial Condition Problems:
Particular solution < the single graph of the Family –
through a given point.>
Ex 4:

17. Initial Condition Problems: EX 5.
B) Initial Condition Problems:
Particular solution < the single graph of the Family –
through a given point.>
Ex 5:

18. Initial Condition Problems: EX 6.
B) Initial Condition Problems:
A particle is moving along the x - axis such that its acceleration is
At t = 2 its velocity is 5 and its position is 10.
Find the function, , that models the particle’s motion.

19. Initial Condition Problems: EX 7.
B) Initial Condition Problems:
EX 7:
If no Initial Conditions are given:
Find if

20. Last Update:
12/17/10
Assignment
Xerox