ANTIDERIVATIVES AND INDEFINITE INTEGRATION AB Calculus

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) =

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.

Family of Graphs +C The Family of Functions whose derivative is given. All same equations with different y intercept

Notation: Differential Equation Differential Form (REM: A Quantity of change) Integral symbol = Integrand = Variable of Integration = small =sum Summing a bunch of little changes

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

The Family of Functions whose derivative is given.

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

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

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

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

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

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

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

Special Considerations

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 and solve for C

through the point (1,1)

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

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

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.

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

Last Update: 12/17/10 Assignment –Xerox