4001 ANTIDERIVATIVES AND INDEFINITE INTEGRATION BC 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) =

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

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

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.

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

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

( 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

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 !!!!!

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 < Initial Condition > 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