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ANTIDERIVATIVES AND INDEFINITE INTEGRATION

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)

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ANTIDERIVATIVES AND INDEFINITE INTEGRATION

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  1. ANTIDERIVATIVES AND INDEFINITE INTEGRATION ABCalculus

  2. ANTIDERIVATIVES AND INDEFINITE INTEGRATION Rem: DEFN: A function F is called an Antiderivativeof 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. ANTIDERIVATIVES Layman’s Idea: A) What is the function that hasf (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.

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

  5. Notation: Differential Equation Differential Form (REM: A Quantity of change) Integral symbol = Integrand = Variable of Integration =   8-8.1 small Summing a bunch of little changes =sum

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

  7. The Family of Functions whose derivative is given.

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

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

  15. Special Considerations

  16. Initial Condition Problems: B) InitialCondition 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 1+c

  17. through the point (1,1)

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

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

  20. Initial Condition Problems: EX 6. B) InitialCondition 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.

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

  22. Last Update: • 12/17/10 • Assignment • Xerox

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