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u-du : Integrating Composite Functions

u-du : Integrating Composite Functions. AP Calculus. Integrating Composite Functions (Chain Rule). Remember: Derivatives Rules Remember: Layman’s Description of Antiderivatives. *2 nd meaning of “ du ” du is the derivative of an implicit “ u ” .

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u-du : Integrating Composite Functions

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  1. u-du:Integrating Composite Functions AP Calculus

  2. Integrating Composite Functions (Chain Rule) Remember: Derivatives Rules Remember: Layman’s Description of Antiderivatives *2nd meaning of “du” du is the derivative of an implicit “u”

  3. Integrating Composite Functions (Chain Rule) Revisit the Chain Rule If letu = inside function du = derivative of the inside becomes u-du Substitution

  4. Development from the layman’s idea of antiderivative “The Family of functions that has the given derivative” must have the derivative of the inside in order to find ---------- the antiderivative of the outside

  5. A Visual Aid USING u-du Substitutiona Visual Aid REM: u = inside function du = derivative of the inside let u = becomes now only working with f , the outside function

  6. Working With Constants: Constant Property of Integration With u-du Substitution REM: u = inside function du = derivative of the inside Missing Constant? u = du = Worksheet - Part 1

  7. Example 1 : du given Ex 1:

  8. Example 2: du given Ex 2:

  9. Example 3: du given Ex 3:

  10. Example 4: du given Ex 4:

  11. Example 5: Regular Method Ex 5:

  12. Working with Constants < multiplying by one> Constant Property of Integration ILL. let u = du = and becomes = Or alternately = =

  13. Example 6 : Introduce a Constant - my method

  14. Example 7 : Introduce a Constant

  15. Example 8 : Introduce a Constant << triple chain>>

  16. Example 9 : Introduce a Constant - extra constant << extra constant>

  17. Example 10: Polynomial

  18. Example 11: Separate the numerator

  19. Formal Change of Variables << the Extra “x”>> Solve for x in terms of u ILL: Let Solve for x in terms of u then and becomes

  20. Formal Change of Variables << the Extra “x”>> Rewrite in terms of u - du

  21. Complete Change of Variables << Changing du >> At times it is required to even change the du as the u is changed above. We will solve this later in the course.

  22. Development must have the derivative of the inside in order to find the antiderivative of the outside *2nd meaning of “dx” dx is the derivative of an implicit “x” more later if x = f then dx = f /

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