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U-du : Integrating Composite Functions AP Calculus.

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Presentation on theme: "U-du : Integrating Composite Functions AP Calculus."— Presentation transcript:

1 u-du : Integrating Composite Functions AP Calculus

2 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”

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

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? Worksheet - Part 1 u = du =

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 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 >

16 Example 9 : Introduce a Constant - extra constant

17 Example 10 : Polynomial

18 Example 11: Separate the numerator

19 Formal Change of Variables > Solve for x in terms of u ILL: Let Solve for x in terms of u then and becomes

20 Formal Change of Variables > Rewrite in terms of u - du

21 Formal Change of Variables > Solve for x in terms of u - du > - could divide or multiply by

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

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


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