The FTC in all its glory… and area approximations

1. The FTC in all its glory…and area approximations

2. The Fundamental Theorem of Calculus The theorem establishes a connection between the two branches of calculus: differential calculus and integral calculus: Theorem: Suppose f is continuous on [a, b]. Example of Part 1: The derivative of is

3. Examples (2) Example Solution

4. Examples (3) Example Solution

5. :

6. (a) By the Fundamental Theorem, (b) Plug in x = 1:

8. Here’s 1995 / BC-6:

9. (a) The domain of h is all x for which is defined: (b) A little Chain Rule:

10. (c) Since is positive from -6 to -1 and negative from -1 to 4, the minimum occurs at an endpoint. By comparing areas, h(4) < h(-6) = 0, so the minimum occurs at x = 4. This “area comparison” genre of problem was pretty common in the early graphing calculator days.

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18. Average value of a function The average value of function f on the interval [a, b] is defined as Note: For a positive function, we can think of this definition as saying area/width = average height Example: Find the average value of f(x)=x3 on [0,2].

19. The Mean Value Theorem for Integrals Theorem: If f is continuous on [a, b], then there exists a number c in [a, b] such that Example: Find c such that fave=f(c) for f(x)=x3 on [0,2]. From previous slide, f(c)=fave=2. Thus, c3=2, so

20. done?