Week Six

This week was very hectic and we learned a lot. We had a test on monday, which didn't really go well for me. On Tuesday we learned about the Extreme Value Theorem, the Rolle's Theorem, and the Mean Value Theorem. On Wednesday, we learned about Optimization. On Thursday and Friday, we did example problems with Optimization.

Things I now understand:

1.) Everything I got wrong on my test, thanks to doing corrections.

2.) Extreme Value Theorem

3.) Rolle's Theorem
F(x) must be continuous on a closed intervale. It must be differentiable on the open interval. And f(a) must equal f(b). If all this is true, then there is at least one number 'c' within (a,b) such that f'(c) = 0.

Example:

You are given: f(x) = x^4 -2x^2 on [-2,2]. Find all values c such that f '(c) = 0.

Since it is continous (polynomial) and differentiable (no corners/discontinuities), you now check that f(a) = f(b).

f(-2) = f(2) = 8

Since all this works, you take the derivative of f(x).

F'(x) = 4x^3 -4x

You then set f '(x) = 0

You get that x = 0, 1, -1
Since all these are within the interval:

c= -1, 0, 1

4.) Mean Value Theorem

Same as Rolle's such that: must be continous and differentiable.

Different because: f '(c) = [f(b) - f(a)] / (b-a).


Something I do not understand is Optimization.

I understand most of the steps, but usually get stuck when trying to find the primary and secondary equations and which is one is which.

And example would be Example 5 on the handout Ms. Robinson gave us.


I am still kicking myself in the butt over not studying hard enough for the test on Monday, but I guess I know now.