posts 33 and 34

happy late easter.

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.

Mean Value Theorem
Same as Rolle's such that: must be continous and differentiable.Different because: f '(c) = [f(b) - f(a)] / (b-a).

EVT
a continuous function on a closed interval [a,b] must have both a mnimmum and a maximum on the interval

Optimization(steps from packet)
1. Identify all quantities
2. Write an equation
3. Reduce equation
4. Determine domain of equation
5. Determine max/min values

Finding absolute max/min
1. First derivative test
2. Plug critical values into the origonal function to get y-values
3. Plug endpoints into the origional function to get y-values
4. The highest y-value is the absolute maximum
5. The lowest y-value is the absolute minimum
*Remember that absolute maximums or minimums are written as a point, or simply as the y-value

First/second derivative tests.
1. Take a derivative
2. Set it equal to 0
3. Solve for x to find max, min, horizontal tangents, extrema (critical points)
4. Set up intervals using step 3.
5. Plug in first derivate
6. To find an absolute max/min plug values from #5 into original function
7. Check endpoints

Volume by disks is used to find the volume of a solid object
The formula for this is: pi S a-b [R(x)]^2 dx

Volume by washers is used to find volume of object with a hole
pi S a-b top^2-bottom^2 dx
*you will always have two equations.

Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in

ANGLE OF ELEVATION:
HELP.