11/27/10

Well since we didn't have school this week, I think I'll just review some simple things like Rolle's Theorem and the Mean Value Theorem..
**First, let's start off with Rolle's Theorem. When using it, it tells you IF there is a max or a min on an interval..and if you DO have one, it is going to be AT LEAST 1 max OR min. Typically, Rolle's Theorem uses x-intercepts. Butttttttt, before you can even apply the theorem, you first have to check the function to make sure it is both continuous and differentiable. Then you have to check to make sure the y-values match (check this by plugging both numbers of your interval into the function you're given). Once you've checked all that, you take the derivative of the function and set it equal to zero to find your x-value(s). Once you get those x-values you have to check to make sure they are on the interval you were given. If not, then you just don't include them in your answer.
Ex 1.) Determine whether Rolle's Theorem can be applied to the function f(x)=x^2-5x+4 on the closed interval [1,4]...and if it can be applied, find all values of c.
*Okay, first check the function for differentiability and continuity....it is continuous and differentiable. Now check to make sure the y-values match, so plug 1 and 4 into the function:
f(1)=1-5+4=0
f(4)=16-20+4=0
andddddddd the y-values do match, so now you can move on to the next step and take the derivative of the function which is:
2x-5
Now set it equal to zero and solve for x:
2x-5=0
x=5/2 ...and that's your value of c
**Now let's move on to the Mean Value Theorem...It is actually very similar to Rolle's Theorem except that it involves slope. When you use this theorem, it means on an interval there must be some x-value where the derivative equals the slope between the numbers of that interval. But before you use the theorem, you have to check the function for differentiability and continuity (but NOT if the y-values are the same; that doesn't matter). Once you do that, you find the slope using this formula: f'(c)=f(b)-f(a)/b-a...where the interval is [a,b]
Once you do that, you take the derivative of the function and then set it equal to the slope and solve for x
Ex 2.) Determine whether the Mean Value Theorem can be applied to f(x)=x^3+2x on the closed interval [-1,1]...if it can be applied, find all values of c.
*Yessssssss, the function is continuous and differentiable so we can move on with our lives...and find the slope of the function!
*So you should get (3+3)/2
=3
*Now take the derivative of the function......and you should get 3x^2+2
*Now set the derivative equal to the slope and solve for x:
3x^2+2=3
x^2=1/3
x=+/-squareroot of 3/3 ....and that's your value of c

**And I SERIOUSLY NEED HELP with drawing the graphs..(like when you're given the graph of f and they ask you questions about the graph of f' and f''..) Those are really tough; I could use a lot more practice with them.