Optimization
1 Identify primary and secondary equations your primary is the one your or maximizing or minimizing and your secondary is the other equation
2. Solve for your secondary variable and plug into your primary equation if your primary only has on variable this isn’t necessary
3. Plug into secondary equation to find the other value check your end points if necessary
Related Rates:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation involving your variables
5. Take the derivative with respect to time
6. Substitute in derivative and solve.
for some things im still having trouble with are rolles theorm and mean value theroem. Can somebody help
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Rolle's Theorem
ReplyDeleteThere are three conditions you must check before applying Rolle's theorem:
1. The function has to be continuous on the interval given
2. The function has to be differentiable
3. f(a) = f(b)
If it passes all three, all you have to do is take the derivative, set equal to zero and solve for x or c.
Example
f(x) = x^2 -3x +2 Show that f'(x)=0 on some interval.
So the first thing you have to do is find your interval. Since it is not given, set your equation equal to 0 and find your x intercepts.
x^2 -3x +2=0
(x-2) (x-1)
x= 2, 1
so your interval is [1,2]
Now check to see if your equation is continuous and it is because it is a parabola and if it is differential which it is.
Next plug in your interval to see if they are equal.
1: (1)^2 -3(1) +2 = 0
2: (2)^2 - 3(2) +2 = 0
f(1) = f(2) = 0
Since it passes all the requirements, Rolle's theorem can be applied
Take the derivative: 2x - 3
Set equal to zero and solve for x: 2x-3 = 0
x= 3/2 so c= 3/2
To further up on Chelsea's comment...
ReplyDeleteRolle's theorem states that differentiable function (thus the requirement continuous and differentiable), which attains equal values at two points (thus f(a)=f(b)), must have a point somewhere between them where the slope of the tangent line to the graph of the function is zero (this is the derivative part and finding c).
If you know the actual theorem, actually doing the problem is quite simple. No need to really remember the steps...
EVT: the EVT states that a continuous function on a close interval [a,b], must have both a minimum and a maximum on the interval. However, the max and min can occur at the endpoints.
ReplyDeleteRolles theorem gives the conditions that guarantee the existance of an extrema in the interior of a closed inteval.
Rolles: Let f be continuous on a closed interval [a,b] and differentiable on the open interval (a,b). If f(a)= f(b) then there is at least one number, “c” in (a,b) such the f '(c)=0
MVT: If f is continuous on the closed interval [a,b] and differentiable on the interval (a,b) then there exists a number c in (a,b) such that f '(c)= (f(b)-f(a))/(b-a).