There are three main theorems that one should know about when doing Calculus. They are the Mean Value Theorem, the Intermediate Value Theorem, and Rolle's Theorem.


Mean Value Theorem


Unformal definition: given a section of a curve, there is at least one point on that section in which the derivative. or slope, of the curve is parallel to the average derivative of the secmvt.pngtion
Formal definition: for any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c
(more information available at Wikipedia)

The equation above is the equation used in conjunction with the MVT.


In order for the MVT to apply, the function must be continuous, and

mvt3.pngmust be a finite number or equals plus or minus infinite. The derivative of the function is the finite number.


This video gives a brief visual description of the MVT.


1. Find the number in the given interval that satisfies the Mean Value Theorem.

First you have to calculate the slope of the secant line through the endpoints.
Than you have to find the derivative of the function.
Last, set the derivative equal to the slope of the secant and solve for x which is the number that satisfies the MVT.
*The MVT applies to this function because it is continuous on the interval and there are no cusps of kinks in the graph which would make it non-differentiable. Make sure you check the graph of the function prior to trying to apply the MVT.*
This is the graph of the function above.

1. Drill Problems on the Mean Value Theorem
2. More Drill Problems (go to Calculus Book I, Techniques and Theory of Differentiation, Theory, Mean Value Theorem)
Applet site (shows how the MVT works using a graph)
Applet site (shows how the IVT works using a graph)
5. Mean Value Theorem Examples and Solutions (home page link, this site also has a real-life application)


Intermediate Value Theoremivt.jpg


Unformal Definition: for each value between the least upper bound and greatest lower bound on the image of a continuous function, there is a corresponding value in the domain mapping the original continuous function
Formal Definition: If the function y=f(x), is continuous on the interval [a,b], and u is a number between f(a) and f(b), then there is a c on [a,b] such that f(c)=u.


The function must be continuous and u must be a real number such that f(a)<u<f(b) or f(a)>u>f(b)and a<c<b. *For instance, given f is continuous on [1, 2], if f(1) = 3 and f(2) = 5 then f must take the value 4 somewhere between 1 and 2.*


This video briefly describes the IVT further. Here is another video.


2. Show that on the interval [0,1], the following function has a root (x=0). (problem from here)

Given equation.
Create a new equation using f(x).

Solve f(x) at each of the endpoints.

Graph the fuction f(x) to show that it is continuous and differentiable on the given interval, which in this case it is.
This graph shows that the function is continuous and differentiable on the given interval.
Because 0 is between f(0) and f(1) and the function is continuous, there must be a point c on the interval [0,1] where f(c)=0.

Now, you have to find the value of c. Simply put c in for x in the f(x) equation and then set f(c) equal to zero and solve for c.
Place c into the equation.

Set the equation equal to zero.

Solve for c by graphing the equation on your calculator and finding the zero.


Rolle's Theorem


rolles_theorem.pngUnformal Definition: A differentiable and continuous function, which attains equal values at two points, must have a point somewhere between them where the slope of the tangent line to the graph of the function is zero.
Formal Definition: If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and if f(a)=f(b), then there is at least one number c in (a,b) such that f'(c)=0.


The function must be continuous on [a,b] and differentiable on (a,b). Also, the endpoints must equal eachother f(a)=f(b).
*This theorem is very similar to the MVT mentioned above.*


3. Find all value of c in the interval (-2,2) such that f'(c)=0

First take the derivative of the function and set it equal to zero.

Factor the derivative to a simpler form to find the values of x easier.
Solve for x.
*This means that on the interval (-2,2) there are three values of x where the derivative will equal zero.*
This is a zoomed in image of the function and the visible maximum and minimum values where the derivative is 0.

*Use the same drill problems for the Mean Value Theorem above for practice.*


The video above provides real-life applications and information on the MVT and Rolle's Theorem.