Prep Session: The Big Theorems

Many theorems in mathematics are expressed as conditional statements in the form “If hypothesis, then conclusion.”

The converse of a conditional statement is found by exchanging the hypothesis and the conclusion of the conditional.

The inverse of a conditional statement is found by negating the hypothesis and conclusion of the statement.

The contrapositive of a conditional statement is found by exchanging AND negating the hypothesis and conclusion.

A counterexample is an example that proves a statement false.

Definitions in mathematics are usually expressed as biconditional statements. Biconditional statements are statements of the form “Hypothesis is true if and only if conclusion is true.” Because of the nature of the biconditional, the following is also true “Conclusion is true if and only if hypothesis is true.”

Consider the following important theorem:

Differentiability implies Continuity

If a function is differentiable at a point, then it is continuous at the point.

What is the converse of the statement? Is the converse true or false?

What is the inverse of the statement? Is the inverse true or false?

What is the contrapositive of the statement? Is the contrapositive true or false?

Intermediate Value Theorem

If f is continuous on a closed interval [a, b] and , then for every value of M between and , there exist at least one value of c in the open interval (a, b) such that

Extreme Value Theorem

If f is continuous on a closed interval [a, b], then f takes on a maximum and a minimum value on that interval.

Mean Value Theorem

If f is continuous on the closed interval [a, b], and differentiable on the open interval (a, b), then there exists a number c in the open interval (a, b) such that

Rolle’s Theorem

If f is continuous on the closed interval [a, b], and differentiable on the open interval (a, b), then there is at least one value of c in the open interval (a, b) such that

and

Increasing/Decreasing Theorem

If f is continuous on the closed interval [a, b], and differentiable on the open interval (a, b), and if for all c in the open interval (a, b), , then f is increasing (decreasing on the closed interval [a, b].

First Derivative Test

If f is differentiable and c is a critical point of f, then if changes from positive to negative at x = c, then is a local maximum of f.If f is differentiable and c is a critical point of f, then if changes from negative to positive at x = c, then is a local minimum of f.

Second Derivative Test

Let f be a function such that and the second derivative of f exists on an open interval containing c, then if , f has a local minimum value at x = c. If , f has a local maximum value at x = c. If , then the second derivative test cannot be used.

Fundamental Theorem of Calculus

If then