AP Calculus

What are the three conditions that must exist for a point to be continuous?

1. The function must be defined at the point. 2. The limit of the function must exist at that point. 3. The limit of the function as it approaches the point must equal the function's value at that point.

What does the intermediate value theorem state?

If a function is continuous on a closed interval [a, b] and k is a number between f(a) and f(b), there is at least one number c in (a,b) such that f(c )=k.

In other words, the intermediate value theorem asserts that for any value between the outputs of a continuous function on a closed interval, there is a corresponding input where the function equals that value.

What is L’Hospital Rule?

It is used to evaluate indeterminate forms, including 0/0 or infinity/infinity. You can take the derivative of the functions in the limit and set the limit equal to those derivatives.

What is the Extreme Value Theorem?

If a function is continuous on a closed interval [a,b], then f(x) must have at least one absolute maximum and absolute minimum in that interval.