Useful Calculus Theorems, Formulas, and Definitions
Following are some of the most frequently used theorems, formulas, and definitions that you encounter in a calculus class for a single variable. The list isn’t comprehensive, but it should cover the items you’ll use most often.
Limit Definition of a Derivative
Definition: Continuous at a number a
The Intermediate Value Theorem
Definition of a Critical Number
A critical number of a function f is a number c in the domain of f such that either f^{ }‘(c) = 0 or f^{ }‘(c) does not exist.
Rolle’s Theorem
Let f be a function that satisfies the following three hypotheses:

f is continuous on the closed interval [a, b].

f is differentiable on the open interval (a, b).

f^{}(a) = f^{}(b).
Then there is a number c in (a, b) such that f^{ }‘(c) = 0.
The Mean Value Theorem
Let f be a function that satisfies the following hypotheses:

f is continuous on the closed interval [a, b].

f is differentiable on the open interval (a, b).
Newton’s Method Approximation Formula
Newton’s method is a technique that tries to find a root of an equation. To begin, you try to pick a number that’s “close” to the value of a root and call this value x_{1}. Picking x_{1} may involve some trial and error; if you’re dealing with a continuous function on some interval (or possibly the entire real line), the intermediate value theorem may narrow down the interval under consideration. After picking x_{1}, you use the recursive formula given here to find successive approximations:
A word of caution: Always verify that your final approximation is correct (or close to the value of the root). Newton’s method can fail in some instances, based on the value picked for x_{1}. Any calculus text that covers Newton’s method should point out these shortcomings.
The Fundamental Theorem of Calculus
Suppose f is continuous on [a, b]. Then the following statements are true:
The Trapezoid Rule
where
Simpson’s Rule
where n is even and