Calculus Concepts and Theorems

A collection of important calculus definitions, theorems, and rules essential for understanding derivatives, integrals, and the properties of continuous functions.

Intermediate Value Theorem

If a function is continuous on an interval, it takes every value between its endpoints.

Definition of Derivative

The limit of the average rate of change of the function as the interval approaches zero.

Power Rule

If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

Product Rule

If $f(x) = u(x)v(x)$, then $f'(x) = u'v + uv'$.

Quotient Rule

If f(x) = rac{u(x)}{v(x)}, then f'(x) = rac{u'v - uv'}{v^2}.

Chain Rule

If $f(g(x))$, then $f'(g(x))g'(x)$.

Extreme Value Theorem

If a function is continuous on a closed interval, it has both an absolute maximum and minimum on that interval.

Mean Value Theorem

If a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change equals the average rate of change.

Fundamental Theorem of Calculus, Part 1

If a function is continuous, then the function has a derivative at every point in its domain.

Fundamental Theorem of Calculus, Part 2

If a function is continuous and F is an antiderivative, then $ext{Area under the curve} = F(b) - F(a)$.

Displacement

The difference between the initial and final position of an object.

Total Distance Traveled

The total length of the path traveled by an object, irrespective of direction.

Derivative of Inverses

If $f$ is the inverse of $g$, then f'(x) = rac{1}{g'(f(x))}.