If $f$ is continuous on $[a, b]$, then the function $F$ defined by $F(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $F'(x) = f(x)$.

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$.

It shows that differentiation and integration are inverse processes.

If $f$ is continuous on $[a, b]$, then there exists a number $c$ in $[a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$.

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

By finding the antiderivative of the function and evaluating it at the limits of integration.

The derivative of the integral of a function is the original function itself.

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$) and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = L$.

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

If a function $f$ is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the open interval $(a, b)$ such that $f'(c) = 0$.

The area under a function's curve is equal to the value of its antiderivative, calculated with the same bounds.

A function is increasing where its first derivative is positive.

A function is decreasing where its first derivative is negative.

Relative extrema occur where the first derivative is zero or undefined and changes sign.

A function is concave up where its second derivative is positive and concave down where it is negative.

Inflection points occur where the second derivative changes sign.

Area below the x-axis is considered negative when calculating definite integrals.

When $f'(x) > 0$, $f(x)$ is increasing. When $f'(x) < 0$, $f(x)$ is decreasing.

The slope of $f'(x)$ indicates the concavity of $f(x)$. Positive slope means concave up, negative slope means concave down.

Evaluate the function at critical points and endpoints; the smallest/largest value is the absolute minimum/maximum.

Critical Point: $f'(x) = 0$ or undefined, potential for max/min. | Point of Inflection: $f''(x)$ changes sign, change in concavity.

Relative Extrema: Local max/min within an interval. | Absolute Extrema: Overall max/min over the entire domain.

Increasing: $f'(x) > 0$, function is rising. | Concave Up: $f''(x) > 0$, function curves upwards.

Decreasing: $f'(x) < 0$, function is falling. | Concave Down: $f''(x) < 0$, function curves downwards.

Function: Represents the value of the function at each point. | Derivative: Represents the rate of change of the function at each point.

Definite Integral: Computes the area under a curve between two limits, resulting in a numerical value. | Indefinite Integral: Finds the antiderivative of a function, resulting in a family of functions.

$f'(x)$: The derivative of $f(x)$, representing the instantaneous rate of change. | $\int f(x) dx$: The antiderivative of $f(x)$, representing the accumulation of $f(x)$.

First Derivative Test: Examines the sign change of $f'(x)$ around a critical point. | Second Derivative Test: Uses the sign of $f''(x)$ at a critical point to determine concavity and thus whether it is a max or min.

Local Extremum: A maximum or minimum within the interior of an interval. | Endpoint Extremum: A maximum or minimum that occurs at the boundary of an interval.

Average Rate of Change: The slope of the secant line between two points. | Instantaneous Rate of Change: The slope of the tangent line at a single point, given by the derivative.