1. Find $G'(x) = f(x)$. 2. Determine where $f(x) > 0$.

1. Find $G'(x) = f(x)$. 2. Find $G''(x) = f'(x)$. 3. Determine where $f'(x) > 0$.

Identify points on the graph of $f'(x)$ where the slope changes sign (from positive to negative or vice versa).

1. Find $g'(x) = f'(x) - 1$. 2. Solve for $f'(x) < 1$. 3. Identify the intervals where $f'(x)$ is less than 1.

1. Find $g'(x) = f'(x) - 1$. 2. Find critical points where $g'(x) = 0$. 3. Evaluate $g(x)$ at critical points and endpoints $a$ and $b$. 4. Choose the smallest value.

1. Identify the interval [a, b]. 2. Integrate the function over the interval: $\int_a^b f(x) dx$. 3. Take the absolute value if the region is below the x-axis.

1. Find $f'(x)$. 2. Find critical points where $f'(x) = 0$ or is undefined. 3. Check if $f'(x)$ changes from positive to negative at the critical point.

1. Find $f'(x)$. 2. Find critical points where $f'(x) = 0$ or is undefined. 3. Check if $f'(x)$ changes from negative to positive at the critical point.

Use the formula: $f(x) = f(a) + \int_a^x f'(t) dt$, where $f(a)$ is the initial value.

1. Find $f''(x)$. 2. Determine where $f''(x) > 0$ (concave up) and $f''(x) < 0$ (concave down).

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.

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.