Glossary

Accumulation functions

Criticality: 3

Functions defined by an integral with a variable upper limit, representing the accumulated change of a rate over an interval.

Example:

If a car's velocity is given by $v(t)$ , then $s(x) = \int_0^x v(t) dt$ is an accumulation function representing the total distance traveled from time 0 to $x$ .

Antiderivatives

Criticality: 3

A function whose derivative is the original function; it is the reverse operation of differentiation.

Example:

If $f(x) = 2x$ , then $F(x) = x^2 + C$ is an antiderivative of $f(x)$ , where C is any constant.

Area under the curve

Criticality: 3

The definite integral of a function over an interval, representing the net signed area between the function's graph and the x-axis.

Example:

The area under the curve of a velocity function from time $t_1$ to $t_2$ gives the total displacement of an object during that time interval.

Concave Down (F(x))

Criticality: 2

A function $F(x)$ is concave down on an interval if its second derivative, $F''(x)$, is negative on that interval, indicating the slope is decreasing.

Example:

If $F''(x) = -x^2$ , then $F(x)$ is always concave down because $-x^2 \le 0$ for all $x$ .

Concave Up (F(x))

Criticality: 2

A function $F(x)$ is concave up on an interval if its second derivative, $F''(x)$, is positive on that interval, indicating the slope is increasing.

Example:

If $F''(x) = e^x$ , then $F(x)$ is always concave up because $e^x > 0$ for all $x$ .

Critical Points

Criticality: 2

Points in the domain of a function where the first derivative is either zero or undefined, which are candidates for local extrema.

Example:

For $F(x) = x^3 - 3x$ , $F'(x) = 3x^2 - 3$ . Setting $F'(x)=0$ gives $x=\pm 1$ , which are the critical points.

Decreasing (F(x))

Criticality: 2

A function $F(x)$ is decreasing on an interval if its first derivative, $F'(x)$, is negative on that interval.

Example:

If $F'(x) = x^2 - 4$ , then $F(x)$ is decreasing on $(-2, 2)$ because $F'(x) < 0$ there.

Endpoints

Criticality: 2

The boundary values of a closed interval over which a function is being analyzed, which must be considered when finding absolute extrema.

Example:

To find the absolute maximum of $F(x)$ on $[0, 5]$ , you must evaluate $F(x)$ at any critical points within $(0,5)$ and at the endpoints $x=0$ and $x=5$ .

Fundamental Theorem of Calculus (FTC)

Criticality: 3

A foundational theorem linking differentiation and integration, stating that the derivative of an accumulation function is the integrand, and that definite integrals can be evaluated using antiderivatives.

Example:

The FTC allows us to find the exact area under $f(x) = x^2$ from 0 to 2 by evaluating its antiderivative $F(x) = x^3/3$ at the bounds: $F(2) - F(0) = 8/3$ .

Increasing (F(x))

Criticality: 2

A function $F(x)$ is increasing on an interval if its first derivative, $F'(x)$, is positive on that interval.

Example:

If $F'(x) = \cos(x)$ , then $F(x)$ is increasing on $(0, \pi/2)$ because $\cos(x) > 0$ there.

Inflection Point (F(x))

Criticality: 3

A point on the graph of a function where the concavity changes (from concave up to down or vice versa), meaning the second derivative changes sign.

Example:

For $F(x) = x^3$ , $F''(x) = 6x$ . At $x=0$ , $F''(x)$ changes sign, so $(0,0)$ is an inflection point.

Relative Maximum (F(x))

Criticality: 3

A point where a function changes from increasing to decreasing, meaning its first derivative changes from positive to negative.

Example:

If $F'(x)$ changes from positive to negative at $x=c$ , then $F(c)$ is a relative maximum of $F(x)$ .

Relative Minimum (F(x))

Criticality: 3

A point where a function changes from decreasing to increasing, meaning its first derivative changes from negative to positive.

Example:

If $F'(x)$ changes from negative to positive at $x=c$ , then $F(c)$ is a relative minimum of $F(x)$ .

Glossary

Accumulation functions

Criticality: 3

Functions defined by an integral with a variable upper limit, representing the accumulated change of a rate over an interval.

Example:

If a car's velocity is given by $v(t)$ , then $s(x) = \int_0^x v(t) dt$ is an accumulation function representing the total distance traveled from time 0 to $x$ .

Antiderivatives

Criticality: 3

A function whose derivative is the original function; it is the reverse operation of differentiation.

Example:

If $f(x) = 2x$ , then $F(x) = x^2 + C$ is an antiderivative of $f(x)$ , where C is any constant.

Area under the curve

Criticality: 3

The definite integral of a function over an interval, representing the net signed area between the function's graph and the x-axis.

Example:

The area under the curve of a velocity function from time $t_1$ to $t_2$ gives the total displacement of an object during that time interval.

Concave Down (F(x))

Criticality: 2

A function $F(x)$ is concave down on an interval if its second derivative, $F''(x)$, is negative on that interval, indicating the slope is decreasing.

Example:

If $F''(x) = -x^2$ , then $F(x)$ is always concave down because $-x^2 \le 0$ for all $x$ .

Concave Up (F(x))

Criticality: 2

A function $F(x)$ is concave up on an interval if its second derivative, $F''(x)$, is positive on that interval, indicating the slope is increasing.

Example:

If $F''(x) = e^x$ , then $F(x)$ is always concave up because $e^x > 0$ for all $x$ .

Critical Points

Criticality: 2

Points in the domain of a function where the first derivative is either zero or undefined, which are candidates for local extrema.

Example:

For $F(x) = x^3 - 3x$ , $F'(x) = 3x^2 - 3$ . Setting $F'(x)=0$ gives $x=\pm 1$ , which are the critical points.

Decreasing (F(x))

Criticality: 2

A function $F(x)$ is decreasing on an interval if its first derivative, $F'(x)$, is negative on that interval.

Example:

If $F'(x) = x^2 - 4$ , then $F(x)$ is decreasing on $(-2, 2)$ because $F'(x) < 0$ there.

Endpoints

Criticality: 2

The boundary values of a closed interval over which a function is being analyzed, which must be considered when finding absolute extrema.

Example:

To find the absolute maximum of $F(x)$ on $[0, 5]$ , you must evaluate $F(x)$ at any critical points within $(0,5)$ and at the endpoints $x=0$ and $x=5$ .

Fundamental Theorem of Calculus (FTC)

Criticality: 3

Example:

The FTC allows us to find the exact area under $f(x) = x^2$ from 0 to 2 by evaluating its antiderivative $F(x) = x^3/3$ at the bounds: $F(2) - F(0) = 8/3$ .

Increasing (F(x))

Criticality: 2

A function $F(x)$ is increasing on an interval if its first derivative, $F'(x)$, is positive on that interval.

Example:

If $F'(x) = \cos(x)$ , then $F(x)$ is increasing on $(0, \pi/2)$ because $\cos(x) > 0$ there.

Inflection Point (F(x))

Criticality: 3

A point on the graph of a function where the concavity changes (from concave up to down or vice versa), meaning the second derivative changes sign.

Example:

For $F(x) = x^3$ , $F''(x) = 6x$ . At $x=0$ , $F''(x)$ changes sign, so $(0,0)$ is an inflection point.

Relative Maximum (F(x))

Criticality: 3

A point where a function changes from increasing to decreasing, meaning its first derivative changes from positive to negative.

Example:

If $F'(x)$ changes from positive to negative at $x=c$ , then $F(c)$ is a relative maximum of $F(x)$ .

Relative Minimum (F(x))

Criticality: 3

A point where a function changes from decreasing to increasing, meaning its first derivative changes from negative to positive.

Example:

If $F'(x)$ changes from negative to positive at $x=c$ , then $F(c)$ is a relative minimum of $F(x)$ .