Glossary

Antiderivative

Criticality: 3

A function whose derivative is the original function. Finding an antiderivative is the reverse process of differentiation.

Example:

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

Definite Integral

Criticality: 3

An integral with upper and lower limits of integration, representing the net signed area under the curve of a function over a specific interval.

Example:

To find the total distance traveled by a car with velocity $v(t)$ from $t=0$ to $t=5$ seconds, we would calculate the definite integral $\int_{0}^{5} v(t) dt$ .

Differentiation

Criticality: 2

The process of finding the derivative of a function, which represents the instantaneous rate of change of the function.

Example:

The process of finding the slope of the tangent line to a curve at any point is called differentiation.

Fundamental Theorem of Calculus (FTOC)

Criticality: 3

A foundational theorem in calculus that links the concepts of differentiating a function and integrating a function. It has two main parts that establish the relationship between antiderivatives and definite integrals.

Example:

The Fundamental Theorem of Calculus allows us to quickly evaluate $\int_{0}^{1} x^2 dx$ by finding the antiderivative and evaluating it at the bounds, rather than using Riemann sums.

Fundamental Theorem of Calculus Part 1

Criticality: 2

States that if $g(x) = \int_{a}^{x} f(t) dt$, then $g'(x) = f(x)$. It shows that differentiation and integration are inverse operations.

Example:

If $g(x) = \int_{1}^{x} \sin(t) dt$ , then by Fundamental Theorem of Calculus Part 1, $g'(x) = \sin(x)$ .

Fundamental Theorem of Calculus Part 2

Criticality: 3

States that if $F$ is an antiderivative of $f$ on $[a,b]$, then $\int_{a}^{b} f(x)dx = F(b) - F(a)$. This part is commonly used to evaluate definite integrals.

Example:

Using Fundamental Theorem of Calculus Part 2, we can evaluate $\int_{0}^{2} 3x^2 dx$ as $x^3 \Big|_0^2 = 2^3 - 0^3 = 8$ .

Integration

Criticality: 3

The process of finding the antiderivative of a function, often used to calculate areas, volumes, or total accumulation.

Example:

To find the area under the curve of $f(x) = x^2$ from $x=0$ to $x=3$ , we perform integration of $x^2$ over that interval.

Lower Bound

Criticality: 2

The bottom limit of integration in a definite integral, indicating the starting point of the interval over which the function is being integrated.

Example:

In the integral $\int_{1}^{5} x^2 dx$ , the number 1 is the lower bound.

Upper Bound

Criticality: 2

The top limit of integration in a definite integral, indicating the endpoint of the interval over which the function is being integrated.

Example:

In the integral $\int_{1}^{5} x^2 dx$ , the number 5 is the upper bound.

Glossary

Antiderivative

Criticality: 3

A function whose derivative is the original function. Finding an antiderivative is the reverse process of differentiation.

Example:

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

Definite Integral

Criticality: 3

An integral with upper and lower limits of integration, representing the net signed area under the curve of a function over a specific interval.

Example:

To find the total distance traveled by a car with velocity $v(t)$ from $t=0$ to $t=5$ seconds, we would calculate the definite integral $\int_{0}^{5} v(t) dt$ .

Differentiation

Criticality: 2

The process of finding the derivative of a function, which represents the instantaneous rate of change of the function.

Example:

The process of finding the slope of the tangent line to a curve at any point is called differentiation.

Fundamental Theorem of Calculus (FTOC)

Criticality: 3

Example:

The Fundamental Theorem of Calculus allows us to quickly evaluate $\int_{0}^{1} x^2 dx$ by finding the antiderivative and evaluating it at the bounds, rather than using Riemann sums.

Fundamental Theorem of Calculus Part 1

Criticality: 2

States that if $g(x) = \int_{a}^{x} f(t) dt$, then $g'(x) = f(x)$. It shows that differentiation and integration are inverse operations.

Example:

If $g(x) = \int_{1}^{x} \sin(t) dt$ , then by Fundamental Theorem of Calculus Part 1, $g'(x) = \sin(x)$ .

Fundamental Theorem of Calculus Part 2

Criticality: 3

States that if $F$ is an antiderivative of $f$ on $[a,b]$, then $\int_{a}^{b} f(x)dx = F(b) - F(a)$. This part is commonly used to evaluate definite integrals.

Example:

Using Fundamental Theorem of Calculus Part 2, we can evaluate $\int_{0}^{2} 3x^2 dx$ as $x^3 \Big|_0^2 = 2^3 - 0^3 = 8$ .

Integration

Criticality: 3

The process of finding the antiderivative of a function, often used to calculate areas, volumes, or total accumulation.

Example:

To find the area under the curve of $f(x) = x^2$ from $x=0$ to $x=3$ , we perform integration of $x^2$ over that interval.

Lower Bound

Criticality: 2

The bottom limit of integration in a definite integral, indicating the starting point of the interval over which the function is being integrated.

Example:

In the integral $\int_{1}^{5} x^2 dx$ , the number 1 is the lower bound.

Upper Bound

Criticality: 2

The top limit of integration in a definite integral, indicating the endpoint of the interval over which the function is being integrated.

Example:

In the integral $\int_{1}^{5} x^2 dx$ , the number 5 is the upper bound.