Glossary

Accumulation of change

Criticality: 3

The total sum of how much a quantity has changed over a specified period of time, often represented by the area under a rate-of-change curve.

Example:

Calculating the total accumulation of change in a bank account balance after a year of deposits and withdrawals.

Anti-derivatives

Criticality: 3

The reverse process of differentiation; a function whose derivative is the original function.

Example:

If the derivative of $F(x)$ is $f(x)$ , then $F(x)$ is an anti-derivative of $f(x)$ . For instance, $x^3$ is an anti-derivative of $3x^2$ .

Approximating (area under the curve)

Criticality: 3

The process of estimating the value of the area under a curve, often using methods like Riemann sums, when an exact calculation is difficult or unnecessary.

Example:

When you use a trapezoidal rule to estimate the distance traveled from a complex velocity graph, you are approximating the area under the curve.

Area under the curve

Criticality: 3

The region bounded by a function's graph, the x-axis, and vertical lines at specified intervals, representing the total accumulation of the quantity.

Example:

The area under the curve of a velocity-time graph gives the total distance traveled.

Definite integral

Criticality: 3

An integral evaluated over a specific interval, yielding a numerical value that represents the net accumulation or signed area under the curve between the given limits.

Example:

The definite integral $\int_0^5 v(t) dt$ would give the total displacement of an object from time $t=0$ to $t=5$ .

Independent variable

Criticality: 2

The variable in a function whose value is freely chosen or manipulated, and which determines the value of the dependent variable.

Example:

In the function $y = f(x)$ , x is the independent variable that we can choose, and it affects the value of y.

Integral

Criticality: 3

A mathematical operation that calculates the total accumulation of a quantity, representing the exact area under a curve.

Example:

To find the total volume of water that has flowed into a tank given its flow rate function, you would compute the integral of the flow rate.

Rate of change

Criticality: 3

A measure of how one quantity changes in relation to another quantity, typically expressed as a derivative or a ratio.

Example:

The rate of change of a car's position is its velocity, measured in miles per hour.

Riemann Sum

Criticality: 3

A method for approximating the area under a curve by dividing the region into a series of simple shapes, typically rectangles, and summing their areas.

Example:

To estimate the total water flow into a reservoir over a day, you could use a Riemann Sum by measuring the flow rate every hour and multiplying by the time interval.

Signed (area)

Criticality: 2

Area that accounts for direction, where area above the x-axis is positive and area below the x-axis is negative, reflecting net change.

Example:

If a car drives forward for a while and then backward, the signed area under its velocity-time graph would represent its net displacement, not total distance.

Glossary

Accumulation of change

Criticality: 3

The total sum of how much a quantity has changed over a specified period of time, often represented by the area under a rate-of-change curve.

Example:

Calculating the total accumulation of change in a bank account balance after a year of deposits and withdrawals.

Anti-derivatives

Criticality: 3

The reverse process of differentiation; a function whose derivative is the original function.

Example:

If the derivative of $F(x)$ is $f(x)$ , then $F(x)$ is an anti-derivative of $f(x)$ . For instance, $x^3$ is an anti-derivative of $3x^2$ .

Approximating (area under the curve)

Criticality: 3

The process of estimating the value of the area under a curve, often using methods like Riemann sums, when an exact calculation is difficult or unnecessary.

Example:

When you use a trapezoidal rule to estimate the distance traveled from a complex velocity graph, you are approximating the area under the curve.

Area under the curve

Criticality: 3

The region bounded by a function's graph, the x-axis, and vertical lines at specified intervals, representing the total accumulation of the quantity.

Example:

The area under the curve of a velocity-time graph gives the total distance traveled.

Definite integral

Criticality: 3

An integral evaluated over a specific interval, yielding a numerical value that represents the net accumulation or signed area under the curve between the given limits.

Example:

The definite integral $\int_0^5 v(t) dt$ would give the total displacement of an object from time $t=0$ to $t=5$ .

Independent variable

Criticality: 2

The variable in a function whose value is freely chosen or manipulated, and which determines the value of the dependent variable.

Example:

In the function $y = f(x)$ , x is the independent variable that we can choose, and it affects the value of y.

Integral

Criticality: 3

A mathematical operation that calculates the total accumulation of a quantity, representing the exact area under a curve.

Example:

To find the total volume of water that has flowed into a tank given its flow rate function, you would compute the integral of the flow rate.

Rate of change

Criticality: 3

A measure of how one quantity changes in relation to another quantity, typically expressed as a derivative or a ratio.

Example:

The rate of change of a car's position is its velocity, measured in miles per hour.

Riemann Sum

Criticality: 3

A method for approximating the area under a curve by dividing the region into a series of simple shapes, typically rectangles, and summing their areas.

Example:

To estimate the total water flow into a reservoir over a day, you could use a Riemann Sum by measuring the flow rate every hour and multiplying by the time interval.

Signed (area)

Criticality: 2

Area that accounts for direction, where area above the x-axis is positive and area below the x-axis is negative, reflecting net change.

Example:

If a car drives forward for a while and then backward, the signed area under its velocity-time graph would represent its net displacement, not total distance.