Glossary

Absolute Value

Criticality: 3

The non-negative value of a number, representing its distance from zero, regardless of its sign. It is crucial for calculating total distance from velocity.

Example:

When calculating the total distance traveled, we use the absolute value of velocity to ensure that movement in any direction contributes positively to the total path length.

Acceleration

Criticality: 2

The rate of change of velocity over time. It is a vector quantity.

Example:

When a car presses the gas pedal, its acceleration increases its velocity; when it brakes, it experiences negative acceleration.

Acceleration

Criticality: 3

Acceleration is the rate of change of velocity with respect to time, indicating how quickly an object's velocity is changing.

Example:

A car pressing the gas pedal experiences positive acceleration, causing its speed to increase.

Accumulation of change

Criticality: 2

This refers to the total change in a quantity over an interval, often represented by the definite integral of its rate of change.

Example:

The total amount of water that has flowed into a tank over a certain period, given its inflow rate, represents the accumulation of change in the water volume.

Area Under Graph

Criticality: 2

The geometric interpretation of a definite integral, where the area between a function's curve and the x-axis over an interval represents the accumulated quantity over that interval.

Example:

The area under a velocity-time graph between two time points gives the total change in displacement during that interval.

Average value

Criticality: 3

The average value of a function f over an interval [a, b] provides a single number that represents the average output of the function over that interval.

Example:

If a car's speed varies over an hour, its average value of speed might be 60 km/h, even if it sped up and slowed down during that time.

Constant function

Criticality: 1

A function whose output value remains the same regardless of the input value, resulting in a horizontal line when graphed.

Example:

The temperature inside a perfectly insulated thermos after reaching equilibrium can be described by a constant function, as it doesn't change over time.

Constant of Integration

Criticality: 3

An arbitrary constant, denoted as $+C$, that arises when computing an indefinite integral, representing the family of functions whose derivative is the integrand. Its specific value is determined by initial conditions.

Example:

When integrating $3x^2$ , the result is $x^3 + C$ ; the constant of integration $C$ accounts for any initial value or vertical shift of the function.

Continuous function

Criticality: 2

A function is continuous if its graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes over its domain or a specified interval.

Example:

The path of a smoothly flying drone can be modeled by a continuous function, as its position changes without sudden, instantaneous jumps.

Definite Integral

Criticality: 3

An integral evaluated over a specific interval, representing the total accumulation of a quantity or the signed area under a curve between two specified limits.

Example:

The definite integral $\int_1^3 (2x) dx$ calculates the exact area under the line $y=2x$ from $x=1$ to $x=3$ .

Definite Integral

Criticality: 3

A mathematical concept that represents the accumulation of a rate of change over a specified interval, used to calculate the net change of a quantity.

Example:

To determine the total volume of water that flowed into a reservoir between 8 AM and 12 PM, given the flow rate, you would compute the definite integral of the flow rate function over that time period.

Displacement

Criticality: 3

A vector quantity that measures the change in position of an object from its starting point to its ending point. It has both magnitude and direction.

Example:

After moving 5 meters forward and 3 meters backward, the robot's displacement from its starting point is 2 meters forward.

Displacement (Position)

Criticality: 3

Displacement is the overall change in position of an object from its starting point, and it can be found by integrating velocity with respect to time.

Example:

If you walk 10 meters forward and then 10 meters backward, your total distance traveled is 20 meters, but your displacement is 0 meters.

Displacement Expression over Time

Criticality: 2

A function that describes an object's position at any given time, derived by integrating the velocity function and using initial conditions to determine the constant of integration.

Example:

If $v(t) = 2t - 1$ , the displacement expression over time might be $s(t) = t^2 - t + C$ , where $C$ is found from an initial position.

Displacement at a Particular Point

Criticality: 3

This refers to the instantaneous position of an object at a specific moment in time, found by adding the total change in displacement from a known point to the initial displacement at that known point.

Example:

If a ball starts at a height of 5 meters and its vertical velocity is known, its displacement at a particular point (e.g., at $t=2s$ ) can be found by integrating the velocity and adding the initial height.

Distance

Criticality: 3

The total length of the path traveled by an object, regardless of direction. It is a scalar quantity and always positive.

Example:

If a robot moves 5 meters forward and then 3 meters backward, the total distance it traveled is 8 meters.

Dummy Variable

Criticality: 1

A variable used within an integral that does not affect the final result of the integration, often employed to avoid confusion when the upper limit of integration is also a variable.

Example:

In the integral $\int_{a}^{x} f(t) , dt$ , $t$ is a dummy variable; the result will be a function of $x$ .

Indefinite Integral

Criticality: 2

An integral without specified limits, representing a family of functions whose derivative is the integrand, always including a constant of integration.

Example:

The indefinite integral of $\cos(x)$ is $\sin(x) + C$ , representing all functions whose derivative is $\cos(x)$ .

Indefinite Integral Approach

Criticality: 2

A method to find a general expression for velocity or displacement by computing the indefinite integral of acceleration or velocity, respectively, and then solving for the constant of integration using given initial conditions.

Example:

Using the indefinite integral approach, you integrate $v(t)$ to get $s(t) + C$ , then use a known position $s(t_0)$ to find the specific value of $C$ .

Integral

Criticality: 3

A mathematical operation that finds the area under a curve, often used to compute accumulated quantities such as total distance or displacement from a rate function.

Example:

To find the total volume of water collected in a tank over time, given the flow rate, you would calculate the integral of the flow rate function.

Interval

Criticality: 2

A set of real numbers between two specified endpoints, which can be inclusive or exclusive of the endpoints.

Example:

When calculating the average temperature for a day, you'd consider the interval from midnight to midnight, or [0, 24] hours.

Marginal Profit/Revenue/Cost

Criticality: 2

The rate of change of profit, revenue, or cost with respect to the number of units sold, indicating the additional profit, revenue, or cost incurred by producing or selling one more unit.

Example:

If a company's marginal revenue for selling smartphones is $500, it means selling one additional smartphone will increase total revenue by approximately$ 500.

Mean Value Theorem for Integrals

Criticality: 3

This theorem states that for a continuous function over an interval, there exists a point within that interval where the function's value equals its average value over the interval. Geometrically, it means a rectangle with the average height over the interval has the same area as the area under the curve.

Example:

If a company's profit rate is given by a function, the Mean Value Theorem for Integrals guarantees there was a specific moment when the instantaneous profit rate was exactly equal to the average profit rate over a quarter.

Net Change

Criticality: 3

The total change in a quantity over a specific interval, obtained by integrating its rate of change over that interval.

Example:

If you know the rate at which a population of bacteria is growing, you can calculate the net change in the bacterial population from the start to the end of an experiment by integrating the growth rate.

Rate of Change

Criticality: 2

How one quantity changes in relation to another, often expressed as a derivative. In kinematics, velocity is the rate of change of displacement, and acceleration is the rate of change of velocity.

Example:

The speedometer in a car measures the rate of change of its position, which is its speed.

Rate of Change

Criticality: 3

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

Example:

The acceleration of a rocket is the rate of change of its velocity, indicating how quickly its speed and direction are altering.

Scalar Quantity

Criticality: 2

A physical quantity that has only magnitude (size) and no direction.

Example:

Your mass of 70 kg is a scalar quantity because it only has a value, not a direction.

Speed

Criticality: 3

The magnitude of velocity, indicating how fast an object is moving without regard to direction. It is a scalar quantity and always positive.

Example:

A car traveling at 60 km/h has a speed of 60 km/h, whether it's moving north or south.

Total Change in Displacement

Criticality: 3

The net change in an object's position over a specific time interval, calculated by the definite integral of velocity over that interval.

Example:

Integrating a runner's velocity from $t=0$ to $t=30$ minutes gives the total change in displacement from their starting point during that half-hour.

Total Change in Velocity

Criticality: 3

The total change in velocity represents the net change in an object's velocity over a specific time interval, calculated by the definite integral of acceleration over that interval.

Example:

To find how much a car's speed increased between $t=5$ and $t=10$ seconds, you would calculate the total change in velocity by integrating its acceleration over that period.

Vector Quantity

Criticality: 2

A physical quantity that has both magnitude (size) and direction.

Example:

The wind blowing at 20 km/h from the north is a vector quantity because it has both a speed and a specific direction.

Velocity

Criticality: 3

A vector quantity that describes the rate of change of an object's position, including both its speed and direction.

Example:

A bird flying east at 15 m/s has a velocity of 15 m/s east, while a bird flying west at the same rate has a velocity of 15 m/s west.

Velocity (as an integral)

Criticality: 3

Velocity is the rate of change of displacement with respect to time, and it can be found by integrating acceleration with respect to time.

Example:

If a rocket's acceleration is given by $a(t) = 2t$ , integrating it yields its velocity function, $v(t) = t^2 + C$ .

Velocity Expression over Time

Criticality: 2

A function that describes an object's velocity at any given time, derived by integrating the acceleration function and using initial conditions to determine the constant of integration.

Example:

If $a(t) = 6t$ , the velocity expression over time might be $v(t) = 3t^2 + C$ , where $C$ is found from an initial velocity.

Velocity at a Particular Point

Criticality: 3

This refers to the instantaneous velocity of an object at a specific moment in time, found by adding the total change in velocity from a known point to the initial velocity at that known point.

Example:

Given a particle's initial velocity and its acceleration function, you can find its velocity at a particular point like $t=7$ seconds by integrating the acceleration and adding the initial velocity.