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  2. AP Calculus
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Glossary

A

Acceleration

Criticality: 3

Acceleration is the rate of change of an object's velocity with respect to time. It is the first derivative of the velocity function or the second derivative of the position function.

Example:

When a car speeds up or slows down, its acceleration is non-zero. If your car's velocity is increasing, you have positive acceleration; if it's decreasing, you have negative acceleration (often called deceleration).

D

Derivative

Criticality: 3

The derivative of a function, $f'(x)$, represents the instantaneous rate of change of the function $f(x)$ with respect to its independent variable $x$. Geometrically, it is the slope of the tangent line to the function's graph at a specific point.

Example:

If C(x)C(x)C(x) is the cost to produce xxx items, then C′(x)C'(x)C′(x) is the marginal cost, representing the approximate cost to produce one additional item. For instance, if C′(100)=5C'(100) = 5C′(100)=5, it means producing the 101st item will cost approximately 5,whichisthe[objectObject]ofthecostfunctionat5, which is the [object Object] of the cost function at5,whichisthe[objectObject]ofthecostfunctionatx=100$.

I

Instantaneous Rate of Change

Criticality: 3

This refers to how quickly a quantity is changing at a precise moment or specific value of the independent variable. It is precisely what the derivative of a function measures at a given point.

Example:

The speedometer in a car shows the instantaneous rate of change of distance with respect to time, which is the car's speed at that exact moment, not its average speed over a trip.

Interpreting the Meaning of the Derivative in Context

Criticality: 3

This involves explaining what the numerical value and sign of a derivative signify in a real-world scenario, paying close attention to the units of both the original function and its independent variable.

Example:

If T(h)T(h)T(h) is the temperature in degrees Celsius after hhh hours, then interpreting the meaning of the derivative in context for T′(3)=−0.5T'(3) = -0.5T′(3)=−0.5 means that at 3 hours after the start, the temperature is decreasing at a rate of 0.5 degrees Celsius per hour.

T

Tangent Line Approximation

Criticality: 2

A method that uses the equation of the tangent line to a function at a known point to estimate the function's value at a nearby point. It leverages the idea that the tangent line closely approximates the function's behavior near the point of tangency.

Example:

If you know the current position of a rocket and its instantaneous velocity, you can use tangent line approximation to estimate its height a few seconds later, assuming its velocity remains relatively constant over that short interval.

U

Units of the Derivative

Criticality: 2

The units of a derivative are always expressed as the units of the dependent variable divided by the units of the independent variable. This indicates the 'per' relationship of the rate of change.

Example:

If a function P(t)P(t)P(t) measures population in 'people' and time ttt in 'years', then the units of the derivative P′(t)P'(t)P′(t) would be 'people per year', indicating how fast the population is changing.

V

Velocity

Criticality: 3

In physics, velocity is the rate of change of an object's position with respect to time. It is the first derivative of the position function and includes both speed and direction.

Example:

If a particle's position is given by s(t)=t2−4ts(t) = t^2 - 4ts(t)=t2−4t, then its velocity at any time ttt is v(t)=s′(t)=2t−4v(t) = s'(t) = 2t - 4v(t)=s′(t)=2t−4. A positive velocity means it's moving in the positive direction, while a negative velocity means it's moving in the negative direction.