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Glossary

D

Derivative

Criticality: 3

The instantaneous rate of change of a function with respect to its independent variable, representing the slope of the tangent line to the function's graph at any given point.

Example:

The derivative of f(x)=x2f(x) = x^2 is f′(x)=2xf'(x) = 2x, which tells us the slope of the parabola at any x-value.

F

Free Response Question

Criticality: 3

A type of exam question where students must show their work and provide a detailed solution, often involving multiple steps and concepts.

Example:

A Free Response Question might require you to find a derivative, then use it to determine the equation of a tangent line, and finally find critical points.

M

Mnemonic (Product Rule)

Criticality: 2

A memory aid for the Product Rule, often stated as 'First d second plus second d first,' which helps recall the order of operations in the formula.

Example:

Using the Mnemonic (Product Rule), 'First d second plus second d first,' helps me remember that for f(x)g(x)f(x)g(x), the derivative is f(x)g′(x)+g(x)f′(x)f(x)g'(x) + g(x)f'(x).

Multiple Choice Question

Criticality: 2

A type of exam question where a student selects the correct answer from a list of provided options.

Example:

On the AP exam, a Multiple Choice Question might ask you to identify the correct derivative of a given function from four choices.

P

Product Rule

Criticality: 3

A rule used to find the derivative of a function that is the product of two differentiable functions, given by the formula: (first function) * (derivative of the second) + (second function) * (derivative of the first).

Example:

To find the derivative of y=x3cos⁡(x)y = x^3 \cos(x), you would apply the Product Rule to get y′=x3(−sin⁡(x))+cos⁡(x)(3x2)y' = x^3(-\sin(x)) + \cos(x)(3x^2).

S

Simplifying Answers

Criticality: 1

The process of algebraically manipulating a derivative expression to its most reduced or compact form, though often not required on the AP exam unless specified.

Example:

After applying the product rule to y=exsin⁡(x)y = e^x \sin(x) to get y′=excos⁡(x)+exsin⁡(x)y' = e^x\cos(x) + e^x\sin(x), you could consider Simplifying Answers by factoring out exe^x to get y′=ex(cos⁡(x)+sin⁡(x))y' = e^x(\cos(x) + \sin(x)).

T

Tangent Line

Criticality: 2

A straight line that touches a curve at a single point and has the same slope as the curve at that point.

Example:

To find the equation of the Tangent Line to f(x)=x2f(x) = x^2 at x=2x=2, you first find f′(2)f'(2) for the slope and f(2)f(2) for the y-coordinate.