Glossary

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 $f(x) = x^2$ is $f'(x) = 2x$ , which tells us the slope of the parabola at any x-value.

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.

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 $f(x)g(x)$ , the derivative is $f(x)g'(x) + g(x)f'(x)$ .

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.

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 $y = x^3 \cos(x)$ , you would apply the Product Rule to get $y' = x^3(-\sin(x)) + \cos(x)(3x^2)$ .

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 $y = e^x \sin(x)$ to get $y' = e^x\cos(x) + e^x\sin(x)$ , you could consider Simplifying Answers by factoring out $e^x$ to get $y' = e^x(\cos(x) + \sin(x))$ .

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 $f(x) = x^2$ at $x=2$ , you first find $f'(2)$ for the slope and $f(2)$ for the y-coordinate.

Glossary

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 $f(x) = x^2$ is $f'(x) = 2x$ , which tells us the slope of the parabola at any x-value.

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.

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 $f(x)g(x)$ , the derivative is $f(x)g'(x) + g(x)f'(x)$ .

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.

Product Rule

Criticality: 3

Example:

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

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 $y = e^x \sin(x)$ to get $y' = e^x\cos(x) + e^x\sin(x)$ , you could consider Simplifying Answers by factoring out $e^x$ to get $y' = e^x(\cos(x) + \sin(x))$ .

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 $f(x) = x^2$ at $x=2$ , you first find $f'(2)$ for the slope and $f(2)$ for the y-coordinate.