Explain the meaning of $f'(x) > 0$ .

$f(x)$ is increasing at $x$ .

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

Explain the meaning of $f'(x) > 0$ .

$f(x)$ is increasing at $x$ .

Explain the meaning of $f'(x) < 0$ .

$f(x)$ is decreasing at $x$ .

Explain the meaning of $f'(x) = 0$ .

$f(x)$ has a stationary point (local max, min, or inflection) at $x$ .

How does the sign of the derivative relate to the function's behavior?

Positive derivative: increasing function. Negative derivative: decreasing function. Zero derivative: stationary point.

What does $A'(5) = 12$ mean if $A(t)$ is the number of ants at time $t$ ?

At $t=5$ , the ant population is increasing at a rate of 12 ants per unit of time.

What does a negative derivative imply in a real-world context?

The quantity is decreasing or being depleted over time.

Explain the difference between $f(x)$ and $f'(x)$ .

$f(x)$ represents the value of the function at $x$ , while $f'(x)$ represents the rate of change of the function at $x$ .

How can derivatives be used to analyze real-world scenarios?

Derivatives can be used to determine the rate of change of quantities, predict future values, and optimize processes.

Define instantaneous rate of change.

The rate of change of a function at a specific point, represented by the derivative.

What does $f'(x)$ represent?

The instantaneous rate of change of $f(x)$ with respect to $x$ .

Define derivative in context.

The rate at which a quantity is changing with respect to another, within a real-world scenario.

What are the units of a derivative?

Units of the dependent variable divided by the units of the independent variable.

How do you interpret $f'(a) = b$ in context?

At $x=a$ , the function $f(x)$ is changing at a rate of $b$ units per unit of $x$ .

Steps to interpret derivative in context?

Identify the function and its variables. 2. Determine the units of the derivative. 3. Explain the meaning of the derivative at a specific point.

How do you check if your interpretation of a derivative is correct?

Ensure the units of the derivative match the context and make logical sense.

Given $V(r)$ is the volume of a sphere, interpret $V'(r)$ .

$V'(r)$ represents the rate of change of the volume of the sphere with respect to its radius.

All Flashcards

Explain the meaning of $f'(x) > 0$ .

$f(x)$ is increasing at $x$ .

Explain the meaning of $f'(x) < 0$ .

$f(x)$ is decreasing at $x$ .

Explain the meaning of $f'(x) = 0$ .

$f(x)$ has a stationary point (local max, min, or inflection) at $x$ .

How does the sign of the derivative relate to the function's behavior?

Positive derivative: increasing function. Negative derivative: decreasing function. Zero derivative: stationary point.

What does $A'(5) = 12$ mean if $A(t)$ is the number of ants at time $t$ ?

At $t=5$ , the ant population is increasing at a rate of 12 ants per unit of time.

What does a negative derivative imply in a real-world context?

The quantity is decreasing or being depleted over time.

Explain the difference between $f(x)$ and $f'(x)$ .

$f(x)$ represents the value of the function at $x$ , while $f'(x)$ represents the rate of change of the function at $x$ .

How can derivatives be used to analyze real-world scenarios?

Derivatives can be used to determine the rate of change of quantities, predict future values, and optimize processes.

Define instantaneous rate of change.

The rate of change of a function at a specific point, represented by the derivative.

What does $f'(x)$ represent?

The instantaneous rate of change of $f(x)$ with respect to $x$ .

Define derivative in context.

The rate at which a quantity is changing with respect to another, within a real-world scenario.

What are the units of a derivative?

Units of the dependent variable divided by the units of the independent variable.

How do you interpret $f'(a) = b$ in context?

At $x=a$ , the function $f(x)$ is changing at a rate of $b$ units per unit of $x$ .

Steps to interpret derivative in context?

Identify the function and its variables. 2. Determine the units of the derivative. 3. Explain the meaning of the derivative at a specific point.

How do you check if your interpretation of a derivative is correct?

Ensure the units of the derivative match the context and make logical sense.

Given $V(r)$ is the volume of a sphere, interpret $V'(r)$ .

$V'(r)$ represents the rate of change of the volume of the sphere with respect to its radius.