How is a function's increasing/decreasing behavior related to its first derivative?

If $f'(x) > 0$ , $f(x)$ is increasing. If $f'(x) < 0$ , $f(x)$ is decreasing.

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How is a function's increasing/decreasing behavior related to its first derivative?

If $f'(x) > 0$ , $f(x)$ is increasing. If $f'(x) < 0$ , $f(x)$ is decreasing.

How is a function's concavity related to its second derivative?

If $f''(x) > 0$ , $f(x)$ is concave up. If $f''(x) < 0$ , $f(x)$ is concave down.

What does $f'(x) = 0$ indicate?

A potential relative maximum or minimum of $f(x)$ .

What does $f''(x) = 0$ indicate?

A potential point of inflection of $f(x)$ .

How does the first derivative test work?

Examines the sign change of $f'(x)$ around a critical point to determine if it's a max or min.

How does the second derivative test work?

Uses the sign of $f''(x)$ at a critical point to determine concavity and thus if it's a max or min.

What is the relationship between the extrema of f(x) and f'(x)?

All relative extrema of $f(x)$ are x-intercepts of $f'(x)$ .

What is the relationship between the points of inflection of f(x) and f'(x)?

All points of inflection of $f(x)$ are relative extrema of $f'(x)$ .

What does it mean if f'(x) is increasing?

The function f(x) is concave up and f''(x) > 0.

What does it mean if f'(x) is decreasing?

The function f(x) is concave down and f''(x) < 0.

If $f'(x)$ is positive, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is increasing.

If $f'(x)$ is negative, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is decreasing.

If $f''(x)$ is positive, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is concave up.

If $f''(x)$ is negative, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is concave down.

What does an x-intercept on the graph of $f'(x)$ represent on the graph of $f(x)$ ?

A potential relative maximum or minimum on the graph of $f(x)$ .

What does a relative maximum on the graph of $f'(x)$ represent on the graph of $f(x)$ ?

A point of inflection where the concavity of $f(x)$ changes from up to down.

What does a relative minimum on the graph of $f'(x)$ represent on the graph of $f(x)$ ?

A point of inflection where the concavity of $f(x)$ changes from down to up.

If $f'(c) = 0$ and $f''(c) > 0$ , what does this mean for the graph of $f(x)$ at $x=c$ ?

There is a relative minimum at $x=c$ .

If $f'(c) = 0$ and $f''(c) < 0$ , what does this mean for the graph of $f(x)$ at $x=c$ ?

There is a relative maximum at $x=c$ .

How can you identify a point of inflection on the graph of f''(x)?

Look for points where f''(x) changes sign (crosses the x-axis).

Define relative minimum.

A point where a function changes from decreasing to increasing.

Define relative maximum.

A point where a function changes from increasing to decreasing.

Define point of inflection.

A point where the concavity of a function changes.

Define concavity.

The direction of the curve of a function (upward or downward).

What does it mean for a function to be increasing?

The function's value is getting larger as x increases; $f'(x) > 0$ .

What does it mean for a function to be decreasing?

The function's value is getting smaller as x increases; $f'(x) < 0$ .

Define the first derivative.

The rate of change of a function with respect to its variable.

Define the second derivative.

The rate of change of the first derivative; indicates concavity.

What is an x-intercept?

The point where a graph crosses the x-axis ( $y=0$ ).

Define extrema

The maximum and minimum values of a function.

All Flashcards

How is a function's increasing/decreasing behavior related to its first derivative?

If $f'(x) > 0$ , $f(x)$ is increasing. If $f'(x) < 0$ , $f(x)$ is decreasing.

How is a function's concavity related to its second derivative?

If $f''(x) > 0$ , $f(x)$ is concave up. If $f''(x) < 0$ , $f(x)$ is concave down.

What does $f'(x) = 0$ indicate?

A potential relative maximum or minimum of $f(x)$ .

What does $f''(x) = 0$ indicate?

A potential point of inflection of $f(x)$ .

How does the first derivative test work?

Examines the sign change of $f'(x)$ around a critical point to determine if it's a max or min.

How does the second derivative test work?

Uses the sign of $f''(x)$ at a critical point to determine concavity and thus if it's a max or min.

What is the relationship between the extrema of f(x) and f'(x)?

All relative extrema of $f(x)$ are x-intercepts of $f'(x)$ .

What is the relationship between the points of inflection of f(x) and f'(x)?

All points of inflection of $f(x)$ are relative extrema of $f'(x)$ .

What does it mean if f'(x) is increasing?

The function f(x) is concave up and f''(x) > 0.

What does it mean if f'(x) is decreasing?

The function f(x) is concave down and f''(x) < 0.

If $f'(x)$ is positive, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is increasing.

If $f'(x)$ is negative, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is decreasing.

If $f''(x)$ is positive, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is concave up.

If $f''(x)$ is negative, what does this mean for the graph of $f(x)$ ?

The graph of $f(x)$ is concave down.

What does an x-intercept on the graph of $f'(x)$ represent on the graph of $f(x)$ ?

A potential relative maximum or minimum on the graph of $f(x)$ .

What does a relative maximum on the graph of $f'(x)$ represent on the graph of $f(x)$ ?

A point of inflection where the concavity of $f(x)$ changes from up to down.

What does a relative minimum on the graph of $f'(x)$ represent on the graph of $f(x)$ ?

A point of inflection where the concavity of $f(x)$ changes from down to up.

If $f'(c) = 0$ and $f''(c) > 0$ , what does this mean for the graph of $f(x)$ at $x=c$ ?

There is a relative minimum at $x=c$ .

If $f'(c) = 0$ and $f''(c) < 0$ , what does this mean for the graph of $f(x)$ at $x=c$ ?

There is a relative maximum at $x=c$ .

How can you identify a point of inflection on the graph of f''(x)?

Look for points where f''(x) changes sign (crosses the x-axis).

Define relative minimum.

A point where a function changes from decreasing to increasing.

Define relative maximum.

A point where a function changes from increasing to decreasing.

Define point of inflection.

A point where the concavity of a function changes.

Define concavity.

The direction of the curve of a function (upward or downward).

What does it mean for a function to be increasing?

The function's value is getting larger as x increases; $f'(x) > 0$ .

What does it mean for a function to be decreasing?

The function's value is getting smaller as x increases; $f'(x) < 0$ .

Define the first derivative.

The rate of change of a function with respect to its variable.

Define the second derivative.

The rate of change of the first derivative; indicates concavity.

What is an x-intercept?

The point where a graph crosses the x-axis ( $y=0$ ).

Define extrema

The maximum and minimum values of a function.