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).

Given the graph of $f'(x)$ , how do you find intervals where $f(x)$ is increasing?

Identify intervals where $f'(x) > 0$ (above the x-axis).

Given the graph of $f'(x)$ , how do you find relative maxima of $f(x)$ ?

Find points where $f'(x)$ changes from positive to negative.

Given the graph of $f'(x)$ , how do you find relative minima of $f(x)$ ?

Find points where $f'(x)$ changes from negative to positive.

Given the graph of $f''(x)$ , how do you find intervals where $f(x)$ is concave up?

Identify intervals where $f''(x) > 0$ (above the x-axis).

Given the graph of $f''(x)$ , how do you find intervals where $f(x)$ is concave down?

Identify intervals where $f''(x) < 0$ (below the x-axis).

Given the graph of $f''(x)$ , how do you find points of inflection of $f(x)$ ?

Find points where $f''(x)$ changes sign (crosses the x-axis).

Given the graph of f(x), how do you determine where f'(x) is positive?

Look for intervals where f(x) is increasing.

Given the graph of f(x), how do you determine where f''(x) is positive?

Look for intervals where f(x) is concave up.

Given the graph of f'(x), how to determine where f''(x) is positive?

Look for intervals where f'(x) is increasing.

Given the graph of f'(x), how to determine where f''(x) is negative?

Look for intervals where f'(x) is decreasing.

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).

Given the graph of $f'(x)$ , how do you find intervals where $f(x)$ is increasing?

Identify intervals where $f'(x) > 0$ (above the x-axis).

Given the graph of $f'(x)$ , how do you find relative maxima of $f(x)$ ?

Find points where $f'(x)$ changes from positive to negative.

Given the graph of $f'(x)$ , how do you find relative minima of $f(x)$ ?

Find points where $f'(x)$ changes from negative to positive.

Given the graph of $f''(x)$ , how do you find intervals where $f(x)$ is concave up?

Identify intervals where $f''(x) > 0$ (above the x-axis).

Given the graph of $f''(x)$ , how do you find intervals where $f(x)$ is concave down?

Identify intervals where $f''(x) < 0$ (below the x-axis).

Given the graph of $f''(x)$ , how do you find points of inflection of $f(x)$ ?

Find points where $f''(x)$ changes sign (crosses the x-axis).

Given the graph of f(x), how do you determine where f'(x) is positive?

Look for intervals where f(x) is increasing.

Given the graph of f(x), how do you determine where f''(x) is positive?

Look for intervals where f(x) is concave up.

Given the graph of f'(x), how to determine where f''(x) is positive?

Look for intervals where f'(x) is increasing.

Given the graph of f'(x), how to determine where f''(x) is negative?

Look for intervals where f'(x) is decreasing.