Explain the relationship between the first derivative and increasing/decreasing intervals.

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

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Explain the relationship between the first derivative and increasing/decreasing intervals.

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

Explain the First Derivative Test.

If $f'(x)$ changes from positive to negative at $x=c$ , then $f(x)$ has a local maximum at $x=c$ . If $f'(x)$ changes from negative to positive at $x=c$ , then $f(x)$ has a local minimum at $x=c$ .

Explain the relationship between the second derivative and concavity.

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

Explain how to find points of inflection.

Find where $f''(x) = 0$ or is undefined, and check if the concavity changes at those points.

How do you determine symmetry about the y-axis?

Check if $f(-x) = f(x)$ . If true, the function is even and symmetric about the y-axis.

How do you determine symmetry about the origin?

Check if $f(-x) = -f(x)$ . If true, the function is odd and symmetric about the origin.

What does a discontinuity in a function indicate?

A point where the function is not continuous, possibly indicating a hole, jump, or vertical asymptote.

What is the significance of critical points?

Critical points are potential locations of local maxima, local minima, or saddle points of a function.

Explain the Second Derivative Test.

Uses the second derivative to determine if a critical point is a local max or min. If $f''(c) > 0$ , local minimum. If $f''(c) < 0$ , local maximum.

What is the domain of a polynomial function?

All real numbers, unless otherwise restricted.

How to find intervals of increasing/decreasing?

Find $f'(x)$ . 2. Find critical points. 3. Test intervals between critical points in $f'(x)$ . 4. Determine if $f'(x)$ is positive (increasing) or negative (decreasing).

How to find local extrema using the First Derivative Test?

Find critical points. 2. Determine intervals of increasing/decreasing. 3. If $f'(x)$ changes from + to -, local max. If $f'(x)$ changes from - to +, local min.

How to find points of inflection?

Find $f''(x)$ . 2. Find possible points of inflection where $f''(x) = 0$ or is undefined. 3. Check if concavity changes at these points.

How to determine the concavity of a function?

Find $f''(x)$ . 2. Find where $f''(x) = 0$ or is undefined. 3. Test intervals to determine if $f''(x)$ is positive (concave up) or negative (concave down).

How to sketch a graph of a function?

Find the domain and discontinuities. 2. Find intercepts and symmetry. 3. Find critical points. 4. Determine increasing/decreasing intervals. 5. Find extrema. 6. Determine concavity and points of inflection. 7. Sketch the graph.

How to use the Second Derivative Test to find extrema?

Find critical points. 2. Find $f''(x)$ . 3. Evaluate $f''(x)$ at each critical point. 4. If $f''(c) > 0$ , local min. If $f''(c) < 0$ , local max.

How to determine if a function is even or odd?

Find $f(-x)$ . 2. If $f(-x) = f(x)$ , the function is even. 3. If $f(-x) = -f(x)$ , the function is odd.

How to find the domain of a rational function?

Identify values of $x$ that make the denominator zero. 2. Exclude those values from the set of all real numbers.

How to find the domain of a polynomial function?

The domain is all real numbers unless there are specific restrictions given.

How to find the x-intercepts of a function?

Set $f(x) = 0$ . 2. Solve for $x$ . 3. The solutions are the x-intercepts.

Formula for testing even function symmetry?

$f(-x) = f(x)$

Formula for testing odd function symmetry?

$f(-x) = -f(x)$

Second Derivative Test Formula

If $f''(c) > 0$ , local minimum at $x=c$ . If $f''(c) < 0$ , local maximum at $x=c$ .

How to find x-intercepts?

Set $f(x) = 0$ and solve for $x$ .

How to find y-intercepts?

Set $x = 0$ and solve for $f(0)$ .

What is the condition when $f$ is increasing?

$f'(x) > 0$

What is the condition when $f$ is decreasing?

$f'(x) < 0$

Condition for concave up?

$f''(x) > 0$

Condition for concave down?

$f''(x) < 0$

How do you find critical points?

Solve $f'(x) = 0$ or where $f'(x)$ is undefined.

All Flashcards

Explain the relationship between the first derivative and increasing/decreasing intervals.

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

Explain the First Derivative Test.

If $f'(x)$ changes from positive to negative at $x=c$ , then $f(x)$ has a local maximum at $x=c$ . If $f'(x)$ changes from negative to positive at $x=c$ , then $f(x)$ has a local minimum at $x=c$ .

Explain the relationship between the second derivative and concavity.

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

Explain how to find points of inflection.

Find where $f''(x) = 0$ or is undefined, and check if the concavity changes at those points.

How do you determine symmetry about the y-axis?

Check if $f(-x) = f(x)$ . If true, the function is even and symmetric about the y-axis.

How do you determine symmetry about the origin?

Check if $f(-x) = -f(x)$ . If true, the function is odd and symmetric about the origin.

What does a discontinuity in a function indicate?

A point where the function is not continuous, possibly indicating a hole, jump, or vertical asymptote.

What is the significance of critical points?

Critical points are potential locations of local maxima, local minima, or saddle points of a function.

Explain the Second Derivative Test.

Uses the second derivative to determine if a critical point is a local max or min. If $f''(c) > 0$ , local minimum. If $f''(c) < 0$ , local maximum.

What is the domain of a polynomial function?

All real numbers, unless otherwise restricted.

How to find intervals of increasing/decreasing?

Find $f'(x)$ . 2. Find critical points. 3. Test intervals between critical points in $f'(x)$ . 4. Determine if $f'(x)$ is positive (increasing) or negative (decreasing).

How to find local extrema using the First Derivative Test?

Find critical points. 2. Determine intervals of increasing/decreasing. 3. If $f'(x)$ changes from + to -, local max. If $f'(x)$ changes from - to +, local min.

How to find points of inflection?

Find $f''(x)$ . 2. Find possible points of inflection where $f''(x) = 0$ or is undefined. 3. Check if concavity changes at these points.

How to determine the concavity of a function?

Find $f''(x)$ . 2. Find where $f''(x) = 0$ or is undefined. 3. Test intervals to determine if $f''(x)$ is positive (concave up) or negative (concave down).

How to sketch a graph of a function?

Find the domain and discontinuities. 2. Find intercepts and symmetry. 3. Find critical points. 4. Determine increasing/decreasing intervals. 5. Find extrema. 6. Determine concavity and points of inflection. 7. Sketch the graph.

How to use the Second Derivative Test to find extrema?

Find critical points. 2. Find $f''(x)$ . 3. Evaluate $f''(x)$ at each critical point. 4. If $f''(c) > 0$ , local min. If $f''(c) < 0$ , local max.

How to determine if a function is even or odd?

Find $f(-x)$ . 2. If $f(-x) = f(x)$ , the function is even. 3. If $f(-x) = -f(x)$ , the function is odd.

How to find the domain of a rational function?

Identify values of $x$ that make the denominator zero. 2. Exclude those values from the set of all real numbers.

How to find the domain of a polynomial function?

The domain is all real numbers unless there are specific restrictions given.

How to find the x-intercepts of a function?

Set $f(x) = 0$ . 2. Solve for $x$ . 3. The solutions are the x-intercepts.

Formula for testing even function symmetry?

$f(-x) = f(x)$

Formula for testing odd function symmetry?

$f(-x) = -f(x)$

Second Derivative Test Formula

If $f''(c) > 0$ , local minimum at $x=c$ . If $f''(c) < 0$ , local maximum at $x=c$ .

How to find x-intercepts?

Set $f(x) = 0$ and solve for $x$ .

How to find y-intercepts?

Set $x = 0$ and solve for $f(0)$ .

What is the condition when $f$ is increasing?

$f'(x) > 0$

What is the condition when $f$ is decreasing?

$f'(x) < 0$

Condition for concave up?

$f''(x) > 0$

Condition for concave down?

$f''(x) < 0$

How do you find critical points?

Solve $f'(x) = 0$ or where $f'(x)$ is undefined.