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Define critical point.

A point where the function's first derivative is zero or undefined.

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Define critical point.

A point where the function's first derivative is zero or undefined.

What is a point of inflection?

A point where the concavity of a function changes.

Define local maximum.

A point where the function's value is greater than or equal to the values at all nearby points.

What is a local minimum?

A point where the function's value is less than or equal to the values at all nearby points.

Define concavity.

The direction in which a curve bends; either concave up or concave down.

What does symmetry mean for a function?

A function is symmetric if it looks the same when reflected across a line or point.

Define even function.

A function where $f(-x) = f(x)$ for all $x$ in the domain.

Define odd function.

A function where $f(-x) = -f(x)$ for all $x$ in the domain.

Define x-intercept.

The point(s) where the graph of a function intersects the x-axis.

Define y-intercept.

The point(s) where the graph of a function intersects the y-axis.

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.

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

Define critical point.

A point where the function's first derivative is zero or undefined.

What is a point of inflection?

A point where the concavity of a function changes.

Define local maximum.

A point where the function's value is greater than or equal to the values at all nearby points.

What is a local minimum?

A point where the function's value is less than or equal to the values at all nearby points.

Define concavity.

The direction in which a curve bends; either concave up or concave down.

What does symmetry mean for a function?

A function is symmetric if it looks the same when reflected across a line or point.

Define even function.

A function where $f(-x) = f(x)$ for all $x$ in the domain.

Define odd function.

A function where $f(-x) = -f(x)$ for all $x$ in the domain.

Define x-intercept.

The point(s) where the graph of a function intersects the x-axis.

Define y-intercept.

The point(s) where the graph of a function intersects the y-axis.

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.

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.