Formula for testing even function symmetry?

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

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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.

What are the differences between local and global extrema?

Local extrema: max/min in a neighborhood. Global extrema: max/min over the entire domain.

What are the differences between the First and Second Derivative Tests?

First Derivative Test: uses sign changes of $f'(x)$ to find extrema. Second Derivative Test: uses the sign of $f''(x)$ at critical points to find extrema.

What are the differences between even and odd functions?

Even functions: symmetric about the y-axis, $f(-x) = f(x)$ . Odd functions: symmetric about the origin, $f(-x) = -f(x)$ .

Compare increasing vs. concave up.

Increasing: $f'(x) > 0$ . Concave Up: $f''(x) > 0$ . A function can be increasing and concave down, or increasing and concave up.

Compare decreasing vs. concave down.

Decreasing: $f'(x) < 0$ . Concave Down: $f''(x) < 0$ . A function can be decreasing and concave up, or decreasing and concave down.

What is the difference between a critical point and a point of inflection?

Critical Point: $f'(x) = 0$ or undefined (potential max/min). Point of Inflection: $f''(x) = 0$ or undefined (concavity change).

What is the difference between a root and a y-intercept?

Root: where the function crosses the x-axis. Y-intercept: where the function crosses the y-axis.

Compare finding critical points of $f(x)$ vs. $f'(x)$ .

Critical points of $f(x)$ : $f'(x) = 0$ or undefined. Critical points of $f'(x)$ : $f''(x) = 0$ or undefined.

Compare even symmetry vs. odd symmetry.

Even symmetry: symmetric about the y-axis. Odd symmetry: symmetric about the origin.

What is the difference between finding where a function is zero versus undefined?

Zero: Find where the function equals zero. Undefined: Find where the function has a discontinuity (e.g., division by zero).

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.

All Flashcards

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.

What are the differences between local and global extrema?

Local extrema: max/min in a neighborhood. Global extrema: max/min over the entire domain.

What are the differences between the First and Second Derivative Tests?

First Derivative Test: uses sign changes of $f'(x)$ to find extrema. Second Derivative Test: uses the sign of $f''(x)$ at critical points to find extrema.

What are the differences between even and odd functions?

Even functions: symmetric about the y-axis, $f(-x) = f(x)$ . Odd functions: symmetric about the origin, $f(-x) = -f(x)$ .

Compare increasing vs. concave up.

Increasing: $f'(x) > 0$ . Concave Up: $f''(x) > 0$ . A function can be increasing and concave down, or increasing and concave up.

Compare decreasing vs. concave down.

Decreasing: $f'(x) < 0$ . Concave Down: $f''(x) < 0$ . A function can be decreasing and concave up, or decreasing and concave down.

What is the difference between a critical point and a point of inflection?

Critical Point: $f'(x) = 0$ or undefined (potential max/min). Point of Inflection: $f''(x) = 0$ or undefined (concavity change).

What is the difference between a root and a y-intercept?

Root: where the function crosses the x-axis. Y-intercept: where the function crosses the y-axis.

Compare finding critical points of $f(x)$ vs. $f'(x)$ .

Critical points of $f(x)$ : $f'(x) = 0$ or undefined. Critical points of $f'(x)$ : $f''(x) = 0$ or undefined.

Compare even symmetry vs. odd symmetry.

Even symmetry: symmetric about the y-axis. Odd symmetry: symmetric about the origin.

What is the difference between finding where a function is zero versus undefined?

Zero: Find where the function equals zero. Undefined: Find where the function has a discontinuity (e.g., division by zero).

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.