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What are the differences between local and global extrema?

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

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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).

What does $f'(x) = 0$ mean on the graph of $f(x)$ ?

It indicates a horizontal tangent line, which could be a local maximum, local minimum, or saddle point.

What does $f'(x) > 0$ mean on the graph of $f(x)$ ?

The function $f(x)$ is increasing.

What does $f'(x) < 0$ mean on the graph of $f(x)$ ?

The function $f(x)$ is decreasing.

What does $f''(x) = 0$ mean on the graph of $f(x)$ ?

It indicates a possible point of inflection where the concavity may change.

What does $f''(x) > 0$ mean on the graph of $f(x)$ ?

The function $f(x)$ is concave up.

What does $f''(x) < 0$ mean on the graph of $f(x)$ ?

The function $f(x)$ is concave down.

How to identify local maxima on a graph?

Look for points where the function changes from increasing to decreasing.

How to identify local minima on a graph?

Look for points where the function changes from decreasing to increasing.

What does a sharp corner on the graph of $f(x)$ indicate about $f'(x)$ ?

It indicates that $f'(x)$ is undefined at that point.

How does the steepness of $f(x)$ relate to $f'(x)$ ?

The steeper the graph of $f(x)$ , the larger the absolute value of $f'(x)$ .

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.