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.
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.
Define relative minimum.
A point where a function changes from decreasing to increasing.
Define relative maximum.
A point where a function changes from increasing to decreasing.
Define point of inflection.
A point where the concavity of a function changes.
Define concavity.
The direction of the curve of a function (upward or downward).
What does it mean for a function to be increasing?
The function's value is getting larger as x increases; $f'(x) > 0$.
What does it mean for a function to be decreasing?
The function's value is getting smaller as x increases; $f'(x) < 0$.
Define the first derivative.
The rate of change of a function with respect to its variable.
Define the second derivative.
The rate of change of the first derivative; indicates concavity.
What is an x-intercept?
The point where a graph crosses the x-axis ($y=0$).
Define extrema
The maximum and minimum values of a function.
