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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.
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.
If $f'(x)$ is positive, what does this mean for the graph of $f(x)$?
The graph of $f(x)$ is increasing.
If $f'(x)$ is negative, what does this mean for the graph of $f(x)$?
The graph of $f(x)$ is decreasing.
If $f''(x)$ is positive, what does this mean for the graph of $f(x)$?
The graph of $f(x)$ is concave up.
If $f''(x)$ is negative, what does this mean for the graph of $f(x)$?
The graph of $f(x)$ is concave down.
What does an x-intercept on the graph of $f'(x)$ represent on the graph of $f(x)$?
A potential relative maximum or minimum on the graph of $f(x)$.
What does a relative maximum on the graph of $f'(x)$ represent on the graph of $f(x)$?
A point of inflection where the concavity of $f(x)$ changes from up to down.
What does a relative minimum on the graph of $f'(x)$ represent on the graph of $f(x)$?
A point of inflection where the concavity of $f(x)$ changes from down to up.
If $f'(c) = 0$ and $f''(c) > 0$, what does this mean for the graph of $f(x)$ at $x=c$?
There is a relative minimum at $x=c$.
If $f'(c) = 0$ and $f''(c) < 0$, what does this mean for the graph of $f(x)$ at $x=c$?
There is a relative maximum at $x=c$.
How can you identify a point of inflection on the graph of f''(x)?
Look for points where f''(x) changes sign (crosses the x-axis).