3
 Find any times when the velocity of the particle is constant.
 Find any intervals where the velocity is increasing.
 Find any intervals where the velocity is decreasing.
 Find the intervals where the particle is speeding up

s(t) = t 3 -2t
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Graph the position function
Mark the points on the position function when the velocity is constant.
Locate the intervals when the velocity is increasing on the graph of the position function.
Locate the intervals when the velocity is decreasing on the graph of the position function.
How does the graph of the position function differ when the velocity is increasing rather than decreasing?

Vocabulary
Know:
Critical pointspossible extrema, where first derivative =
0 or derivative doesn’t exist, but original function does exist
Now:
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Relative extremacritical points where change in sign in first derivative before or after
Intervals of Increaseintervals where first derivative is positive
Intervals of Decreaseintervals where first derivative is negative
Possible inflection pointswhere second derivative = 0 or doesn’t exist, but original function does exist
Inflection pointspossible inflection points where there is a change in sign in the second derivative before or after
Concave upintervals where second derivative is positive
Concave downintervals where second derivative is negative

For f(x) = 5x 3 -5x+6
 Find intervals of increase, decrease, relative extrema (label as max. or min.)
 Find inflection points, intervals of increase or decrease.

2 x
2 
4 x
f(x) =
 Find intervals of increase, decrease, relative extrema (label as max. or min.)
 Find inflection points, intervals of increase or decrease.