For the function f(x) below, draw a graph of f ' (x). Don't worry too much about whether f ' is straight or curvy—focus on getting it to cross the x axis in the right places.
Answer
f has a constant, negative slope:
This means f' will be constant and negative:
Example 2
For the function f(x) below, draw a graph of f ' (x). Don't worry too much about whether f ' is straight or curvy—focus on getting it to cross the x axis in the right places.
Answer
f has a negative slope to the left of zero, a positive slope to the right of zero, and a slope of zero at zero:
This means its derivative will be negative to the left of zero, positive to the right of zero, and zero at zero:
Example 3
For the function f(x) below, what would the graph of f ' (x) look like?
Answer
f has a positive slope everywhere, except at zero where its slope is zero:
This means its derivative is always positive except at zero, where it's 0.
Example 4
For the function f(x) below, what would a rough sketch of f ' (x) look like?
Answer
This function f is all confused. Its slope bounces all around between negative, positive, and zero. It's easiest to see what's going on by labeling the graph:
Respecting the sign of f ' gets us this graph of f ':
Example 5
From the graph of f ' (x), what might the graph of f(x) look like? Make sure to label each graph.
Answer
Wherever f' is positive, f will be increasing. Wherever f' is negative, f will be decreasing: This means f will look something like one of these:
Right now we don't have enough information to know which it would be.
Example 6
From the graph of f ' (x), draw a graph of f(x). Make sure to label each graph.
Answer
f' is positive everywhere except 0:
This means f will be increasing everywhere except 0, so f will look something like this, possibly shifted up or down:
Example 7
From the graph of f ' (x), draw a graph of f(x). Make sure to label each graph.
Answer
Wherever f' is positive, f will be increasing. Wherever f' is negative, f will be decreasing:
This means f will look something like this, possibly shifted up or down:
Example 8
From the graph of f ' (x), what will the graph of f(x) look like? Make sure to label each graph.
Answer
Here's f', labeled with what it's doing:
f' starts out as a positive constant, meaning f will start as a straight line with positive slope. Next f' gets closer to zero, therefore the slope of f gets shallower. Finally, f' becomes negative, meaning f will decrease:
This means f will look something like one of these: 
