2.2c - The Derivative Function (f 0 ! f )
(Going Backwards)
Miss. Gomero
Calculus
Introduction
OBJECTIVE: To do the reverse of what we did in the previous
lesson: the graph of the derivative will be given and we will sketch
a possible graph of the original function.
To do this, we must pay attention to four behaviors of the graph of
y = f 0 (x).
f 0 (x) = 0.
f 0 (x) > 0 or f 0 (x) < 0.
f 0 (x) is increasing or f 0 (x) is decreasing.
f 0 (x) has a maximum or minimum.
Our goal is to translate these four behaviors of the f 0 (x) graph to
the corresponding behaviors of f (x) graph.
Gomero
2.2c - The Derivative Function (f 0 ! f ) (Going Backwards)
Important Properties to Note
Properties: Relationship b/w f’ and f graphs
If f 0 (a) = 0, then the graph of y = f (x) has a horizontal tangent line at
x = a.
i.e., y = f (x) has a max or min at x = a.
If f 0 (x) > 0 over an interval, then y = f (x) is increasing over that
interval.
If f 0 (x) < 0 over an interval, then y = f (x) is decreasing over that
interval.
If f 0 (x) is increasing over an interval, then y = f (x) is concave up over
that interval.
If f 0 (x) is decreasing over an interval, then y = f (x) is concave down
over that interval.
If the f 0 -graph has a max/min at x = a, then the f -graph has an
inflection point at x = a.
Gomero
2.2c - The Derivative Function (f 0 ! f ) (Going Backwards)
Example 1
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) =
3x + 3
y
x
Example 2
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) = x 2 + 4x
3
Example 3
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) =
x3
+ 7x
3
20
3
y
x
Practice 1
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) =
x2
2x + 4
Practice 2
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) = x 2 + 2x + 3
Practice 3
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) = x 3 + 2x 2 + 3x + 4
Practice 4
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) =
x 3 + 2x 2 + 3x + 4
Practice 5
The graph of y = f 0 (x) is given. In the same coordinates system,
sketch a possible graph of y = f (x).
f 0 (x) =
x 3 + 5x
Example/Practice 6
The graph of y = f 0 (x) is given. In the same coordinates system, sketch a
possible graph of y = f (x). Assume that f (0) = 1 and f (1) = 1.5.
HW: Complete hmwk#25 handout.
Gomero
2.2c - The Derivative Function (f 0 ! f ) (Going Backwards)