We will learn how to visualize a function by looking at its derivatives. Try to picture the function (the third graph) by looking at the first two graphs.
Signs
The signs of
f
,
f'
, and
f''
reveal much of the function.
Memorize the terms in bold.
Sign of
f
(Sign of
f
):
Positive:
f
is above the x-axis.
Negative:
f
is below the x-axis.
Zero:
f
is on the x-axis.
Sign of
f'
(Monotonicity of
f
):
Positive:
f
is increasing.
Negative:
f
is decreasing.
Zero: Critical Point:
f
is flat.
Sign of
f''
(Concavity of
f
):
Positive: Concave Up: The slope of
f
is increasing.
Negative: Concave Down: The slope of
f
is decreasing.
Zero: Inflection Point: The slope of
f
is not changing.
Inflection points are the critical points of the slope. Thus, the local minimums and maximums of the slope are at the inflections points. This is where the function is steepest.
Critical and Inflection Points
Here are the signs that
f
has a critical or inflection point:
Critical Point:
f
is flat.
f'
is zero.
f''
cannot tell.
Inflection Point:
f
is at its steepest.
f'
is flat.
f''
is zero.
Practice
Below are some derivatives (red) and second derivatives (orange).
Try to sketch the function, following these steps:
1. Mark the starting point of the function. Eg:
f(0) = 1
.
2. Find the critical points, and mark the function as flat.
3. Mark between the critical points whether the function is increasing or decreasing.
4. Find the inflection points, and mark those locations as where the function is
steepest.
5. Sketch the function.
6. Reveal the answer.
7. Check your critical points, slopes, and inflection points.