MATH 133 { Calculus with Fundamentals 1
Curve Sketching
December 1, 2015
Background
When mathematicians and scientists want to understand graphs of functions and other curves
these days, they almost always turn to graphing calculators and mathematical software to gen-
erate the plots. So learning to plot functions by hand can seem like a waste of time. However,
in my experience, some experience with curve sketching by hand is a great way to solidify your
understanding of the First and Second Derivative Tests, concavity, asymptotes, etc. So while you
will probably never need to do this again \in real life," it's still a worthwhile educational exercise.
Here's a checklist of things to do and look for, given y = f(x) to plot:
1. Determine the domain of your function and note whether the graph has any vertical asymp-
totes. Draw those in if there are any.
2. Compute f0(x) and nd the critical points x = c, and the critical values f(c).
3. On each interval between adjacent critical points, determine the sign of f0(x). Recall that
intervals where f0(x) > 0 are intervals where f is increasing and intervals where f0(x) < 0
are intervals where f is decreasing.
4. Compute f00(x) and determine any x where f00(x) = 0 or where f00(x) does not exist.
5. Determine the intervals where f is concave up and concave down using the sign of f00 { recall
that in
ection points are places where the concavity changes. If your function has in
ection
points, nd them and the function values there.
6. Determine horizontal asymptotes (if any), by taking limx!1 f(x).
7. The critical points and the in
ection points can be thought of as transition points as described
in Section 4.6 of the text.
8. You should now be prepared to sketch the graph using the transition points, the information
about where the function is increasing, decreasing, concave up, and concave down. See Figures
1 and 2 on page 231 for the four basic local shapes of graphs and an example of how they can
be put together.
Questions
Find the transition points (critical points and points of in
ection), intervals of increase and decrease,
intervals of upward and downward concavity, asymptotes, and sketch the graph:
1. y = x5 􀀀 15x3 (Hint: This one has three critical points, three points of in
ection, and no
asymptotes.)
2. y = x 􀀀 2 ln(x) (Hint: limx!0+ ln(x) = 􀀀1.)
3. y =
1
x2 􀀀 2x
(Hint: This one has both horizontal and vertical asymptotes. Be sure to simplify
f0(x) before computing the second derivative.)
