Chapter 4
Project A: Graphing
Introduction
The goal of this project is to offer a more engaging opportunity for you to learn about the
application of Calculus to “curve sketching.” In particular you will learn how to improve on
sketching a graph of a function by using information about its ¿rst and second derivative.
You know that we can get rough graphs of functions by plotting points and looking for
asymptotes or exploring with a graphing calculator. However, using calculus can both
make the general task easier and provide insights about the function that otherwise are
dif¿cult to obtain.
Even people in possession of graphing calculators need to learn the skills discussed in
this project. Some of the reasons include:
(i) There may be situations in which you have information about the shape of
a function but not its formula. Calculus can be useful either to ¿nd a suitable
formula for the function or to develop more insight about the function without
knowing its formula.
(ii) It is important to understand the ideas behind what a graphing calculator
does. If you don’t have a handle on this, then you will not be able to determine
when an answer doesn’t make sense.
(iii). Both graphing calculators and computers are limited by the scale used to
display the graph. It may be necessary to display a graph at different scales to
see the details of a particular function. You can use your knowledge of calculus
to help you decide if this is necessary.
In this project, it is assumed that you will use your text as necessary for de¿nitions,
theorems, and as a general reference. We will indicate when you should stop and read a
0 h1994
c
by CaRP, Department of Mathematics, University of California, Davis
207
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CHAPTER 4. PROJECT A: GRAPHING
section of the text. The exercises in the project, however, are written so that you can learn
by discovery – that is, you should master the material covered by completing the project.
Your instructor and teaching assistants will be available for help, but they will not do the
exercises for you.
To complete this project on time, it is essential that you start this project at the
beginning of the project week. It is long and some parts are fairly challenging. Since
the normal workload expectation for this course is 2-3 hours outside of class for each hour
inside, project completion and discussion of the material should take 12-20 hours. You may
¿nd that you’ll make the best progress if you work on it for a few hours, take a break and
do something else for awhile, and then return to it (we’ve even indicated a few good places
for breaks in the pages that follow). You may ¿nd that you get so engrossed in a problem
that you think about it while walking to your other classes, brushing your teeth, etcetera
(this may seem odd, but it’s quite normal – at other institutions, students even have arrived
at important insights while chowing down on pizza).
Finally, even if you have studied calculus previously: DON’T SKIP THE PRELIMINARY STUFF and jump to the last few problems, on the assumption that you already
“know it all”. One of the goals of Cameos is for you to gain pro¿ciency in writing mathematics, making conjectures, and proving or justifying mathematical statements. The exercises that follow are designed to help you do this. Failure to do them would both deprive
you of an important part of the intended experience and put you at risk of not knowing the
material with the depth of understanding needed for successful completion of exams and/or
other assignments.
What to Expect
In this project, we will lead you through an exploration of the steps that you should follow
when sketching the graph of a function. When curve sketching, you are going for an indication of the basic shape and “key points.” In fact, a minimum number of points on the
curve are to be labelled namely, key points where something mathematically signi¿cant
happens. Intercepts are an example of key points.
This project begins with a review of a few concepts that you should have seen in precalculus. Next, you’ll learn how the derivative of a function can provide information about the
shape of a curve. Finally, you’ll ¿ne tune your understanding of these ideas and practice
using all the information you glean about a given function to sketch its graph. We refer
to this graphing process as “discuss and sketch”, i.e., discuss the properties of the function and provide a fairly accurate sketch that reÀects information determined during the
discussion.
The discussion should be systematic and should include properties such as symmetry,
intercepts, asymptotes, whether the function is increasing or decreasing, relative and global
maxima/minima, points of inÀection, and concavity. We will discuss each of these properties in the pages that follow and, by the end of this project, show you how to use all of the
PAINLESS GRAPHING WITHOUT CALCULUS
209
information.
SHOW YOUR WORK ON THESE PAGES UNLESS OTHERWISE INDICATED.
Your completed project should serve as a nice set of notes that can be referred to when
studying for exams. It also should serve as an alternative to the approach that is offered in
the text.
Painless Graphing without Calculus
A library of functions
A lot of information can be gleaned about the shape of a function by applying deeper
thinking to what you know about a “core set” of functions. For example, you should
know the rough shape of graphs of polynomial, logarithmic, exponential, and trigonometric functions. That is, if asked to sketch the graph of f x 0003x 8 0009x 3 10,
you should recognize that the function roughly has the shape of a letter “n” by relating
it to a parabola that opens down. Though it may have more wiggles, its basic shape will
not change. The ¿gures below depict f x 0003x 8 0009x 3 10 at two different
scales of display (look at the axes).
-9.9999
-20
-9.99992
-9.99994
-40
-9.99996
-9.99998
-60
-10
-80
-10
-10
-100
-10.0001
-10.0001
-120
-4
-2
0
Fig. 1
2
x
4
-0.4
-0.2
0
0.2
x
0.4
Fig.2
1. For the following library of functions, sketch a set of axes and a labelled graph for
each of the three functions for which the graph is omitted. Each should require no
more than 10 seconds. Then, for each of the twelve functions, note carefully the
domain and range of the function, as well as any intercepts.
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CHAPTER 4. PROJECT A: GRAPHING
(a) f x x 2
(b) f x x 3
25
100
20
50
15
-4
10
-2
00
2
x
4
-50
5
-100
-4
-2
00
2
x
4
(c) f x x n where n is an even integer
(d) f x x n , where n is an odd integer
(e) f x x 1n , where n is an even
integer
(f) f x x 1n , where n is an odd integer
PAINLESS GRAPHING WITHOUT CALCULUS
(g) f x 1
x
211
(h) f x log10 x
0.6
0.4
-4
-2
0.2
0
2
x
4
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
(i) f x 10x
(j) f x sin x
1
0.5
-4
-2
00
2
x
4
-0.5
-1
(k) f x cos x
(l) f x tan x
1
4
0.5
-4
-2
00
-0.5
2
2
x
4
-4
-2
00
2
x
4
-2
-4
-1
Remark. The sketches that were given to you were done with the student version of Maple.
Look carefully at the graphs of (b), (d), (f), and (h) and note that there is something misleading about their appearance.
When curve sketching, having a library of functions can help you anticipate the basic
shape of the graph you seek.
Keep these functions in mind as you work. Use them as needed to give yourself a jump start
towards the rough shape of a graph. We will not make reference to them in our discussion
212
CHAPTER 4. PROJECT A: GRAPHING
of process. They should simply be part of a repertoire of functions about which you have a
general awareness.
Checking for the Obvious: Intercepts, Symmetry and Asymptotes
Intercepts
Points where the graph of a function crosses an axis are called intercepts. Whenever
possible, you should ¿nd the intercepts of any function you are asked to graph — these
provide you with some landmarks from which to start your discussion.
Read 1 , pp. 46-47 now.
When curve sketching, mark and label intercepts carefully.
Symmetry
Symmetry is a concept that you may recognize without thinking about it. For example, you
are bilaterally symmetrical — you have a right half and a left half that are mirror images
(this is a biological simpli¿cation, of course — for example, if you still have your appendix,
you only have one and it’s to the right of your abdomen), whereas a bicycle wheel is radially
symmetrical. In mathematics, we often speak of the symmetry of even and odd functions.
A function f is said to be even if f x f x for each x in its domain, and it is said
to be odd if f x f x for each x in its domain. (Read SB, pp. 45-46 now.)
2.
(a) The function f x x 6 is an even function because it ¿ts the de¿nition. To
prove this, using the de¿nition of an even function, we note that
f x 6 f
6
Thus, f is even. Moreover, this function, like any even function, is symmetric
about the line
.
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213
(b) In contrast, f x x 7 is an odd function. Use the de¿nition of an odd function
and a procedure similar to (2a) to prove this statement.
Note that, like other odd functions, f x x 7 is symmetric about
.
3. Some functions are neither even nor odd. One example is f x x 6 2x 5 . State
three more examples.
4. Indicate, on the lines provided, whether the functions given in the library (1) are
even, odd, or neither. You need not do the proofs here, but you should know how to
do them.
(a)
(c)
(e)
(g)
(i)
(k)
(b)
(d)
(f)
(h)
(j)
(l)
5. For (4a) – (4f), you should have decided that f x x n is an odd function if n is odd
and an even function if n is even. A proof for these assertions would look somewhat
like what follows. Fill in the blanks.
Claim: if n is even, then f x x n is an even function.
Proof. We will use the de¿nition of even functions to prove this. First, note that
if n is an even integer, then we can write n 2k k 0 1 2 to help with
214
CHAPTER 4. PROJECT A: GRAPHING
simpli¿cation. Now using the de¿nition,
f x r
sn
k
x
Using properties of exponents, we have
r
f x r
x
f
s
2
s
Hence, f x x 2k is an even function.
AN ASIDE ABOUT PROVING: If you did a bunch of examples for different even
values of n, this would not be suf¿cient to prove the claim in general. In mathematics,
one counterexample is enough to disprove a theorem, but examples alone cannot prove a
theorem. Also, notice that for each of the proofs that we have written, we have started
with the de¿nition and worked with one side to obtain the desired statement (look over the
proofs again to see that this indeed is what we did). This is a standard method of writing a
proof in mathematics.
Conversely, if you want to show that a function is not even (for example), you can
either choose a number a in the domain of f and show that f a / f a, or show that
the de¿nition is not satis¿ed for any x. Both methods work and there isn’t a magic formula
that tells you which to use. You may need to do some initial playing (or scratchwork) in
order to choose the one that seems easiest for the particular problem.
6.
(a) Using the de¿nition of an odd function, prove that the product of two odd functions is even. To start off, Let f and g be odd functions. We want to show that
.
hx f xgx is
(b) Is (a) true if “product” is replaced by “sum”? Explain your position.
Position:
PAINLESS GRAPHING WITHOUT CALCULUS
215
Explanation:
7. Suppose that the graph in part a is what is known about the graph of a function f that
is de¿ned for all reals.
(a) Sketch the rest of the graph of f if f is even.
(b) Sketch the rest of the graph of f if f is odd.
216
CHAPTER 4. PROJECT A: GRAPHING
When curve sketching, knowing that a function is even or odd means that
you only have to do the analysis (i.e., information search) about the function
in * 0 or 0 *. You can then use symmetry to complete the sketch.
Asymptotes
8. Congratulations! You have just been appointed to be a counselor at the Summer
Mathmagic Camp for inner city children who are about to enter junior high school.
Your ¿rst task will be to assist them with their introduction to systematic function
graphing. Luckily for you, the students will have learned function notation and how
to graph polynomial functions (via methods similar to what you used above). To help
your students move to the next step, examine the graph below and write a paragraph
that explains the behavior of f in terms that they can understand.
5
4
3
2
1
0
5
10
15
20
In your paragraph, you should have included something indicating that the value
of the function f x approaches 3 as x gets larger and larger. This is represented
graphically by having the curve representing f get closer and closer to the line y 3.
We call the line y 3 a horizontal asymptote. More precisely:
We say that a function f has y L as a horizontal asymptote if
lim f x L
x*
or if
lim
x*
f x L We say that a function f has x a as a vertical asymptote if
lim f x *
xa lim f x *
xa lim f x * or
xa lim f x *
xa Read SB, pp. 47 through the top of p. 48, then take a break.
PAINLESS GRAPHING WITHOUT CALCULUS
217
OK, now that you’re relaxed, refreshed, and mentally alert...practice ¿nding asymptotes
for rational functions. We’ll consider vertical and horizontal asymptotes separately.
9. Use limit computations to ¿nd all horizontal asymptotes for each of the following.
(a)
x 3 5x
x 4 5x 2 4
2x 3 3x
(b)
x3 1
(c)
2x 3 3x
x3 1
(d)
2x 52 4x
x3 1
In many cases, it is useful to know whether the graph of a function lies above or
below its horizontal asymptote. For example, in (9b) and (9c), the function had a
218
CHAPTER 4. PROJECT A: GRAPHING
horizontal asymptote at y 2. In (9b), the numerator is larger than the denominator
for large positive values of x because of the 3x term in the numerator. Hence, the
graph of f will lie above the horizontal asymptote for large positive values of x. In
(9c), the 3x term was subtracted so that the numerator of f will be smaller than the
denominator for large positive values of x. Therefore, for the function in (9c), the
graph of f will lie below the horizontal asymptote for large positive values of x.
10. Do a similar analysis relating the location of the graph of f to its horizontal asymptote for (9b) and (9c) when x is negative with large absolute value.
3x 8 4x 5 3
11. Let gx . Find all the horizontal asymptotes for g and determine
x8 x4 2
whether the graph of the function lies above or below any horizontal asymptotes that
are found.
In each of (9) and (1111), the graph of the function had a single horizontal asymptote.
In general, rational functions will have at most one horizontal asymptote (although the
graph of the function may lie on different sides of a horizontal asymptote for positive vs.
negative x).
PAINLESS GRAPHING WITHOUT CALCULUS
219
In contrast, other types of functions can have more than one horizontal asymptote.
Recall the de¿nition of x:
|
x if x o 0
x x otherwise
Alternatively,
T
x x 2 .
12. Fill in what’s missing.
T
7x 2 5x
(a) For lim
,
x*
x
T
7x 2 5x
x*
x
T T
2
x 7
lim
x*
T x
x 7 lim
x*
T x
lim 1 7 f x lim
x*
In contrast,
T T
T
x2 7 2
7x 5x
lim
lim
x*
x*
x
x
T
lim 7
x*
T
7
T
(b) Where did the negative sign in the second part of (12a), i.e.,on 7, come
from?
220
CHAPTER 4. PROJECT A: GRAPHING
(c) Explain what the results above mean for the graph of the function.
13. Now, how about vertical asymptotes? One important thing to keep in mind when
looking for vertical asymptotes is that they generally occur at values of x that are not
in the domain of the function. On the other hand,“a” not being in the domain of a
function doesn’t automatically mean that x a is an asymptote. To con¿rm whether
a function has a vertical asymptote at x a, you need to take the limit of f as x
approaches
from the left and as x approaches
from the
. Now ¿nd all vertical asymptotes for the functions in (9a) and
(9b).
There is one more type of asymptote that warrants study: tilted asymptotes. These
occur in a rational function when the degree of the polynomial in the numerator is larger
than the degree of the polynomial in the denominator.
To ¿nd the equation of a tilted asymptote, divide the denominator into the numerator.
The non-remainder part of the result gives the equation of the tilted asymptote.
The next problem justi¿es the process that is described in the box.
1
14. Consider the functions f x x 3 and g x x 3 .
x
PAINLESS GRAPHING WITHOUT CALCULUS
221
(a) Compute the values of f and g for x 100 20 10 1 1 10 20 100.
(b) Make a conjecture about the graph of g compared to the graph of f for large
values of x.
15. Consider the function f x 3x 2 2x 5
.
x 3
(a) Use long division to write f as the sum of a polynomial and the quotient of the
remainder and the divisor. If you need a review on how to do this, see SB, p.
52.
222
CHAPTER 4. PROJECT A: GRAPHING
(b) Use the description given in the box and the intuition developed from (14) to
¿nd the equation of the tilted asymptote of this function.
16. Are asymptotes always linear?
Position:
Explanation:.
When discussing a function to be graphed, ¿nd all asymptotes.
When curve sketching, indicate asymptotes with broken lined curves (i.e., - - - -). You
need not indicate tilted asymptotes whose equations are of degree two or higher.
Pulling it Together (thus far):
Discussing and Sketching the Graphs of Functions, Part I
Now, take a break if you need one. Feel refreshed? Good.
Before seeing what Calculus can do to improve our curve sketching, let’s make sure
that we have a grasp of the process that we have uncovered thus far.
Given a function to graph, with what we have thus far, your discussion needs to include
checking if the function is even or odd, ¿nding all intercepts, ¿nding asymptotes, and
looking for where the graph will be in relationship to the asymptotes.
Your sketch needs to include labelled intercepts and asymptotes indicated with broken
lines, and should depict a function that accurately and consistently reÀects the information obtained in the discussion.
PAINLESS GRAPHING WITHOUT CALCULUS
223
17. Suppose that you are asked to discuss and sketch the graph of a function such as
f x x sin x, and that you want to do this as painlessly as possible. Before reading
on, think about it.
The function f x x sin x is an even function because it satis¿es the de¿nition.
This function has an in¿nite number of x-intercepts they occur at
. There
is only one y-intercept it occurs when
. Now recall that, as x varies over all
real values, the sin x takes on all real values from
to
. Thus, it must
be true that, for any real x, the graph of f x x sin x lies between the graph of
y x and y Putting it all together, the graph of f x x sin x must look
something like the following.
15
10
5
-20
-10
00
-5
-10
-15
Now you try it.
18. Let g x sin 2x
for x / 0 .
x
(a) Determine if g is even, odd, or neither.
(b) Find the intercepts of gx.
10
x
20
224
CHAPTER 4. PROJECT A: GRAPHING
(c) Use the squeeze principle (SB p. 68) to ¿nd the horizontal asymptote of gx.
(d) Sketch the graph of gx.
(e) How many times does gx cross the horizontal asymptote?
19. Consider the function from (9b) and (10): f x 2x 3 3x
.
x3 1
than
(a) Notice that f is neither even nor odd. Furthermore, the sign of f is
0 if x is negative. If x is greater than
, then f is positive. Finally, f is
negative for values of x
.
(b) Find all intercepts of f .
PAINLESS GRAPHING WITHOUT CALCULUS
225
(c) You’ve already been asked to ¿nd the asymptotes of f . List them here.
(d) Use (a) - (c) to sketch the graph of f .
20. Discuss and sketch the graph of the function that was given in (15):
3x 2 2x 5
f x . (Remember to present your discussion with appropriate headx 3
226
CHAPTER 4. PROJECT A: GRAPHING
ings. The reader should not have to guess about what you are doing.)
Graphing Using Calculus, Part 1
In Search of Peaks and Valleys
The general premise of this section is that the sign of the ¿rst and second derivative of
a function provides useful information about its qualitative behavior. More precisely, you
will learn how to determine when a function has peaks and valleys, when it bends up or
down, and how this is related to derivatives.
The First Derivative and Extrema
21. You learned in SB 41 about the relationship between the ¿rst derivative and increasing and decreasing functions. Write a brief paragraph that summarizes that relation-
GRAPHING USING CALCULUS, PART 1
227
ship.
Suppose now that some continuously differentiable function f is increasing for values of x that are less than some number a and decreasing for values of x that are
greater than some number a (see the ¿gure below). What does this tell you about the
derivative of f exactly at x a?
DID
\
D[
22. Think about it and look at the ¿gure again – it should be apparent that f ) a 0.
Why is continuity of f ) important for this conclusion?
A number c in the domain of the function f is called a critical number for f if either
f ) c 0 or f ) is unde¿ned at x c. The corresponding point c f c is called a
critical point on the graph.
Note: this de¿nition differs slightly from that given in SB, p. 176.
You saw a graphical example of a critical point at which the derivative was 0. The ¿gure
below depicts a function that has a critical point at a value of x where the ¿rst derivative is
unde¿ned. Without doing any computations, write a brief statement describing why f ) is
not de¿ned at x c.
228
CHAPTER 4. PROJECT A: GRAPHING
\
FIF
F
23. Find all the critical points of the function f x 3x 5 50x 3 135x
24. (adapted from Hughes-Hallett et al.
1993) The points depicted at the right
are the only critical points of a function
f.
(a) Sketch a graph for a function f that is
continuous for all real numbers, satis¿es
lim f x * and
x*
lim
x *
f x *
and has its only critical points where
depicted above.
[
GRAPHING USING CALCULUS, PART 1
229
(b) Sketch a graph for a function f that is
continuous for all real numbers, satis¿es
lim f x * and
x*
lim
x *
f x 0
and has its only critical points where
depicted above.
25. It should be apparent from your work on (24) that one of three things can occur at a
critical point: the graph of f can have a peak, a valley, or neither. Examine the graph
below.
DFE
\
[
FIF
Fill in what’s missing.
(a) Write a sentence that describes the graph of the function depicted above. Specifically, what do you notice about the point c f c relative to other points on
the graph?
Now that you have completed writing your description, look at the graph again.
Consider values of f near the point c f c.
interval a b that contains c, notice that f c is
than f x for any other x in a b in the domain of f . When
this occurs, we say that f has a relative maximum at x c (The word
“maximum” originally is Latin its plural form is maxima).
On the
Intuitively speaking, relative maxima correspond to peaks of a graph.
230
CHAPTER 4. PROJECT A: GRAPHING
\
FIF
F
[
(b) On the other hand, examine the ¿gure above. Once again, write a sentence that
describes the graph of the function.
Speci¿cally, we want to be aware of what happens regarding the point
c f c relative to other points on the graph. Consider values of f near the
point c f c.
On the open interval a b that contains c, notice that f c is
than f x for any other x in a b in the domain of f . When
this occurs, we say that f has a relative minimum at x c (analogous to
above, the plural for minimum is minima).
Intuitively speaking, relative minima correspond to (the lowest points in the)
valleys of a graph.
(c) The fact that f x o f c at values of x outside the interval a b is irrelevant
– we are interested only in points near c f c. That’s why the adjective
“relative” must be included. Return to your graphs from (24). On the graphs,
label each of the extrema as a relative maximum or a relative minimum.
When you labelled the graphs from (24), you should have noticed that
there are critical points that are neither relative maxima nor relative minima.
We will discuss the terms used to describe such points in the next section.
If f c o f x for all x in the domain of f , then we say that f has
a global maximum at x c (and we call c f c the global maximum).
Geographically speaking, on earth, places such as the summit of Mt. Whitney
or Mt. Kilimanjaro are relative maxima whereas the summit of Mt. Everest is
the global maximum. Global maxima also may be called absolute maxima.
GRAPHING USING CALCULUS, PART 1
231
(d) One can de¿ne global minimum in an analogous way. Write a de¿nition for
global minimum.
(e) Identify the apparent global maxima and global minima on your graphs from
(24).
(f) Notice that a point can be both a relative extremum and a global extremum.
Moreover, a point can be a global extremum but not a relative extremum. Examine the graph below. For each of the labelled points B, C, D, E and F,
write the term that best describes the point, where the choices are: relative
maximum, relative minimum, global maximum, global minimum, or not an extremum. Write out explanations for these claims, using the de¿nitions of global
and relative extrema (read them carefully).
\
&
(
[
%
'
)
26. For all of the functions depicted below, observe that, for a relative extremum at x c,
either f ) x 0 at x c or f ) x is unde¿ned at x c.
232
CHAPTER 4. PROJECT A: GRAPHING
\
\
\
\
[
F
F
[
F
[
F
[
Is it possible for a function to have a relative maximum or a relative minimum at
x c without the derivative of f at c being zero or unde¿ned? Explain in intuitive
terms.
We have now focused in on an important idea:
To ¿nd relative extrema of a function, ¿nd the critical points of the function. These give
candidates for relative extrema. However, you must do a little more work to decide if
the extrema are relative maxima, relative minima, or neither.
27. Examine the graphs below and ¿ll in what’s missing in the table to classify the extrema depicted.
(a)
\
F
values of x:
f x:
f ) x:
* x c
increasing
[
x c
rel max
0
cx *
GRAPHING USING CALCULUS, PART 1
(b)
233
\
[
F
values of x:
f x:
f ) x:
* x c
0
x c
rel min
not de¿ned
cx *
increasing
(c) Write a sentence or two that describes your results from (a) and (b) in terms of
classifying critical points.
(d) Congratulations! You have just described the First Derivative Test, which is
used to classify relative extrema.
Read SB, p. 177 and write out the First Derivative Test in the space provided below.
Then take a break before going on.
Classifying Extrema
One way to organize the information from the ¿rst derivative test is to use a sign chart such
as was shown in (27a) and (27b). A sign chart is constructed by the following procedure.
First, the critical numbers of a function must be determined. These are used to subdivide the
real number line into open intervals. On each of these intervals, the ¿rst derivative of the
function must have the same sign because
.
Hence, to determine the sign of the ¿rst derivative in any interval, it is suf¿cient to choose a
234
CHAPTER 4. PROJECT A: GRAPHING
single test number in the interval and to evaluate the sign of the
at the
test number. In turn, this is used to determine intervals in which the function is increasing
or
, and the ¿rst derivative test is used to classify the critical points.
28. Construct a sign chart and use the ¿rst derivative test to classify the relative extrema
of f x 3x 5 50x 3 135x by ¿lling in what’s missing (do computations somewhere other than on the chart!). Note that we are only doing a chart for positive reals
because f is odd.
(a) From work on (23), the only critical numbers of f are at x 1 3.
crit. number:
intervals:
test points:
f ):
f:
classi¿cation:
0x 1
x 1
x 3
3x *
2
increasing
rel
rel
Thus, using the ¿rst derivative test, we have determined that, in x
0, the
5
3
function f x 3x 50x 135x has a relative
at x and a relative minima at x . Since f is odd, we also have a relative
maximum at x and a relative minimum at x .
(On any problem like this, make sure that you always include a sentence or two
at the end indicating what you found.)
(b) Is there anything special about the test numbers used above? For example,
could you have used x 043789571 for a test number in the interval 0 1?
1
Why is x a better choice of test number? What advice would you give to
2
a colleague for choosing test numbers?
29. Find the critical points for gx x 3 3x 2 1, make the corresponding sign chart,
and use it to classify the extrema.
GRAPHING USING CALCULUS, PART 1
235
Problems (28a) and (29) illustrate what the new piece of Discussing and Sketching the
Graphs of Functions looks like.
Given a function to graph, with what we have thus far, your discussion needs to include checking if the function is even or odd, ¿nding all intercepts, ¿nding asymptotes,
looking for where the graph will be in relationship to the asymptotes, and ¿nding and
classifying critical points.
Your sketch needs to include labelled intercepts, asymptotes indicated with broken lines,
labelled relative minima, labelled relative maxima, and should depict a function that
accurately and consistently reÀects the information obtained in the discussion.
For the next problem, we lead you through the work so that you can see what a discuss
and sketch looks like, thus far. For (31), you are expected to systematically and neatly
present the discussion as well as to conclude with a graph.
30. Let’s return to the function that you’ve been investigating,
f x 3x 5 50x 3 135x.
(a) First, check for symmetry in the function (i.e., if the function is even, odd, or
neither), or any general observations concerning when it is positive or negative.
Do that now.
(b) Next, ¿nd all intercepts and asymptotes. Show them on the axes provided in
part (d).
(c) From (28a), we know the values of x at which f has relative extrema. Calculate
the corresponding values of f . Put those on your sketch in part (d) and label
236
CHAPTER 4. PROJECT A: GRAPHING
them.
(d) Now, connect the points on your sketch with a smooth curve. Ta-da!! You have
a picture of the function.
\
[
(e) Explain the mathematical reasons why the curve from (d) should be smooth.
(f) If you have a graphing calculator, enter the function and compare that graph to
the one you just obtained.
2
31. Let f x 4x 4 16x . Discuss and sketch f (i.e., check for symmetry, asymptotes, ¿nd intercepts, critical points, and classify extrema then use these to sketch a
A PAUSE TO REFLECT ON F VS F )
237
graph of the function).
A Pause to ReÀect on f vs f )
Before adding the last piece of information that Calculus contributes towards curve sketching, the next few problems explore the relationship between the graph of f ) and the graph of
f . The goal is to develop a deeper understanding of the phenomena that we have observed.
32. A function f has a relative maximum at x 4 and a relative minimum at x 10.
The function also has a horizontal asymptote at y 3.
(a) Sketch the graph of a function that satis¿es these properties.
238
CHAPTER 4. PROJECT A: GRAPHING
\
[
(b) Suppose that f has an x intercept at x 2. How does this affect your graph
from part (a)?
\
[
(c) Suppose that you were told that f 11 1, with the conditions above still
holding. Can this occur? Explain why or why not.
33.
(a) For the function from (31), the graph of f and f ) are depicted below. Study the
graphs and try to discern the relationship between the graph of f and the graph
A PAUSE TO REFLECT ON F VS F )
239
of f ) .
15
10
5
-2
00
-1
1
x
2
-5
-10
-15
(b) The ¿gure below represents the graph of the derivative of a function, g ) . On the
axes provided, draw a sketch of a curve that could be the original function, g.
\
\
[
[
34. The graph of the derivative f ) of a function is shown below. Indicate on the graph
the values of x that are critical numbers of f itself. At which of these does f have
relative maxima, relative minima, or neither? Explain.
\
f’
[
35. On the coordinate axis below, the graphs of f , f ) , and f )) are offered. Identify which
is which and justify your choice.
240
CHAPTER 4. PROJECT A: GRAPHING
5
3
1
2
-4
-2
0
2
x
4
4
3
0
1
2
-1
-4
-2
00
2
x
1
4
-2
-1
-4
-2
-3
-2
00
-3
-1
-4
-2
2
x
4
-3
36. Earlier in this project, you described in writing the analytical relationship between the
¿rst derivative of a function and the extrema of that function. Now write a paragraph
that describes the relationship between the graph of the ¿rst derivative of a function
and the function itself. Use your work from the previous three problems to aid you.
Graphing Using Calculus, Part 2
In Search of Dips and Bumps
While the ¿rst derivative of a function gave us information concerning when a function
is increasing or decreasing, we still are missing a critical piece of information namely,
the basic “shape” of the curve while it is going up or coming down. For example, if f is
increasing from a f a to b f b we would like to know if it looks like
or
DID
EIE
EIE
EIE
or
DID
.
DID
GRAPHING USING CALCULUS, PART 2
241
The third case is a straight line, which we recognize from examination. The second
derivative will allow us to distinguish between the other two.
The Second Derivative and Concavity
37. Examine the graph of the function f that is shown on the ¿rst set of coordinate axes.
y
A
C
x
B
y
x
y
x
(a) On the graph of f , draw the tangent lines that pass through each of the relative
extrema.
(b) On the graph of f , choose a point to the left of A and draw the line that is
tangent to the graph while passing through your chosen point. Repeat for a
point between A and C, a point between C and B, and a point to the right of B.
(c) What do you notice about the slope of the tangent lines to f as you go from
points just to the left of A to points just to the right of A? As you go from
242
CHAPTER 4. PROJECT A: GRAPHING
points just to the left of B to points just to the right of B?
(d) On the set of axes below the graph of f , sketch the graph of f ) , taking care that
the x-values on the new graph line up with the x values of the original graph.
(e) On the set of axes below the graph of f ) , sketch the graph of f )) , taking care that
the x-values on the new graph line up with the x values of the original graph.
(f) Use your answers to (37a) - (37e) to help you ¿ll in what’s missing.
On intervals in which the second derivative of f is negative, the derivative f )
is
creasing and the function f itself is cupped downward. Moreover, the
relative extremum of f in this interval is a relative
.
On intervals in which the second derivative of f is positive, the derivative f )
is
and the function f itself is cupped
. Moreover, the
relative extremum of f in this interval is a relative minimum.
You have just described the graphical relationship between the second derivative of a
function and the function itself.
We call a function that is cupped downward concave down and a function that is cupped
upward concave up.
38. Rewrite the sentences in (37f), replacing “cupped” with “concave”.
GRAPHING USING CALCULUS, PART 2
243
Examine the ¿gure at the right. Notice that a function can be concave
down (or concave up) on an interval
without having a relative extremum
on that interval. Moreover, notice that
the sign of the second derivative tells
you whether the ¿rst derivative is increasing or decreasing, but it does not
tell you whether the ¿rst derivative is
positive or negative.
\
[
39. Sketch the graph of a function that is:
(a) Increasing and concave up
\
(b) Increasing and concave down
\
[
(c) Decreasing and concave up
\
[
(d) Decreasing and concave down
\
[
[
Checking Possible Points of InÀection (PI)
Because the sign of the second derivative indicates concavity of a given function, a sign
chart is a nice tool to use to check a function for changes in concavity. We’ll illustrate the
set up with a problem that we have seen several times.
40. Back to your ol’ pal, f x 3x 5 50x 3 135x.
244
CHAPTER 4. PROJECT A: GRAPHING
(a) Fill in the entries in the table and ¿nd all intervals where the graph of f is
concave up and where the graph of f is concave down. Please do your work
somewhere other than on the table.
T
T
possible PI
x
5
x
0
x
5
T
T
intervals:
* x 5
0x 5
test points:
1
))
f :
concavity:
concave down
(b) Why was it “extra work” to make our sign chart over all the reals?
If a function is continuous on an interval and if the function changes concavity at a
number c on the interval, then the point c f c is called a point of inÀection (or
sometimes, inÀection point).
At a value of x where a function changes concavity either the second derivative of the
function is zero or it is unde¿ned.
If a function f is not de¿ned at a point x a, is continuous on an open interval containing a, and has one concavity in some c a that is different from its one concavity
in some a b, then x a is called an inÀection number for f .
41. Draw a graph of a function whose second derivative is zero at the inÀection point,
and a graph of a function whose second derivative is unde¿ned at the inÀection point.
\
\
[
[
42. Suppose that g is a function for which g)) c 0 for some number c. Give an
example of a function that satis¿es this condition and for which:
GRAPHING USING CALCULUS, PART 2
245
(a) g has a relative maximum at x c
(b) g has a point of inÀection at x c
(c) g has neither a relative extremum nor an inÀection point at x c.
(d) From the preceding, does g )) c 0 yield any information? Explain.
43. Based on your recent experience, take a position on the following statement.
If a function f is twice differentiable, then f has a point of inÀection at x c
whenever f )) x 0 at x c.
Position (true or false):
Justi¿cation of your position:
The point of the work that we have just completed is summarized in the following:
246
CHAPTER 4. PROJECT A: GRAPHING
Finding values of x for which f )) x 0 or is unde¿ned provides you with candidates
for points of inÀection or inÀection numbers. You must check to see if the de¿nition is
satis¿ed, i.e., determine whether appropriate changes in concavity occur.
You also can use second derivatives to classify extrema. Namely, we have the:
Second Derivative Test: Let f )) exist on the interval [a b], c be a number such that
a c b, and f have a critical point at x c.
if f )) c
0, then the point c f c is a relative minimum
if f )) c 0, then the point c f c is a relative maximum
if f )) c 0, then the test yields no information.
44. Consider the function f t 5t 5 3t 3 .
(a) Classify the relative extrema of f .
(b) Which test did you use for part (a), and why? Could another test have been
used? Explain.
Take a well-deserved break now — stretch, get some air, and exercise something other
than your brain for a little while.
Pulling it All Together
The work done in this project, thus far, leads to the observation that information about the
relationship between the graph of a function and its derivatives can be used in at least two
ways. First, given a formula for a function, you can use information about its derivatives
to obtain a sketch that indicates the key or important features of its graph. Second, given
GRAPHING USING CALCULUS, PART 2
247
information about the derivatives of a function, you can sketch its graph without knowing
its formula.
Read SB pp. 202-203 now. Then do the problems that follow.
It is a good time to think about the last part of sketching a graph. There is one function
for which we already have collected most of the needed information. Let’s use that to think
about the ¿nal part of the process.
45. Once more, with feeling: consider f x 3x 5 50x 3 135x Identify all the inÀection points of this function, using the information gained in (40). Now, improve
upon the graph you drew for (30). (For purposes of study later, it might be useful to
write below all the information that you gathered on this function when you looked
at it for problems 23, 28, 30, and 40.)
Information
Improved Graph
46. For the function f x x 4 4x 3 :
(a) Determine whether f is even, odd, or neither.
248
CHAPTER 4. PROJECT A: GRAPHING
(b) Find the intercepts of the graph of f .
(c) Find and classify all relative extrema of f .
(d) Use the information from parts (a) - (c) to make a rough sketch of the graph of
f.
(e) Find all open intervals where f is concave up and all open intervals where f is
THE APEX
249
concave down. Determine all inÀection points of f .
(f) Use the information from (e) to modify the graph that you drew in (d).
Now, we want to incorporate all the parts of graphing functions that have been discussed
in this project.
The Apex
The process of discussing and sketching the graph of a function pulls together some intuition, your knowledge of graphing without calculus, and your newly developed insights
about graphing with calculus. The challenge is to be systematic and careful during the collection of information so that you will be able to make meaningful use of the information.
The following gives a complete description of what you are expected to do when asked to
Discuss and Sketch the graph of a function.
Given a function to graph, your discussion needs to include checking if the function is
even or odd, ¿nding all intercepts, ¿nding asymptotes whose equation is of degree less
than 2, looking for where the graph will be in relationship to the asymptotes, ¿nding
and classifying critical points, ¿nding inÀection points, and determining concavity of
the graph of f .
Your sketch needs to include labelled intercepts, asymptotes indicated with broken lines,
labelled relative minima, labelled relative maxima, and labelled points of inÀection: It
should depict a function that accurately and consistently reÀects the information obtained in the discussion.
Next, we want you to practice the entire process of discussing and sketching a function.
With the ¿rst part of the next problem, the outline indicates a format to follow. Note that we
250
CHAPTER 4. PROJECT A: GRAPHING
omitted showing the computations that gave the derivatives of the function. When you work
problems from scratch, you will need to show intermediate computations. Consequently,
for (b) - (d), attach extra pages to show your full treatment of those problems.
47. Discuss and sketch each of the following functions, i.e., determine whether the function is even or odd, ¿nd any intercepts and asymptotes of the graph of the function, ¿nd and classify all relative extrema, ¿nd all inÀection points and describe the
concavity on appropriate intervals. Use this information to sketch the graph of the
function. Organize your work carefully so that another person would be able to understand it without dif¿culty.
4x 13
27x 22
4x 2
c. f x 2
x 1
a. hx (a) For h x b. r t 12 2t 2 t 4
d. f x x 23
12x 3
4x 13
,
27x 22
i. Check for h being even or odd:
hx Therefore,
ii. Find intercepts:
hx 0 for
Therefore,
the intercepts are
and h0 127.
and
iii. Asymptotes:
hx * as
hx Hence, asymptotes are
.
. But using long division,
4
4x 1
x 7 2
.
27
x 4x 4
and x .
THE APEX
251
iv. Critical points and relative extrema: differentiating the function and simplifying the resulting expression yields
h ) x 4x 12 x 8
27x 23
This gives
crit numbers:
intervals:
test points:
h):
h:
classi¿cation:
as critical numbers.
rel
rel
rel
v. Check for concavity: the second derivative
h )) x gives
8x 1
9x 24
as possible points where concavity changes.
crit numbers for f ) :
intervals:
test points:
f )) :
concavity:
vi. Sketch the graph, indicating key points and obvious asymptotes.
252
CHAPTER 4. PROJECT A: GRAPHING
\
[
Notice that we have drawn number lines beneath the axes. These are used to indicate intervals where the function is increasing, decreasing, concave up, and concave
down. Filling in this information before sketching the function can make things a
little easier.
Now apply the process detailed in part (a) to the functions given in (b) - (d). Attach
extra pages with your carefully written up solutions.
In working (47), you may have noticed that putting all the information together in the
“sketch” part is a lot harder than doing the computations in the “discuss” part. Here are a
few problems to provide you with practice in assembling information.
48. Sketch the graph of a continuous function that has all of the following characteristics.
f 0 4 f 2 2 f 5 6
f ) x 0 if x 1 1
f )) x 0 if x 1 or x 4 1
f ) 0 0 f ) 2 0
f ) x 0 if x 1 1
f )) x 0 if x 2 1 or x
5
DIGGING DEEPER AND FINE TUNING
253
\
[
49. Sketch the graph of a function that has all the following characteristics.
f has only two x intercepts.
f has a root at x 0.
f has horizontal asymptotes, y 1 y 2.
f is continuous except for a vertical asymptote at x 3.
f is decreasing on 0 2 and 3 5.
f is increasing on * 0 2 3, and 5 *.
f is concave up on * 1 1 3, and 3 6.
f is concave down on 1 1 and 6 *.
\
[
Digging Deeper and Fine Tuning
For the problems remaining on this project, you will focus on various parts of discussing
and sketching the graph of a function and work towards developing deeper insights concerning properties of functions and their derivatives. Remember to take breaks as needed
and to feel free to discuss your work with other students taking MAT21A. On the other
hand, make sure that you assimilate information independently and that the completed
254
CHAPTER 4. PROJECT A: GRAPHING
work is your own.
50. The following two passages describe the growth and size of a population of N individuals, with parameter K (Hutchinson 1978). Sketch a graph of the function that
corresponds to each.
(a) “ the rate of increase rises slowly
to a maximum as N reaches K2 , and
then falls asymptotically to zero as N
approaches K .”
(b) “ the curve of a growing
population is at ¿rst rising slowly
but increasingly fast from any arbitrarily small starting population, inÀecting when N K2 and then ever more
slowly approaching the asymptote K .”
\
[
51. Write a letter to a friend explaining how you use calculus to sketch the graph of
a function as accurately and ef¿ciently as possible. You may use an example to
illustrate your work, but an example alone is insuf¿cient.
After straight lines, probably the simplest graphs arise from quadratics: The vertex and
a few points lead to a reasonably accurate picture of a parabola. Now we want to use
a simple quadratic f x xx 2 to explore the effect of incorporating a fractional
exponent.
DIGGING DEEPER AND FINE TUNING
255
52. On the set of axes provided, sketch a graph of f x x x 2 that you will be able
to use for reference in the problems that follow.
\
[
53. Now consider the function f x x x 2 13 .
(a) Find the critical numbers of f . For each, identify whether f has a horizontal
tangent, a vertical tangent, or whether f is unde¿ned at the critical number.
(b) Classify the relative extrema of f .
(c) Compute the second derivative of f . Are there values of x at which this cannot
256
CHAPTER 4. PROJECT A: GRAPHING
be done? Explain.
(d) Determine intervals in which f is concave down and intervals in which f is
concave up. Also, ¿nd all of the inÀection points and/or inÀection numbers for
f.
(e) Sketch the graph of f .
\
[
54. Repeat (a) - (e) for f x x x 2 23 . Attach your work on separate pages and
just show a labelled sketch of the function here.
DIGGING DEEPER AND FINE TUNING
257
\
[
55. Write a paragraph that describes and explains similarities and differences in the behavior of the functions from (52), (53), and (54).
56. Prove that the graph of a rational function has no vertical tangent lines.
57. Earlier in this project, we showed you a graph of f x 0003x 8 0009x 3 10
that was produced using Maple. Use calculus to critique Fig. 1 and Fig. 2 on page 209.
Some important details about the graph of the function are missing and, in fact, there
258
CHAPTER 4. PROJECT A: GRAPHING
are some misleading features suggested by the given ¿gures.
58. Sketch the graph of a function that has all of the following characteristics.
f is increasing on * 0 and 0 1
f is decreasing on 1 *
f is concave up on * 2, 1 0, and 2 *
f is concave down on 2 1 and 0 2
f has a horizontal asymptote, y 0
f has a vertical asymptote, x 0, and is continuous everywhere else
f has a tilted asymptote, y x and
f has only two x intercepts.
\
[
DIGGING DEEPER AND FINE TUNING
259
59. For the function f x x 13 x 323 , we can show that
f ) x x 1
.
313
x 23 x
(a) Find f )) .
(b) Use the function and its derivatives to ¿nd the intercepts, asymptotes, critical
points, relative extrema, and points of inÀection of f . Use this information to
sketch the graph of f .
60. It is October, 1996. You have been hired as an undergraduate teaching assistant for
MAT21A. Your boss, Professor Fermat, has asked you to design a function for him
to use as a graphing problem on an exam. He would like the function to have the
following characteristics. The function must:
have exactly one horizontal asymptote
have exactly two vertical asymptotes
be neither even nor odd
have at least two x-intercepts
have at least one relative extremum
260
CHAPTER 4. PROJECT A: GRAPHING
have at least one point of inÀection.
Find a formula for a function that will please Prof. Fermat. Sketch its graph and
convince the professor that it ¿ts the speci¿ed requirements.
Mathstory
Prior to the work of Newton and Leibniz, many mathematicians had worked on problems
such as maximization or minimization of a function. Pierre de Fermat was one such
person. A lawyer who practiced mathematics as a hobby, Fermat was the ¿rst person to
obtain a procedure for differentiating polynomials. He is best known, however, for his work
in the ¿eld of number theory — the study of properties and relationships among whole
numbers. The most infamous of his results came to be called Fermat’s Last Theorem.
This theorem states that no integral exponents satisfy x n y n z n for n 2. Fermat wrote
this theorem in the margin of a book, then added, “I have discovered a truly marvelous proof
of this, which however the margin is not large enough to contain” (Kline 1972). Fermat’s
proof was never found, and, in spite of efforts by numerous mathematicians, the theorem
remained unproven until recently. In June 1993, Andrew Wiles, using an array of
methods developed over the last 30 years, presented the outline of a proof in three onehour lectures. When he concluded, his mesmerized audience was said to have erupted
in applause (Cipra 1993). After the announcement of his proof, a glitch was found that
recently yielded to correction.
Partial List of Answers:
9. Both (a) and (b) have asymptote y 0.
11. y 3 The graph is above the asymptote for x large positive and below the asymptote
for x large positive.
x
12b. We know that
is negative for x 0.
x
38
15a. f x 3x 11 15b. y 3x 11
x 3
22. Without continuity of f ) , we wouldn’t be able to claim a value for f ) .
25f. We have the following: B is a relative minimum, C is a relative maximum, D is a
relative minimum, E is a relative and global maximum, and F is not an extremum.
LITERATURE CITED
261
28a. From the table: rel. max. at x 1, rel. min. at x 3 from f being odd: rel. min. at
x 1, rel. max. at x T
3.
T
T
T
252 55
252 55
33b. f is 0 for x 0, ,
the last four are approximately equal to
3
3
856 and 399.
T
T
44a. Points of inÀection at x 5, x 0, and x 5 44b. Since the function is odd,
the points of inÀection in * 0] will be at the negatives of the x values where the points
of inÀection occur in [0 *.
43. False. Your examples from (42) should serve to justify the position.
46. Your discussion should deduce: intercepts at 0 0 and 4 0 no asymptotes not even
or odd critical numbers at x 0 and x 3 rel. min. at 3 27 points of inÀection at
0 0 and 2 16.
53. Your discussion should lead to: intercepts at 0 0 and 2
not even
‚0 no asymptotes
t u13 3 3 1
3
a vertical
or odd critical numbers at x and x 2 rel. min. at
2
2 2 2
tangent at x 2 points of inÀection at 2 0 and 3 3.
57. Your work should lead you to discover that there is no relative extrema to the left of
x 0 and there is not a point of inÀection at x 0.
2
59a. f )) x 53
.
x x 343
Acknowledgments:
Preparation of this document was supported in part by the Regents of the University of California through the Undergraduate Instructional Improvement Program that is administered
by the Teaching Resources Center of the University of California, Davis. We thank Doyle
Cutler, Carole Hom, Lawrence Marx, Evelyn Silvia, John Thoo, and Neil Willits for their
contributions and comments on previous versions of this document. Any errors that remain
are the sole responsibility of the co-directors for UCDCaRP.
Literature Cited
Cipra, B. 1993. Fermat’s last theorem ¿nally yields. Science (261):32-33.
Hughes-Hallett, D., A. M. Gleason, S. P. Gordon, D. O. Lomen, D. Lovelock, W. G. McCallum, B. G. Osgood, A. Pasquale, J. Tecosky-Feldman,
J. B. Thrash, K. R. Thrash, and T. W. Tucker. 1993. Calculus. Preliminary
edition. John Wiley & Sons, Inc. New York.
Hutchinson, G. E. 1978. Population Ecology. Harvard University Press, Boston.
Kline, M. 1972. Mathematical Thought from Ancient to Modern Times. Oxford
University Press, New York.
262
CHAPTER 4. PROJECT A: GRAPHING
Stein, Sherman K. and Anthony Barcellos, 1992. Calculus and Analytic Geometry (5th Edition). McGraw-Hill, Inc., New York.