164 4. CURVE SKETCHING
Example 4.48 Make a full report and sketch for
y D
x
2
9
x
:
Solution:
e roots are easy, x D ˙3; the vertical asymptote is clearly x D 0; and, since the top is
higher degree, there are no horizontal asymptotes. Notice that:
y D
x
2
9
x
D
x
2
x
9
x
D x
9
x
From this, it is easy to see that f
0
.x/ D 1 C
9
x
2
, which means there are no critical points, and
f
0
.x/ > 0 where it exists. is makes the first derivative sign chart
.1/ C C C .0/ C C C .1/
e second derivative is f
00
.x/ D
18
x
3
, so there are no inflection points. Plugging in f
00
.˙1/ D
18, we get a second derivative sign chart like this:
.1/ C C C .0/ .1/
ese sign charts make the various ranges obvious and we get:
Roots: x D ˙3
Vertical asymptotes: x=0
Horizontal asymptotes: none
Critical points: none
Increasing on:
.
1
; 0/
[
.0;
1
/
Decreasing on: never
Inflection points: none
Concave up on: .1; 0/
Concave down on: .0; 1/
e corresponding sketch looks like this:
4.3. THE FULL REPORT FOR CURVE SKETCHING 165
8
-8
-5 5
y D
x
2
9
x
2
Alert students will notice that a non-standard asymptote appears in this picture. Recall that
y D x
9
x
. When jxj is large, y D x something very small, which means that as jxj grows
large, the function gets very close to the line
y
D
x
. is is called a
diagonal asymptote
.
˙
Knowledge Box 4.10
Diagonal asymptotes
If f .x/ D ax C b C g.x/ and
lim
x!˙1
g.x/ D 0;
then the graph of f .x/ approaches the line y D ax C b as jxj gets large.
166 4. CURVE SKETCHING
PROBLEMS
Problem 4.49 Make a full report with a sketch for each of the following functions.
1. f .x/ D x
3
5x
2. g.x/ D x
3
6x
2
C 11x 6
3. h.x/ D
x
3
2x
2
x C 2
x
4. r.x/ D
x
2
9
x
2
1
5. s.x/ D x C
1
x 1
C
1
x C 1
6. q.x/ D ln.x
2
C 5/
7. a.x/ D tan
1
.x
2
/
8. b.x/ D
x
2
1
3 x
2
s.x/
5
-5
-5 5
Problem 4.50 Based on the sketch above, do your best to fill out a full report.
t.x/
5
-1
-5 5
Problem 4.51 Based on the sketch above, do your best to fill out a full report.
4.3. THE FULL REPORT FOR CURVE SKETCHING 167
q.x/
5
-5
-5 5
Problem 4.52 Based on the sketch above, do your best to fill out a full report.
Problem 4.53 Make a full report with a sketch for:
f .x/ D
3x
x
2
C 1
Problem 4.54 Find a function that is concave down everywhere that it exists and has a range
of .1; 1/.
a.x/
5
-1
-5 5
Problem 4.55 Based on the sketch above, do your best to fill out a full report.
168 4. CURVE SKETCHING
Problem 4.56 Make a full report with a sketch for each of the following functions. Include
diagonal asymptotes if any.
1. f .x/ D
x
2
C 2x C 1
x 1
2. g.x/ D
2x
2
C 3x C 1
x
3. h.x/ D 5x 3 C
1
x
2
4. r.x/ D
x
3
x
2
C 2
5. s.x/ D
x
2
x
2
C 1
6. q.x/ D
x
2
x
2
4
b.x/
5
-5
-5
6
Problem 4.57 Based on the sketch above, do your best to fill out a full report.
Problem 4.58 Make a full report with a sketch for
f .x/ D
e
x
1 C e
x
Problem 4.59 Find a function that has two vertical asymptotes and approaches the diagonal
asymptote y D 2x C 1.
Problem 4.60 Find a function with two horizontal and two vertical asymptotes and demon-
strate that your solution is correct.
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