MTH 121 — Fall — 2007
Essex County College — Division of Mathematics
WebAssign # 4.51 — Created November 6, 2007
Name:
Signature:
This should be completed to the best of your ability and handed in before 11/16/2007. Show
all work clearly and in order, and box your final answers. Justify your answers algebraically
whenever possible.
1. Given that
x
.
−8
Here you are given both the first and second derivative to be used to analyze the supplied
graph of f .
2 4 + x3
6x2 16 + x3
0
00
f (x) = −
and f (x) =
(x3 − 8)2
(x3 − 8)3
f (x) =
x3
2
1
-4
-3
-2
-1
0
1
2
-1
-2
Figure 1: Partial Graph of f
(a) What is f ’s domain.
Answer:
R, x 6= 2
1
or
(−∞, 2) ∪ (2, ∞)
This document was prepared by Ron Bannon using LATEX 2ε .
1
3
(b) What is f ’s range.
Answer:
R
(c) Use limits to determine the vertical asymptote.
Answer:
lim f (x) = +∞
⇒
x=2
lim f (x) = −∞
⇒
x=2
x→2+
and
x→2−
(d) Use limits to determine the horizontal asymptote.
Answer:
lim f (x) = 0+
x→∞
⇒
y=0
and
lim f (x) = 0+
x→−∞
⇒
y=0
(e) List any intercepts. They should be in point form.
Answer:
(0, 0)
(f) Use the supplied derivative to determine the intervals where f is increasing.
Answer: You need to identify the critical numbers
and then analyze the sign of the
√
first derivative. The critical numbers are: − 3 4 and 2. The first derivative is positive
on
√
3
−∞, − 4
and therefore f is increasing on this interval.
(g) Use the supplied derivative to determine the intervals where f is decreasing.
Answer: You need to identify the critical numbers
and then analyze the sign of the
√
3
first derivative. The critical numbers are: − 4 and 2. The first derivative is negative
on
√
3
− 4, 2 ; (2, ∞)
and therefore f is decresing on these interval.
2
(h) List any local maximums. They should be in point form.
Answer: Using the graph and the above analysis.
√
√ 3
3
− 4, f − 4
or
!
√
3
4
− 4,
4
√
3
(i) List any local minimums. They should be in point form.
Answer: None .
(j) Use the supplied second derivative to determine the intervals where f is concave-up.
Answer: You need to identify the critical numbers
and then analyze the sign of the
√
3
second derivative. The critical numbers are: −2 2, 0 and 2. The second derivative
is positive on
√
3
−∞, −2 2 ; (2, ∞)
and therefore f is concave-up on these interval.
(k) Use the supplied second derivative to determine the intervals where f is concave-down.
Answer: You need to identify the critical numbers
√ and then analyze the sign of the
second derivative. The critical numbers are: −2 3 2, 0 and 2. The second derivative
is negative on
√
3
−2 2, 2
and therefore f is concave-down on this interval.
(l) List any points points of inflection. They should be in point form.
Answer: Using the graph and the above analysis.
√
3
√
3
−2 2, f −2 2
or
!
√
3
2
−2 2,
12
√
3
2. Given that
f (x) = esin x .
Here you are given both the first and second derivative to be used to analyze the supplied
graph of f .
f 0 (x) = esin x cos x and f 00 (x) = esin x cos2 x − sin x
(a) What is f ’s domain.
Answer:
R
or
3
(−∞, ∞)
3
2
1
-1
0
1
2
3
4
Figure 2: Partial Graph of f
(b) What is f ’s range.
Answer: You should do the analysis to see what the global extrema are first.
−1 e , e
(c) For each indicated point on f , determine the exact coordinates and what it is by
name.
Answer: Using the derivatives and the graph. Here I am listing the points left to
right as they appear on the graph.
First Point: Global and local minimum , −π/2, e−1 .
Second Point: y-intercept , (0, 1) .
Third Point: Okay, this is a tough one. The easy part is knowing that this is a
point-of-inflection. The tough part is finding it,
√
arcsin
5−1
2
√
!
, f
arcsin
5−1
2
Fourth Point: Global and local maximum , (π/2, e) .
4
!!!
.
Fifth Point: Okay, this is another tough one. The easy part is knowing that this is
a point-of-inflection. The tough part is finding it,
√
π − arcsin
5−1
2
√
!
, f
arcsin π −
5−1
2
!!!
.
3. Given that
f (x) = x5/3 − 15x2/3 .
Here you are given both the first and second derivative to be used to analyze the supplied
graph of f .
10x + 30
5x − 30
√
√
f 0 (x) =
and f 00 (x) =
3
33x
9 x4
Figure 3: Partial Graph of f
(a) What is f ’s domain.
Answer:
R
or
(−∞, ∞)
R
or
(−∞, ∞)
(b) What is f ’s range.
Answer:
5
(c) For each indicated point on f , determine the exact coordinates and what it is by
name.
Answer: Using the derivatives and the graph. Here I am listing the points left to
right as they appear on the graph.
First Point: Point-of-inflection , (−3, f (−3)) .
Second Point: x-intercept, y-intercept, local maximum , (0, 0) .
Third Point: Local minimum , (6, f (6)) .
6