Math 220, T Pilachowski, Fall 2013
Fall 2013 MATH 220 – TEST 1(B) (0.3 – 2.7) [Pilachowski] ANSWER KEY
1.a. (14 points) Use calculus to determine whether the function xxxy 3
3
123 −+= is increasing or
decreasing at x = –1.
( ) ( ) ( )
432131211,32 2
2−=−−=−−+−=−
′
−+=
′yxxy , so the function is decreasing.
1.b. (12 points) During a thunderstorm, the rise and fall of the water level (in feet) in a drainage pond
follows the formula
( )
2
2
15
2
π
++−= t
ttm where t is time in hours. Find an equation to express how
quickly the water level is changing. Don’t forget to put the correct label on your answer.
“How quickly” refers to a rate of change:
2
15
2+−=
′tm feet per hour.
Note that
π
/2 is a constant, so its derivative equals 0.
1.c. (14 points) Given 2
1
32
x
xxy +−= , find y
′
.
Rewrite: 2
2
2
1
32
1
32
−
+−=+−= xxx
x
xxy
. Then 3
2
2
3
22
1−
−−−==
′xxy .
2.a. (20 points) Given the function
( )
x
xxh 9
16 += , with a domain limited to x > 0,
i. Find both coordinates of any x-intercepts. 22
16
9
9160
9
160 xx
x
x=−⇒+=⇒+= which
has no real solution, so there are no x-intercepts.
ii. Find the x-coordinate of any relative extrema (within the limited domain x > 0), stating whether each
is a maximum or minimum. Use calculus tools, show your steps, and verify your conclusion.
4
3
9160
9
16916,916 2
2
21 ±=⇒=⇒=−=−=
′
+= −− xx
x
xyxxy
The negative value is not in the domain and is
therefore extraneous.
interval
4
3
0<< x
4
3
=x x<
4
3
sign of h
′
( – ) 0 ( + )
You could also have shown that the second
derivative > 0 at this x-value.
iii. Find the x-coordinate of any points of inflection (within the limited domain x > 0). Use calculus
tools, show your steps, and verify your conclusion.
( ) ( )
3
32 18
29,916
x
xyxy =−∗−=
′′
−=
′−− which never = 0, so there are no points of inflection.
iv. Find the equation of any asymptotes. Show how you found your answers.
x in denominator
⇒
vertical asymptote is x = 0; as 016,
+
→
∞
→
xyx , slant asymptote is y = 16x
241212
3
4
9
4
3
16 =+=
+
=y
At
24,
4
3 the graph of h(x) has a minimum.
