Part 1 Math 1400 - Test #2A Part 2 Fall 2009
Row
Part 2: 80
Name
Decimal answers should be rounded to three places to the right of the decimal point.
Word problems should include units in your answers.
Ql:
The price-demand and total cost functions for the production and sale of x sweatshirts are
p = 36 — 0.03x
10
C(x) = 6x + 750
A. What is the total revenue function?
e.o3,2--
lz • 10
R(x)
Poo
What is the total profit function?
p(x),
J&K)-
C.G 3 )cL-
=
3o le - o. o 3 X_ 2' -
7--
X
C- cx)
30)6-
b.02)C-2-----.
Y-+ SV)
5o
50
Use only the values below and marginal analysis to find the answers to 1- 3
P(250) = 4,875
P'(250) = 15
P(250) = 19.50
P'(250) = -.018
Approximate the profit from the sale of the 251 5t sweatshirt. Answer: Approximate the average profit from the sale of 251 sweatshirts. Answer: /61.
z
(-,o /or,
115-0
3. Approximate the total profit from the sale of 255 sweatshirts. Answer: p SS)
1) /5
y. 9 5- 0
? C2 So) -4- P'(2. so) • "C
g 3-5 4.
Q2: The function Z(t) models the ozone level Z, measured in parts per billion (ppb), on a hot summer day,
where t is time in hours and
values:
t = 0 corresponds to 9 am. Write a single statement that interprets these
Z(3) = 107
Z'(3) = 6
/ 63
Ltb,,
8
Page 2A of 4
A. A cesium isotope has a half-life of 30 years. What is the continuous compound rate of decay?
Q = Qo ert
e-
r(3
.3br-
..0,"
—, 0.2
r
3o r
311
30
Jam"`-
2-
r
30
B. Circle all true statements about the function f (t) =
IC) The domain of f (t) is all Real numbers.
e-t
f ' (t) < 0 on the entire domain of f (t).
3. f "(t) < 0 on the entire domain of f (t).
f (t) = 0
Related Rates: A block of ice, in the shape of a cube, is hung from a hook in a room. If the ice is melting
uniformly, so that the shape of the block is always a cube, both the length of a side of the cube and the
volume of the cube are changing.
If the length of a side is decreasing at a rate of 2 inches per hour, at what rate is
the volume of the cube changing when each side is 10 inches long?
What is the equation that relates the volume,
V,
and s, the length of each side?
3
V=
Equation: (If you don't know, I will sell it to you for two points).
Rewrite the statement of the problem using correct notation:
d 5
//),1.
Given d76
C. Solve.
find
dt
when S =
i°
S
V
v
(000 .f,.„
dv
=
Page 3A of 4
Q5: Given the information about f (x), f (x) and f " (x) in the three charts below,
sketch a graph of the function
f (x), which is continuous on (-00, 00).
x
-4
-2
0
2
4
f (x)
-2
3
-1
2
4
f ' (x) ++
0—
f (x)
-2
—
ND
0
++
0
+
+
2
f " (x) — — — —
ND — — 0 + +
0
•
f (x)
0
2
+
10
Page 4A of 4
Q6: Answer the questions about the unknown function f (x) whose first and second derivatives are :
(x) = 4x 3 — 12x 2f " (x) = 12x 2 — 24x
o
o
3)
2 X(x- 2-)
x
X
3
A. Compete a sign chart for f (x) = 4x 3 — 12x2
f
12
X---- 2-
B. Compete a sign chart for f " (x) =12x 2 — 24x
-4—
--f-
,---,
3
2.
‘___}
C - 1)
3
t)
C. Using your information in A & B, answer the following. Your answers must be consistent with A & B.
At what value(s) of x does f
On what interval(s) is
(x) have a local maximum?
(3,
f (x) increasing?
At what value(s) of x does f
(x) have a point of inflection?
On what interval(s) is f (x) concave up ?
(
-
o
0)
U
)
cc
Part 1
Math 1400 Test #213 Part 2
Fall 2009
Row
Part 2: 80
Name
Decimal answers should be rounded to three places to the right of the decimal point.
Word problems should include units in your answers.
Ql:
The price-demand and total cost functions for the production and sale of
x sweatshirts are
C (x) = 6x + 750
p = 36 — 0.03x
A. What is the total revenue function?
. d 3 "X:2-
S
-to
R(x) = 10
gcx)
PCx)
B. What is the total profit function?
3oy-
P(x) = 3 0- D 03 )c l" - 4-5o
C. Use only the values below and
P(300) = 5,550
.ox-'*- (40 )c+-q-s,f))
marginal analysis to find the answers to 1 -3
P(300) = 18.50
P'(300) = 12
P'(300) = -.022
12
Approximate the profit from the sale of the 301 5 ' sweatshirt. Answer: Approximate the average profit from the sale of 301 sweatshirts. Answer: p( 301 )
3 D
I
13 (3c,c) 4- P1(3°6)
- . 0 22I
'$,
3. Approximate the total profit from the sale of 305 sweatshirts. Answer: r
a
-
$
Le)
.5
P Ca 0 C,)
c.)
5-
Q2: The function Z (t) models the ozone level Z, measured in parts per billion (ppb), on a hot summer day,
where t is time in hours and
values:
t = 0 corresponds to 9 am. Write a single statement that interprets these
Z'(3) = 4
Z(4) = 112
/ m 44c-
a--t
io 7. A
/241--t_c_
1
p
b
pizA,
64,...,2
.
8
Page 2B of 4
Q3: A. A cesium isotope has a half-life of 25 years. What is the continuous compound rate of decay?
Q = Qoert
Qe, e-
ci70
e • 2 5-
.2 5
10
r
r-
c,,2.
. a .2
9- 3-
I-
r
B. Circle all true statements about the function f (t) = e-t
0
The domain of f
2. f ' ( t)
(t) is all Real numbers.
> 0 on the entire domain of f (t).
f "(t) >
0 on the entire domain of
f (t).
f(t) = co
Q4: Related Rates: A block of ice, in the shape of a cube, is hung from a hook in a room. If the ice is melting
uniformly, so that the shape of the block is always a cube, both the length of a side of the cube and the
volume of the cube are changing.
If the length of a side is decreasing at a rate of 2 inches per hour, at what rate is
the volume of the cube changing when each side is 10 inches long?
What is the equation that relates the volume,
V,
and s, the length of each side?
3
Equation: (If you don't know, I will sell it to you for two points).
V- s
Rewrite the statement of the problem using correct notation:
dS
_ 1/\
find Given C. Solve.
v
ge when S 7-)D
v s'
= 3 s •
‘,00(n(13
a t
E
citE
-3(10)1=
(066
Page 3B of 4
Q5: Given the information about f (x), f ' (x) and f " (x) in the three charts below,
sketch a graph of the function f (x), which is continuous on (-00, CO).
x
-4
-2
0
2
4
f (x)
4
2
-1
3
-1
f ' (x) --
0 — — ND + + 0 — —
f (x)
-2
f " (x) ++
0-- ND — - 0
•
f (x)
10
n
-2
0
2
n
0
A Y
4
3
x
-4
-3
2
-2
—2
—3
>
\4
,
Page 4B of 4
Q6: Answer the questions about the unknown function f (x) whose first and second derivatives are :
f'(x) =
f (x) =
4x 3 — 12x2
0
12x 2 — 24x
1 2,x (x-2)
O
3)
12
0 X 7:
= O, 3
A. Compete a sign chart for f
' (x) = 4x 3 — 12x2
B. Compete a sign chart for
f ' (x) = 12x 2 — 24x
TV
C- 3)
-I-
(-2)
3(0 CO
3
C. Using your information in A & B, answer the following. Your answers must be consistent with A & B.
At what value(s) of x does f
On what interval(s) is f
(x) decreasing? At what value(s) of x does f
On what interval(s) is f
(x) have a local minimum? (.
c
JC
3)
(x) have a point of inflection?
a, )
(x) concave down ? R
ell.
-
3
(—c 0 , o) U ( 0 , 3)
,