Download MATH 115 Test 1 (Sec: R

MATH 115 Test 1 (Sec: R - 1.4) Year: 20161
Name: KEY
Sec Number:
Answer the following questions to the best of your ability showing all appropriate work to obtain full
credit. Questions that involve word problems should be answered using a sentence, and appropriate
units provided, in order to receive full credit.
1. (4 pts) Write the following expression as an equivalent expression in the form xn .
q
√
8
x8
8 1
8
2
=⇒ x 8
= x 16 = x0.5
2. (4 pts) Use factoring to solve the polynomial equation:
2x3 + 6x2 − 20x = 0
=⇒
2x(x2 + 3x − 10) = 0
=⇒
2x(x + 5)(x − 2) = 0
=⇒
roots x = 0, x = −5, and x = 2
3. (4 pts) Solve the following equation:
17
2x2/5
=
6
x1/2
9
10
=⇒ 102 = 2x1/2 x2/5 =⇒ 102 = 2x 10 =⇒ x = (51) 9 ≈ 78.94056
1
4032
4. (6 pts) Use the following figure to evaluate the given limits. Write D.N.E. for a limit that
does not exist.
6
4
2
-10
-5
5
10
-2
-4
-6
lim f (x) =
4
lim
0.0
x→−8−
f (x) =
x→−4.0−
lim f (x) =
lim f (x) =
lim
x→−4.0+
2
x→0−
x→4−
lim f (x) =
x→−8+
3
−2
f (x) =
0.0
lim f (x) =
2
x→0+
lim f (x) =
x→4+
lim f (x) =
x→−8
−1
5. (4 pts) Simplify the following expression:
(x8 )3
x3 x5
=⇒
x24
= x24−8 = x16
x8
D.N.E.
lim f (x) =
0.0
x→−4.0
lim f (x) =
x→0
lim f (x) =
x→4
2
D.N.E.
6. (5 pts) Calculate the following limits:
(a) lim
x→6+
13
=
x−6
∞
x−7
=
x→7 x2 − 7x
1
≈ 0.143
7
(b) lim
(c) lim
y→9
9−y
√ =
3− y
6
(a + 3)5
=
a→−5 a + 1
8 ≈ 8.0
80x
lim
=
−
6 − 4x
x→( 64 )
∞
(d) lim
(e)
7. (10 pts) Find the instantaneous rate of change function f 0 (x) for:
f (x) = 2x2 + 7x + 4
Using the definition
f 0 (x) =
=
=
=
=
f (x + h) − f (x)
h→0
h
2x2 + 22xh + 2h2 + 7x + 7h + 4 − 2x2 − 7x − 4
lim
h→0
h
4xh + 2h2 + 7h
lim
h→0
h
h
lim (4x + 2h + 7)
h→0 h
4x + 7
lim
Thus,
f 0 (x) = 4x + 7
8. (8 pts) Bobby decides to mow lawns to earn money. The initial cost of his lawn mower is
$300. Gasoline and maintenance costs are $5 per lawn.
• Find a cost function C(x) for the total cost of mowing x lawns.
The cost function for mowing x lawns is:
C(x) = 5 x + 300
• Bobby determines that the total-profit for the lawn-mowing business is given by
P (x) = 11x − 300.
Find a function for the total revenue for mowing x lawns. How much does Bobby charge
per lawn?
As profit is revenue minus cost the revenue function can be found by adding P (x) and
C(x):
R(x) = 16 x
Bobby is charging $16 per lawn.
• How many lawns must Bobby mow before he begins to make a profit?
11x − 300 = 0 −→ x =
300
= 27.27
11
Bobby will make a profit once he has mowed 28 lawns.
• What is the average rate of change in Bobby’s revenue per lawn?
Bobby’s revenue is increasing at a rate of $16 per lawn.
9. (4 pts) Let
g(x) =
1x − 6,
3x + 2,
x < −9
x ≥ −9
Use the definition of continuity to decide if g(x) is continuous at x = −9. Note you must use
the limit definition in order to receive full credit.
(a) g(−9) = −25. Thus, we can evaluate g at x = −9.
(b) limx→−9− g(x) = −15 and limx→−9+ g(x) = −25. Thus, the limit does not exist.
(c) There is no need to check further.
Not Continuou