MTH 151
Exam 1
Fall 2013
Show all work in a neat and organized fashion.
Clearly indicate your answers.
100 points possible.
Problems 4 and 8 are intentionally omitted.
Problems 1 and 6 are graphical problems on separate pages.
Seven problems, 15 points each, maximum possible score 105 out of 100.
2. (15 pts.) Let
f(x)=
⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩
x2−5,
x ≤ 1
x+2,
1 < x < 3
7,
x=3
8−x,
x > 3.
State each of the following (or write DNE, if it does not exist).
(a) limx→1− f(x)
(b) limx→1+ f(x)
(c) limx→1 f(x)
(d) limx→3− f(x)
(e) limx→3+ f(x)
(f) limx→3 f(x)
3. (15 pts.) Evaluate this
limit symbolically (algebraically), neatly showing all
significant algebraic steps.
lim
x→ −8
1
8
+
1
x
8+x
5. (15 pts.) Find a and b such that f is continuous everywhere.
f(x)=
⎧ ⎪ ⎨
⎪ ⎩
8−x2,
x < 2
ax+b,
2 ≤ x ≤ 6
x2−4x−3,
x > 6.
7. (15 pts.) The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash
is set off. The data in the table describe the charge Q remaining (in microcoulombs, or μC) at time t, where
t is the number of seconds after the flash is set off.
t (seconds)
0.00
0.02
0.04
0.06
0.08
0.10
Q(t) (μC)
100.00
81.87
67.03
54.88
44.93
36.76
(a) Find the average rate at which the charge Q changed from t=0.00 to t=0.06. (You do not have to write a sentence.)
Include the units in your answer.
(b) Use the data in the table (without graphing) to estimate the rate at which the charge was changing
at t=0.06. (You do not have to write a sentence.) Include the units in your answer.
(c) Use the data in the table (without graphing) to estimate the rate at which the charge was changing
at t=0.025. (You do not have to write a sentence.) Include the units in your answer.
9. (15 pts.) Let
f(x)=
1
√
5x
.
Find f′(x) by using the definition of
derivative
f′(x) =
lim
h→0
f(x+h)−f(x)
h
.
Neatly show all significant algebraic steps. You may do parts (a), (b), and (c) separately, if you label them.