MTH 151
Exam 1
Fall 2011
Show all work in a neat and organized fashion.
Clearly indicate your answers.
100 points possible.
cos(s+t)=cos s cos t−sin s sin t
cos(s−t)=cos s cos t+sin s sin t
sin(s+t)=sin s cos t+cos s sin t
sin(s−t)=sin s cos t−cos s sin t
1. (7 pts.) Find the domain of the function.
f(x)=
1
√
1−4x
2. (6 pts.) Classify each function as a power function, root function,
polynomial (state its degree), rational function, algebraic function, trigonometric function,
exponential function, or logarithmic function.
(a)
h(x)=
2x3
1−x2
(b)
u(t)=1−1.1t+2.54 t2
(c)
v(t)=5t
3. (7 pts.) A weight attached to a vertical spring is bouncing up and down. Its distance from the floor
is given by a transformed sine or cosine graph. A stopwatch is started at time t=0 when the weight is at its high point,
80 cm above the floor.
At a time 1.6 seconds later, the weight reaches its next low point, 50 cm above the floor. Find a function that models
the distance from the floor to the weight as a function of time.
4. (7 pts.) On an attached page, the graph of h is given. Use it to graph the following function.
y=h(
1
2
x)
You may draw your graph here or on the attached page.
5. (7 pts.) Given F(x)=cos(sin2 x), find (nontrivial) functions f, g, and h such that F=f°g°h.
6. (7 pts.) A tank holds 500 gallons of water, which drains from the bottom of the tank in
half an hour. The values in the table show the volume V(t) of water remaining in the tank
(in gallons) after t minutes.
t (min)
5
10
15
20
25
30
V(t) (gal)
347
222
125
56
14
0
(a) Find the average rate at which the volume V changed from t=10 to t=30. Include the units in your answer.
(b) Use the data in the table (without graphing) to estimate the rate at which the volume was changing
at t=17.5. Include the units in your answer.
7. (12 pts.) On the attached page, the graph of a function g is given.
State each of the following (or write DNE, if appropriate).
(a) limx→2− g(x)
(b) limx→−4+ g(x)
(c) g(2)
(d) limx→7+ g(x)
(e) limx→7− g(x)
(f) The equation(s) of the vertical asymptote(s).
8. (7 pts.) Let
g(x)=
|x−1|
x2+5x−6
.
(a) Find limx→ 1+ g(x).
(b) Find limx→ 1− g(x).
(c) Does limx→ 1 g(x) exist?
9. (7 pts.) Is f continuous at a? Show the work that leads to your answer, using the mathematical
definition of continuous.
f(x) = 15−x2, if x < 3
4x−6, if x ≥ 3
a=3
10. (7 pts.) Prove that there is a root of the given equation in the specified interval.
x3+5x−8=0 (1,2)
11. (7 pts.) If secx=[5/3] and siny=[1/4], where x and y lie between 0 and π/2,
evaluate the expression.
cos(x+y)
12. (6 pts.) The limit represents the derivative of some function f at some number a. State such an f and a in each case.
(a)
lim
x→4
3x−81
x−4
(b)
lim
h→0
(5+h)3−125
h
13. (6 pts.) Use the given graph on the attached page to estimate the value of each derivative.
(a) f′(0)
(b) f′(2)
14. (7 pts.) Find the derivative of the function f using the definition of derivative.
f(x)=
√
7−x
, f′(x)=
lim
h→0
f(x+h)−f(x)
h
Show important algebraic steps neatly to justify your answer.
Hopefully you will find that
f′(x)=
−1
2
√
7−x
.
If your answer doesn't work out, don't fake it (because that would
be an additional error); just do your best.