[Note: This was Exam 1 given
September 19, 1995, in Math 106.]
Instructions: Show all of your work and clearly
explain your answers.
This is particularly important on problems
with a numerical answer, to allow the possibility
of partial credit. No books are allowed
during the exam, but you may use your calculator.
Problem [1] (8 pts) Suppose the Big Muddy Water Co. charges its customers
$10.00 a month plus $2.00 per 100 gallons of monthly water usage.
(a) Write down a formula for the monthly cost
of water service, as a function of the number of gallons of water used.
(b) What is the domain of this cost function?
(c) What is the range of this cost function?
Answer:
(a) The monthly cost for g gallons is C(g) = 10 + 0.02g.
(b) The domain is: 0 <= g.
(c) The range is: 10 <= C.
Problem [2] (9 pts) The table gives some values for two functions, f(x) and g(x),
one of which is linear, the other exponential.
(a) Indicate which function is linear;
explain how you decided.
x f(x) g(x)
0.2 2.6 2.6
0.4 4.2 3.9
0.6 5.8 5.8
(b) Find the values f(0) and g(0) of each function
at 0; explain how you get your answers.
Answer:
(a) Since f(.4) - f(.2) = 1.6 = f(.6) - f(.4), f is linear.
(b) Since f increases 1.6 for each increase of 0.2 in x,
we have: f(0) = f(0.2) - 1.6 = 2.6 - 1.6 = 1.
Since g is multiplied by g(0.4)/g(0.2) = 3.9/2.6 = 1.5
for each increase of 0.2
in x, we see g(0) = g(0.2)/1.5 = 2.6/1.5.
Problem [3] (8 pts) The graph of y = x2 is shifted up
one unit, then right 2 units,
then reflected across the x-axis, to give the graph of y = f(x).
(a) Sketch the graph of y = x2.
(b) Sketch the graph of y = f(x).
(c) Fill in the blanks: f(x) = _____(x + _____)2 + _____
Answer:
(a) (b)
(c) f(x) = -(x + (-2))2 - 1
Problem [4] (7 pts) Suppose you take a 5 hour trip by car, stopping once,
to eat lunch. Let d = f(t) be the distance covered
during the trip as a function of time t, where t is the number of hours
since the trip started.
(a) Draw a possible graph for f(t).
(b) Indicate whether f(t) is invertible and explain why or why not.
Answer:
(a)
(b) Since different values of t (i.e., times during lunch
when you're stopped)
can give the same value for d, f(t) is
not invertible.
Problem [5] (7 pts) Bank A offers a savings account with an 8.3% annual interest
rate, compounded daily. Bank B offers a savings account with an 8.4% annual interest
rate, compounded twice a year, and Bank C offers a savings account with an 8.5% annual interest
rate, compounded continuously. Which bank offers the best deal and which offers
the worst deal? Explain how you decided.
Answer:
Bank C has the largest rate compounded the fastest, so it is the best.
To compare Banks A and B, compute the annual yields.
For Bank A we have (1 + 0.083/365)365 = 1.0865
which gives 8.65% per year.
For Bank B we have (1 + 0.084/2)2 = 1.0858 or 8.58% per year.
Thus Bank B is the worst.
Problem [6] (11 pts) A sinusoidal function y = f(x)
and a polynomial y = g(x) are graphed below.
(a) Find the period of f(x).
(b) Find the amplitude of f(x).
(c) Give a formula for f(x).
(d) Give a formula for g(x).
Answer:
(a) The period is 4-2 = 2.
(b) The amplitude is (5-2)/2 = 1.5.
(c) The formula is f(x) = 1.5sin(Pi*x - Pi/2) + 3.5; just in case
you should check your answer on the calculator.
(d) Here we have g(x) = kxn(x-2)m,
where n > 1 is even and m > 1 is odd,
and where k is chosen so g(1) = 2. Thus
g(x) = -2x2(x-2)3 works; as a check you can
graph it on the calculator.
(b) 
