[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  (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?
(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  (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.
(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  (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 + _____
(c) f(x) = -(x + (-2))2 - 1
Problem  (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.
(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
Problem  (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.
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  (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).
(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.