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 = x

- (a) Sketch the graph of y = x
^{2}. - (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))

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)

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) = kx