M106 Practice Exam 1 Solutions

[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?

(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.

(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 + _____

(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.

(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.