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) 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) 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) Graph goes here. (b) Graph goes here.
(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) Graph goes here.
(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.

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. Graph goes here. Graph goes here.

(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.