MATH 180 FALL 1995 10:00 SECTION FONG

ASSIGNMENT #2

1.3, #3. The graph should have the following features: i) a first part which is a line segment of positive slope since “the quantity in the blood increases linearly”; ii) a second part which is part of an exponential decay curve since “the quantity then decays exponentially”. So this second part is decreasing and concave-up, and approaches the x-axis asymptotically from above; iii) whether the graph starts at (0, 0) depends on the initial amount of drug already in the blood when the injection is made, so there is no single right answer here.

1.3, #5. For the inputs t = 1, 2, 3, 4, 5, 6, the changes in the values of g(t) are 1, 2, 3, 4, 5; the changes in the values of h(t) are 10, 9, 8, 7, 6; and the changes in the values of k(t) are equal to the constant .3. So g(t) best fits the curve (c) since the changes are increasing, h(t) best fits the curve (b) since the changes are decreasing, and k(t) fits the curve (a) since the changes are constant.

1.3, #7. h, f, g are given to be exponential functions. So consider the ratios h(s+1)/h(s), f(s + 1)/f(s), g(s + 1)/g(s) for the given inputs. The ratios for each function should equal the base of the exponential function (up to errors given by approximations). We find that h(s) = c(1.03)^{s}, f(s) = a(1.1)^{s}, g(s) = b(1.05)^{s}. (The constants a, b, c can also be found by plugging in s = 2 and using the given values of h, f, g at s = 2.)

1.3, #9. The ratios g(t + 1)/g(t) in this case are all .8. So g behaves like an exponential function C(.8)^{t}. To determine t, we can use g(0) = 5.50 to see that C = 5.50.

1.3, #11. The exponential function y = Ca^{x} has two parameters C and a. In principle knowing two points on the graph of the function should determine C and a. In this case the points are (1, 6) and (2, 18), giving 6 = Ca and 18 = Ca^{2}. These two equalities force a = 3 and C = 6,

Comments on #5 and #7: It is a good principle to try simple calculations before doing complicated ones. So in #5 one would consider differences of values to see if this data is enough to settle the problem. In fact, this is the case. In #7 we are given that the three functions are exponential functions. So there is no point in looking at differences of values, and one should look immediately at ratios of values.

A general comment: An answer without explanations is not an adequate answer. No credit (on homework, quizzes, or exams) will be given for such answers. Write enough so that the reader can follow your reasons for the answer.