MATH 180 FALL 1995

ASSIGNMENT #3

1.3, #13. One possible function with the given graph is the exponential function y = 4 − Ca^{x}. Note (0, 0) and (1, 2) are points on the graph. So plugging these values into y = 4 − Ca^{x} gives C = 4 and a = \frac{1}{2}.

1.4, #2. The graphs of x^{2/3} and x^{1/2} cross at (1, 1), so a sketch of the two should contain a region about this point. x^{2/3} dominates x^{1/2} as x → ∞. The general fact: If Cx^{p} and Dx^{q} are power functions such that C, D, p, q are positive and p < q, then Dx^{q} dominates Cx^{p} as x → ∞.

1.4, #17. The three functions are of the form ab^{t}, at^{2}, and bt^{3}. The exponential function can be detected by computing ratios of consecutive values. For h(t), these ratios are

\frac{3.06}{2.04} = \frac{4.59}{3.06} = \frac{6.89}{4.59} = \frac{10.33}{6.89} = \frac{15.49}{10.33} = 1.50

up to 2 decimal places. So h(t) is exponential with base 1.5. Since h(0) = 2.04, h(t) = 2.04(1.5)^{t}. The remaining two functions are power functions. A power function k(t) = at^{p} has the following property:

\frac{k(t_{i+1})}{k(t_{i})} = \frac{at_{i+1}^{p}}{at_{i}^{p}} = (\frac{t_{i+1}}{t_{i}} )^{p} .

So the ratio of outputs is the p-th power of the ratio of inputs. In the problem

\frac{f(2.2)}{f(2.0)} = \frac{5.32}{4.40} = 1.21, (\frac{2.2}{2.0})^{2} = 1.21.

So f(t) is quadratic and g(t) is necessarily the cubic function.

1.5, #10, #11. The sketches tell the story (given appropriate window parameters, say −10 ≤ x ≤ 10, −50 ≤ y ≤ 50). x^{3} − 5x + 10 has a graph where horizontal lines can meet the graph as many as 3 times; x^{3} + 5x + 10 has a graph where every horizontal line meets the graph only once. So x^{3} − 5x + 10 is not invertible, but the x^{3} + 5x + 10 is. We will have other ways of seeing this using calculus.

Comments on 1.4, #17: Doing calculations is unavoidable in this problem. The question is when to stop. To find the exponential function, one needs ratios of consecutive values given in the table. Ratios of values are also needed to distinguish the quadratic and the cubic power functions. How many ratios are needed? The question as posed in the text makes matters easy by saying the functions are exponential, quadratic, and cubic. So it is not necessary to calculate all ratios to answer the question. But had the question been reworded as “Are any of these functions exponential, quadratic, or cubic?”, then one would probably want to check all ratios.