Chapter 5

In the year 2009 , Olivia's bank balance is 1000 dollars. In the year 2010 , her balance is 1100 dollars. (a) If her balance is growing exponentially, in what year will it reach 2500 dollars? (b) If her balance is instead growing linearly, in what year will it reach 2500 dollars ?

Look back at Section \(4.4,\) where the basic properties of graphs of polynomial functions were discussed. Then review the basic properties of the graph of \(f(x)=a^{x}\) discussed in this section. Using these various properties, give an argument to show that for any fixed positive number \(a(\neq 1),\) it is not possible to find a polynomial function \(g(x)=c_{n} x^{n}+\dots+c_{1} x+c_{0}\) such that \(a^{x}=g(x)\) for \(a l l\) numbers \(x .\) In other words, no exponential function is a polynomial function. However, see Exercise \(81 .\)

According to one theory of learning, the number of words per minute \(N\) that a person can type after \(t\) weeks of practice is given by \(N=c\left(1-e^{-k t}\right),\) where \(c\) is an upper limit that \(N\) cannot exceed and \(k\) is a constant that must be determined experimentally for each person. (a) If a person can type 50 wpm (words per minute) after four weeks of practice and 70 wpm after eight weeks, find the values of \(k\) and \(c\) for this person. According to the theory, this person will never type faster than \(c\) wpm. (b) Another person can type 50 wpm after four weeks of practice and 90 wpm after eight weeks. How many weeks must this person practice to be able to type 125 wpm?

Beef consumption in the United States (in billions of pounds) in year \(x\) can be approximated by the function $$ f(x)=-154.41+39.38 \ln x \quad(x \geq 90) $$ where \(x=90\) corresponds to \(1990 .\) (a) How much beef was consumed in 1999 and in \(2002 ?\) (b) According to this model when will beef consumption reach 35 billion pounds per year?

Kate has been offered two jobs, each with the same starting salary of 32,000 dollars and identical benefits. Assuming satisfactory performance, she will receive a 1600 dollars raise each year at the Great Gizmo Company, whereas the Wonder Widget Company will give her a \(4 \%\) raise each year. (a) In what year (after the first year) would her salary be the same at either company? Until then, which company pays better? After that, which company pays better? (b) Answer the questions in part (a) assuming that the annual raise at Great Gizmo is 2000 dollars.

This exercise provides a justification for the claim that the function \(M(x)=c(.5)^{x / h}\) gives the mass after \(x\) years of a radioactive element with half-life \(h\) years. Suppose we have \(c\) grams of an element that has a half-life of 50 years. Then after 50 years, we would have \(c\left(\frac{1}{2}\right)\) grams. After another 50 years, we would have half of that, namely, \(c\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=c\left(\frac{1}{2}\right)^{2}\) (a) How much remains after a third 50-year period? After a fourth 50 -year period? (b) How much remains after \(t 50\) -year periods? (c) If \(x\) is the number of years, then \(x / 50\) is the number of 50-year periods. By replacing the number of periods \(t\) in part (b) by \(x / 50,\) you obtain the amount remaining after \(x\) years. This gives the function \(M(x)\) when \(h=50\) The same argument works in the general case (just replace 50 by \(h\) ). Find \(M(x)\).

Students in a precalculus class were given a final exam. Each month thereafter, they took an equivalent exam. The class average on the exam taken after \(t\) months is given by $$ F(t)=82-8 \cdot \ln (t+1) $$ (a) What was the class average after six months? (b) After a year? (c) When did the class average drop below \(55 ?\)

One person with a flu virus visited the campus. The number \(T\) of days it took for the virus to infect \(x\) people was given by: $$ T=-.93 \ln \left[\frac{7000-x}{6999 x}\right] $$ (a) How many days did it take for 6000 people to become infected? (b) After two weeks, how many people were infected?

The population of St. Petersburg, Florida (in thousands) can be approximated by the function $$ g(x)=-127.9+81.91 \ln x \quad(x \geq 70) $$ where \(x=70\) corresponds to 1970 (a) Estimate the population in 1995 and 2003 . (b) If this model remains accurate, when will the population be \(260,000 ?\)

A bicycle store finds that the number \(N\) of bikes sold is related to the number \(d\) of dollars spent on advertising by \(N=51+100 \cdot \ln (d / 100+2)\) (a) How many bikes will be sold if nothing is spent on advertising? If \(\$ 1000\) is spent? If \(\$ 10,000\) is spent? (b) If the average profit is \(\$ 25\) per bike, is it worthwhile to spend \(\$ 1000\) on advertising? What about \(\$ 10,000 ?\) (c) What are the answers in part (b) if the average profit per bike is \(\$ 35 ?\)

Approximating Logarithmic Functions by Polynomials. For each positive integer \(n,\) let \(f_{n}\) be the polynomial function whose rule is $$ f_{n}(x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}-\cdots \pm \frac{x^{n}}{n} $$ where the sign of the last term is \(+\) if \(n\) is odd and \(-\) if \(n\) is even. In the viewing window with \(-1 \leq x \leq 1\) and \(-4 \leq y \leq 1,\) graph \(g(x)=\ln (1+x)\) and \(f_{4}(x)\) on the same screen. For what values of \(x\) does \(f_{4}\) appear to be a good approximation of \(g ?\)

A harmonic sum is a sum of this form: $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{k} $$ (a) Compute \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}, 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5},\) and $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6} $$ (b) How many terms do you need in a harmonic sum for it to exceed three? (c) It turns out to be hard to determine how many terms you would need for the sum to exceed 10. It will take thousands of terms, more than you would want to plug into a calculator. Using calculus, we can derive this lower-bound formula: \(\sum_{i=1}^{n} \frac{1}{i} > \ln n .\) It means that the harmonic sum with \(n\) terms is always greater than \(\ln n .\) Use this formula to find a value of \(n\) such that the harmonic sum with \(n\) terms is greater than ten. (d) Calculus also gives us an upper-bound formula: \(\sum_{i=1}^{n} \frac{1}{i} < \ln n+1 .\) Estimate the harmonic sum with 100,000 terms. How close is your estimate to the real number?