Chapter 15

Suppose you put \(\$ 100,000\) in a bank account with \(6 \%\) interest and leave it for one year. How much money will there be in the account if the interest is compounded (a) annually? (b) monthly? (c) daily? (d) hourly?

(a) Suppose a population grows at a rate of \(5 \%\) per year: \(P=P_{0}(1.05)^{t}\). i. Express this in the form \(P=P_{0} e^{r t}\). ii. Compute \(\frac{d P}{d t}\). iii. Find the proportionality constant \(k\) so that \(\frac{d P}{d t}=k P\). (b) Suppose a population grows according to \(P=P_{0} e^{0.05 t}\) i. Find the proportionality constant \(k\) so that \(\frac{d P}{d t}=k P\). ii. By what percent does the population grow each year? Look back over this problem and think about it. Do your answers make sense to you?

In 1996 , inflation in Russia was \(22 \%\). This was a decline from the \(131 \%\) inflation rate in 1995 and the \(2600 \%\) inflation rate in \(1994 .\) By contrast, the inflation rate in the United States in 1996 was about \(3 \%\). (Boston Globe, November \(2,1996 .\) ) Compute the amount of time it would take for prices to double under each of the four inflation rates listed.

A Boston Globe article on January 1, 1997, said that the best stock of 1996 was Information Analysis, Incorporated, which closed the year at a price of \(\$ 63\) per share, an increase of \(1525 \%\) during the year. The worst stock of 1996 was Mobilemedia Corporation, which closed the year at \(\$ 7 / 16\) per share, a decrease of \(97.6 \%\). What was the price of each of these stocks at the beginning of the year?

Solve the following differential equations. Give a general solution and then a particular solution corresponding to the initial condition given. \(\begin{array}{ll}\text { (a) } \frac{d y}{d t}=3 y & \text { initial condition: } y(0)=5\end{array}\) (b) \(\frac{d y}{d x}=-0.01 y \quad\) initial condition: \(y(2)=1\) (c) \(\frac{d w}{d s}=w\) initial condition: \(w(0)=\pi\)

Compute the following limits. In each case stop to think of a strategy, and use whatever strategy seems simplest to you. For several of these limits there are different approaches. (a) \(\lim _{x \rightarrow 3} \frac{3}{x-3}\) (b) \(\lim _{x \rightarrow 3} \frac{3}{(x-3)^{2}}\) (c) \(\lim _{x \rightarrow \infty}\left(1-\frac{1}{3 x}\right)^{7 x}\) (d) \(\lim _{t \rightarrow 0^{+}}(1-2 t)^{1 / t}\)

Solve the following differential equations. For each differential equation, find the general solution and then find a solution passing through the point \((0, \sqrt{2})\). (a) \(\frac{d y}{d t}=-2 y\) (b) \(\frac{d y}{d t}=-2 t\) (c) \(\frac{d y}{d t}=-2\)

Evaluate the following limits. (a) \(\lim _{x \rightarrow \infty} \frac{e^{-x}}{x^{2}}\) (b) \(\lim _{x \rightarrow \infty} x^{3} e^{-3 x}\)

(a) Show that \(P=C e^{2 t}\) (where \(C\) is any constant) is a solution to the differential equation \(\frac{d P}{d t}=2 P .\) That is, show that if you compute \(\frac{d P}{d t}\), you get \(2 P\). (b) Show that \(P=e^{2 t}+C\) is not a solution to the differential equation \(\frac{d P}{d t}=2 P\).

Suppose you invest \(\$ 10,000\) in an account with a nominal annual interest rate of \(5 \%\). How much money will you have 10 years later if the interest is compounded (a) quarterly? (b) daily? (c) continuously?

Consider the differential equation \(\frac{d y}{d t}=y-2 .\) Which of the functions below are solutions? There could be more than one answer. (a) \(y=e^{t}+2\) (b) \(y=e^{t}+3\) (c) \(y=C e^{t}+2\) (d) \(y=C\left(e^{t}+2\right)\)

(a) A certain amount of money is put in an account with a fixed nominal annual interest rate, and interest is compounded continuously. If 70 years later the money in the account has doubled, what is the nominal annual interest rate? (b) Answer the same question if the interest is compounded only once a year.

(a) Is \(y=e^{t}+\ln t\) a solution to the differential equation \(\frac{d y}{d t}=y-\frac{y}{t}\) ? (b) Is \(y=t e^{t}\) a solution to the differential equation \(\frac{d y}{d t}=y-\frac{y}{t}\) ?

(a) Kevin has deposited money in a bank account that compounds interest quarterly. If the nominal interest rate is \(5 \%\), what is the effective interest rate? (b) Ama has deposited money in a bank account that compounds interest quarterly. If the effective interest rate is \(5 \%\) per year, what is the nominal rate of interest?

A wet dish towel is put on the back of a kitchen chair to dry. It dries at a rate proportional to the difference in moisture content between the dishtowel and the kitchen air. Assume that the moisture content in the air is fixed and is given by \(M\). (a) Set up the differential equation involving \(W=W(t)\), the amount of water in the dish towel at time \(t\). (b) Find and sketch the solution.

Suppose that a person invests \(\$ 10,000\) in a venture that pays interest at a nominal rate of \(8 \%\) per year compounded quarterly for the first 5 years and \(3 \%\) per year compounded quarterly for the next 5 years. (a) How much does the \(\$ 10,000\) grow to after 10 years? (b) Suppose there were another investment option that paid interest quarterly at a constant interest rate \(r\). What would \(r\) have to be for the two plans to be equivalent, ignoring taxes? (c) If an investment scheme paid \(3 \%\) interest compounded quarterly for the first 5 years and \(8 \%\) interest compounded quarterly for the next 5 years, would it be better than, worse than, or equivalent to the first scheme?

When a population has unlimited resources and is free from disease and strife, the rate at which the population grows is proportional to the population. Assume that both the bee and the mosquito populations described below behave according to this model. In both scenarios you are given enough information to find the proportionality constant \(k\). In one case, the information allows you to find \(k\) solely using the differential equation, without requiring that you solve it. In the other scenario, you must actually solve the differential equation in order to find \(k\). (a) Let \(M=M(t)\) be the mosquito population at time \(t, t\) in weeks. At \(t=0\), there are 1000 mosquitoes. Suppose that when there are 5000 mosquitoes, the population is growing at a rate of 250 mosquitoes per week. Write a differential equation reflecting the situation. Include a value for \(k\), the proportionality constant. (b) Let \(B=B(t)\) be the bee population at time \(t, t\) in weeks. At \(t=0\), there are 600 bees. When \(t=10\), there are 800 bees. Write a differential equation reflecting the situation. Include a value for \(k\), the proportionality constant.

Evaluate the following. Substitution may be helpful; these problems are variations on the theme \(\lim _{n \rightarrow \infty}\left(1+\frac{r}{n}\right)^{n}\). (a) \(\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}\) (b) \(\lim _{w \rightarrow \infty}\left(\frac{w+2}{w}\right)^{w}\) (c) \(\lim _{x \rightarrow \infty}\left(\frac{x-1}{x}\right)^{2 n}\) (d) \(\lim _{n \rightarrow \infty}\left(\frac{n}{n+1}\right)^{n}\) (e) \(\lim _{x \rightarrow 0^{+}}(1+2 x)^{3 /(2 x)}\)

Newton's law of cooling in its more general form tells us that the rate at which the temperature between an object and its environment changes is proportional to the difference in temperatures. In other words, if \(D(t)\) is the temperature difference, then \(\frac{d D}{d t}=k D\) (a) Solve the differential equation \(\frac{d D}{d t}=k D\) for \(D(t)\). (b) Suppose a hot object is placed in a room whose temperature is kept constant at \(R\) degrees. Let \(T(t)\) be the temperature of the object. Newton's law says that the hot object will cool at a rate proportional to the difference in temperature between the object and its environment. Write a differential equation reflecting this statement and involving \(T\). Explain why this differential equation is equivalent to the previous one. (c) What is the sign of the constant of proportionality in the equation you wrote in part (b)? Explain. (d) Suppose that instead of a hot object we now consider a cold object. Suppose that we are interested in the temperature of a cold cup of lemonade as it warms up to room temperature. Let \(L(t)\) represent the temperature of the lemonade at time \(t\) and assume that it sits in a room that is kept at 65 degrees. At time \(t=0\), the lemonade is at 40 degrees. 15 minutes later it has warmed to 50 degrees. i. Sketch a graph of \(L(t)\) using your intuition and the information given. ii. Is \(L(t)\) increasing at an increasing rate, or a decreasing rate?

Suppose you put \(\$ 6000\) in a bank account at \(5 \%\) (nominal) annual interest compounded continuously. (a) How much money do you have at the end of 7 years? (b) How much money do you have at the end of \(t\) years? (c) What is the instantaneous rate of change of money in the account with respect to time? (Find \(\frac{d M}{d t} .\) ) (d) True or False: \(\frac{d M}{d t}=0.05 M\). Explain your reasoning! (e) Write your answer to part (b) in the form \(M=C a^{t}\) and use your calculator to approximate the value of " \(a\) " numerically. (f) Each year, by what percent does your money grow? (This is called the effective annual yield and, if interest is compounded more than once a year, it is always bigger than the nominal annual interest rate.)

Suppose that in a certain country the population grows at a rate proportional to itself with proportionality constant \(0.02 .\) Further suppose that due to a drought people are leaving the country at a constant rate of 1000 people per year. Let \(P=P(t)\) be the population of the country at time \(t\), where \(t\) is in years. Write a differential equation modeling the situation.

Evaluate the following limits. Keep in mind that limit calculations can be subtle-don't ad lib, but instead keep the limits we looked at firmly in your mind and use substitution in order to make the transfer to the problems here. You can determine whether or not your answer is in the ballpark by using your calculator. (a) \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\) (b) \(\lim _{s \rightarrow \infty}\left(1+\frac{1}{s}\right)^{3 s}\) (c) \(\lim _{r \rightarrow \infty}\left(1+\frac{0.3}{r}\right)^{r}\) (d) \(\lim _{w \rightarrow \infty}\left(1+2 w^{-1}\right)^{w}\) (e) \(\lim _{w \rightarrow \infty}\left(1+(2 w)^{-1}\right)^{w}\)

Money in a certain trust-fund account is earning \(5 \%\) interest per year compounded continuously. Suppose money is being withdrawn from the account at a constant rate of \(\$ 2000\) per year. For the sake of our model, assume that money is being withdrawn continuously. The account begins with \(\$ 30,000\). Let \(M=M(t)\) be the amount of money in the account at time \(t\), where \(t\) is in years. Write a differential equation modeling the situation. What is the initial condition?

Which is a better deal, an account offering \(4 \%\) annual interest compounded continuously or an account offering \(4.2 \%\) interest compounded annually? What is the effective annual yield of the former account?

Solve the following differential equations. Use substitution to convert them to the form \(\frac{d y}{d t}=k y\) (a) \(\frac{d y}{d t}=3 y-6\) (b) \(\frac{d y}{d t}=y+1\) (c) \(\frac{d y}{d t}=4-2 y\)

If \(M(t)=M_{0} e^{r t}\), find \(\frac{d M}{d t}\) and show that \(\frac{d M}{d t}=r M\). \(\left(\frac{d M}{d t}=r M\right.\) is called a differential equation because it is an equation with a derivative in it. You have just shown that \(M(t)=M_{0} e^{r t}\) is a solution to this differential equation.)

\text { Solve } \frac{d y}{d t}=2 y-6 \text { with the initial condition } y(0)=2000 \text { . }

A boarding school with 800 students has been hit by a flu epidemic. If we assume that every student is either sick or healthy, that sick students will infect healthy ones, and that the disease is quite long in duration (it's a nasty flu) then we can model the epidemic using the following assumption. The rate at which students are getting sick is proportional to the product of the number of sick students and the number of healthy ones. (a) Let \(S=S(t)\) be the number of sick students at time \(t\). Translate the statement above into mathematical language. (b) \(S(t)\) is an increasing function. The rate at which students are getting sick is a quadratic function of \(S .\) When the rate at which students are getting sick is highest, how many students are sick?

In the beginning of a chemical reaction there are 800 moles of substance A and none of substance \(\mathrm{B}\). Over the course of the reaction, the \(800 \mathrm{moles}\) of substance \(\mathrm{A}\) are converted to 800 moles of substance B. (Each molecule of A is converted to a molecule of \(\mathrm{B}\) via the reaction.) Suppose the rate at which \(\mathrm{A}\) is turning into \(\mathrm{B}\) is proportional to the product of the number of moles of A and the number of moles of \(\mathrm{B}\). (a) Let \(N=N(t)\) be the number of moles of substance \(\mathrm{A}\) at time \(t .\) Translate the statement above into mathematical language. (Note: The number of moles of substance \(\mathrm{B}\) should be expressed in terms of the number of moles of substance A.) (b) \(N(t)\) is a decreasing function. The rate at which \(N\) is changing is a function of \(N\), the number of moles of substance A. When the rate at which A is being converted to \(\mathrm{B}\) is highest, how many moles are there of substance \(\mathrm{A}\) ?