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Calculus for Business, Economics, Life Sciences, and Social Sciences 13th

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 3

Additional Derivative Topics

Educators


Problem 1

Solve for the variable to two decimal places.
$$A=1,200 e^{0.04(5)}$$

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Problem 2

Solve for the variable to two decimal places.
$$A=3,000 e^{0.07(10)}$$

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Problem 3

Solve for the variable to two decimal places.
$$9827.30=P e^{0.025(3)}$$

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Problem 4

Solve for the variable to two decimal places.
$$50,000=P e^{0.054(7)}$$

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Problem 5

Solve for the variable to two decimal places.
$$6,000=5,000 e^{0.0325 t}$$

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Problem 6

Solve for the variable to two decimal places.
$$10,0000=7,500 e^{0.085 t}$$

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Problem 7

Solve for the variable to two decimal places.
$$956=900 e^{1.5 r}$$

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Problem 8

Solve for the variable to two decimal places.
$$4,840=3,750 e^{4.25 r}$$

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Problem 9

Use a calculator to evaluate A to the nearest cent in Problems 9 and 10.
$$A=\$ 1,000 e^{0.1 t} \text { for } t=2,5, \text { and } 8$$

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Problem 10

Use a calculator to evaluate A to the nearest cent in Problems 9 and 10.
$$A=\$ 5,000 e^{0.08 t} \text { for } t=1,4, \text { and } 10$$

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Problem 11

If $\$ 6,000$ is invested at $10 \%$ compounded continuously, graph the amount in the account as a function of time for a period of 8 years.

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Problem 12

If $\$ 4,000$ is invested at $8 \%$ compounded continuously, graph the amount in the account as a function of time for a period of 6 years.

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Problem 13

In Problems $13-18$, solve for t or $r$ to two decimal places.
$$2=e^{0.06 t}$$

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Problem 14

In Problems $13-18$, solve for t or $r$ to two decimal places.
$$2=e^{0.03 t}$$

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Problem 15

In Problems $13-18$, solve for t or $r$ to two decimal places.
$$3=e^{0.1 r}$$

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Numerade Educator

Problem 16

In Problems $13-18$, solve for t or $r$ to two decimal places.
$$3=e^{0.25 t}$$

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Numerade Educator

Problem 17

In Problems $13-18$, solve for t or $r$ to two decimal places.
$$2=e^{5 r}$$

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Problem 18

Solve for t or $r$ to two decimal places.
$$3=e^{10 r}$$

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Problem 19

Use a calculator to complete each table to five decimal places.

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Problem 20

Use a calculator to complete each table to five decimal places.

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Problem 21

Use a calculator and a table of values to investigate
$$\lim _{n \rightarrow \infty}(1+n)^{1 / n}$$
Do you think this limit exists? If so, what do you think it is?

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Problem 22

Use a calculator and a table of values to investigate
$$\lim _{s \rightarrow 0^{+}}\left(1+\frac{1}{s}\right)^{s}$$
Do you think this limit exists? If so, what do you think it is?

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Problem 23

Do you think this limit exists? If so, what do you think it is?
It can be shown that the number $e$ satisfies the inequality
$$\left(1+\frac{1}{n}\right)^{n}<e<\left(1+\frac{1}{n}\right)^{n+1} \quad n \geq 1$$
Illustrate this condition by graphing
$$\begin{array}{l}y_{1}=(1+1 / n)^{n} \\y_{2}=2.718281828 \approx e \\y_{3}=(1+1 / n)^{n+1}\end{array}$$

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Problem 24

It can be shown that
$$e^{s}=\lim _{n \rightarrow \infty}\left(1+\frac{s}{n}\right)^{n}$$
for any real number $s$. Illustrate this equation graphically for $s=2$ by graphing
$$\begin{array}{l}y_{1}=(1+2 / n)^{n} \\y_{2}=7.389056099 \approx e^{2}\end{array}$$
in the same viewing window, for $1 \leq n \leq 50$.

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Problem 25

Provident Bank offers a 10-year CD that earns $2.15 \%$ compounded continuously.
(A) If $\$ 10,000$ is invested in this $\mathrm{CD},$ how much will it be worth in 10 years?
(B) How long will it take for the account to be worth $\$ 18,000 ?$

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Problem 26

Provident Bank also offers a 3-year CD that earns $1.64 \%$ compounded continuously.
(A) If $\$ 10,000$ is invested in this $\mathrm{CD},$ how much will it be worth in 3 years?
(B) How long will it take for the account to be worth $\$ 11.000 ?$

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Problem 27

A note will pay $\$ 20,000$ at maturity 10 years from now. How much should you be willing to pay for the note now if money is worth $5.2 \%$ compounded continuously?

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Problem 28

A note will pay $\$ 50,000$ at maturity 5 years from now. How much should you be willing to pay for the note now if money is worth $6.4 \%$ compounded continuously?

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Problem 29

An investor bought stock for $\$ 20,000 .$ Five years later, the stock was sold for $\$ 30,000$. If interest is compounded continuously, what annual nominal rate of interest did the original $\$ 20,000$ investment earn?

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Problem 30

A family paid $\$ 99,000$ cash for a house. Fifteen years later, the house was sold for $\$ 195,000 .$ If interest is compounded continuously, what annual nominal rate of interest did the original $\$ 99,000$ investment earn?

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Problem 31

Solving $A=P e^{r l}$ for $P,$ we obtain
$$P=A e^{-r t}$$
which is the present value of the amount $A$ due in $t$ years if money earns interest at an annual nominal rate $r$ compounded continuously.
(A) Graph $P=10,000 e^{-0.08 t}, 0 \leq t \leq 50 .$
(B) $\lim _{t \rightarrow \infty} 10,000 e^{-0.08 t}=?$ [Guess, using part (A).
[Conclusion: The longer the time until the amount $A$ is due, the smaller is its present value, as we would expect.

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Problem 32

Referring to Problem 31 , in how many years will the $\$ 10,000$ be due in order for its present value to be $\$ 5.000 ?$

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Problem 33

How long will it take money to double if it is invested at $4 \%$ compounded continuously?

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Problem 34

How long will it take money to double if it is invested at $5 \%$ compounded continuously?

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Problem 35

At what nominal rate compounded continuously must money be invested to double in 8 years?

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Problem 36

At what nominal rate compounded continuously must money be invested to double in 10 years?

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Problem 37

A man with $\$ 20,000$ to invest decides to diversify his investments by placing $\$ 10,000$ in an account that earns $7.2 \%$ compounded continuously and $\$ 10,000$ in an account that earns $8.4 \%$ compounded annually. Use graphical approximation methods to determine how long it will take for his total investment in the two accounts to grow to $\$ 35.000$

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Problem 38

A woman invests $\$ 5,000$ in an account that earns $8.8 \%$ compounded continuously and $\$ 7,000$ in an account that earns $9.6 \%$ compounded annually. Use graphical approximation methods to determine how long it will take for her total investment in the two accounts to grow to $\$ 20,000$.

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Problem 39

(A) Show that the doubling time $t$ (in years) at an annual rate $r$ compounded continuously is given by$$t=\frac{\ln 2}{r}$$
(B) Graph the doubling-time equation from part (A) for $0.02 \leq r \leq 0.30 .$ Is this restriction on $r$ reasonable? Explain.
(C) Determine the doubling times (in years, to two decimal places) for $r=5 \%, 10 \%, 15 \%, 20 \%, 25 \%,$ and $30 \%$.

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Problem 40

Doubling rates
(A) Show that the rate $r$ that doubles an investment at continuously compounded interest in $t$ years is given by
$$r=\frac{\ln 2}{t}$$
(B) Graph the doubling-rate equation from part (A) for $1 \leq t \leq 20$. Is this restriction on $t$ reasonable? Explain.
(C) Determine the doubling rates for $t=2,4,6,8,10,$ and 12 years.

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Problem 41

A mathematical model for the decay of radioactive substances is given by
$$Q=Q_{0} e^{r t}$$
where
$\begin{aligned} Q_{0} &=\text { amount of the substance at time } t=0 \\ r &=\text { continuous compound rate of decay } \\ t &=\text { time in years } \\ Q &=\text { amount of the substance at time } t \end{aligned}$
If the continuous compound rate of decay of radium per year is $r=-0.0004332$, how long will it take a certain amount of radium to decay to half the original amount? (This period is the half-life of the substance.)

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Problem 42

The continuous compound rate of decay of carbon- 14 per year is $r=-0.0001238$. How long will it take a certain amount of carbon- 14 to decay to half the original amount? (Use the radioactive decay model in Problem $41 .$

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Problem 43

A cesium isotope has a half-life of 30 years. What is the continuous compound rate of decay? (Use the radioactive decay model in Problem $41 .$

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Problem 44

A strontium isotope has a half-life of 90 years. What is the continuous compound rate of decay? (Use the radioactive decay model in Problem $41 .)$

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Problem 45

A mathematical model for world population growth over short intervals is given by
$$P=P_{0} e^{r t}$$
where
$$\begin{aligned}P_{0} &=\text { population at time } t=0 \\r &=\text { continuous compound rate of growth} \\t &=\text { time in years } \\P &=\text { population at time } t\end{aligned}$$
How long will it take world population to double if it continues to grow at its current continuous compound rate of $1.3 \%$ per year?

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Problem 46

How long will it take for the U.S. population to double if it continues to grow at a rate of $0.975 \%$ per year?

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Problem 47

Some underdeveloped nations have population doubling times of 50 years. At what continuous compound rate is the population growing? (Use the population growth model in Problem $45 .$ )

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Problem 48

Some developed nations have population doubling times of 200 years. At what continuous compound rate is the population growing? (Use the population growth model in Problem $45 .$

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