Solve for the variable to two decimal places.

$$A=1,200 e^{0.04(5)}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$A=3,000 e^{0.07(10)}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$9827.30=P e^{0.025(3)}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$50,000=P e^{0.054(7)}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$6,000=5,000 e^{0.0325 t}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$10,0000=7,500 e^{0.085 t}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$956=900 e^{1.5 r}$$

Kaylee M.

Numerade Educator

Solve for the variable to two decimal places.

$$4,840=3,750 e^{4.25 r}$$

Kaylee M.

Numerade Educator

Use a calculator to evaluate A to the nearest cent in Problems 9 and 10.

$$A=\$ 1,000 e^{0.1 t} \text { for } t=2,5, \text { and } 8$$

Kaylee M.

Numerade Educator

Use a calculator to evaluate A to the nearest cent in Problems 9 and 10.

$$A=\$ 5,000 e^{0.08 t} \text { for } t=1,4, \text { and } 10$$

Kaylee M.

Numerade Educator

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.

Kaylee M.

Numerade Educator

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.

Kaylee M.

Numerade Educator

In Problems $13-18$, solve for t or $r$ to two decimal places.

$$2=e^{0.06 t}$$

Narayan H.

Numerade Educator

In Problems $13-18$, solve for t or $r$ to two decimal places.

$$2=e^{0.03 t}$$

Narayan H.

Numerade Educator

In Problems $13-18$, solve for t or $r$ to two decimal places.

$$3=e^{0.1 r}$$

Narayan H.

Numerade Educator

In Problems $13-18$, solve for t or $r$ to two decimal places.

$$3=e^{0.25 t}$$

Narayan H.

Numerade Educator

In Problems $13-18$, solve for t or $r$ to two decimal places.

$$2=e^{5 r}$$

Kaylee M.

Numerade Educator

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?

Check back soon!

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?

Kaylee M.

Numerade Educator

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

$$\begin{array}{l}y_{1}=(1+1 / n)^{n} \\y_{2}=2.718281828 \approx e \\y_{3}=(1+1 / n)^{n+1}\end{array}$$

Check back soon!

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 $1 \leq n \leq 50$.

Kaylee M.

Numerade Educator

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 $\$ 18,000 ?$

Kaylee M.

Numerade Educator

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 $\$ 11.000 ?$

Kaylee M.

Numerade Educator

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?

Kaylee M.

Numerade Educator

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?

Kaylee M.

Numerade Educator

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?

Kaylee M.

Numerade Educator

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?

Kaylee M.

Numerade Educator

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.

Kaylee M.

Numerade Educator

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

Kaylee M.

Numerade Educator

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

Kaylee M.

Numerade Educator

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

Kaylee M.

Numerade Educator

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

Kaylee M.

Numerade Educator

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

Kaylee M.

Numerade Educator

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 $\$ 35.000$

Kaylee M.

Numerade Educator

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 $\$ 20,000$.

Kaylee M.

Numerade Educator

(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 $30 \%$.

Kaylee M.

Numerade Educator

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.

Kaylee M.

Numerade Educator

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.)

Kaylee M.

Numerade Educator

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 $41 .$

Kaylee M.

Numerade Educator

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 $41 .$

Kaylee M.

Numerade Educator

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 $41 .)$

Kaylee M.

Numerade Educator

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?

Kaylee M.

Numerade Educator

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?

Kaylee M.

Numerade Educator

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 $45 .$ )

Kaylee M.

Numerade Educator

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 $45 .$

Kaylee M.

Numerade Educator