Q: What will be the amount in an account after 5 years if $$100 is $$invested at an annual nominal rate of $$8\%$$ compounded semiannually?

$$A = 100 \left(1 + \frac{0.08}{2}\$$right)^{2 \times 5}$$

Q: What is the amount in an account after 5 years if $$100 is $$invested at an annual nominal rate of $$8\%$$ compounded continuously?

$$A = 100 e$$^{0.08 \times 5}$$

Q: A savings account compounds interest continuously at a rate of $$3\%$$ per year. If you deposit $$P = 2500$$, what is the value after 10 years?

$$A = 2500 e$$^{0.03 \times 10}$$

Question 1

The function $$y = e^x is $$called the natural exponential function. What is the value of $$e as $$defined by the following limit?

$$e = \lim_{n 	o \infty} \left(1 + \frac{1}{n}ight)^n$$

Accepted Answer

$$e \approx 2.718$$

Question 2

Lydia is the manager of a company that manufactures small appliances. She determines that $$x$$ units of a particular appliance will be produced and sold at a price of $$p$$ dollars per unit, where $$p(x) = 200e$$^{-0.01x}$$. How much revenue should she expect when 100 units of the appliance are produced?

Accepted Answer

$$R = 20000e$$^{-1}$$

Question 3

If the price per unit of an appliance is given by $$p(x) = 200e$$^{-0.01x}$$, how many units is Lydia selling if her appliance is priced at $150?

Accepted Answer

$$x = 35.67$$

Question 4

Suppose a principal $$P is $$invested at an annual interest rate $$r$$ for $$t$$ years and interest is compounded $$m$$ times per year. Which formula gives the future value $$A$$?

Accepted Answer

$$A = P \left(1 + \frac{r}{m}\$$right)^{mt}$$

Question 5

If the compounding is continuous, which formula gives the future value $$A of a $$principal $$P$$ invested at rate $$r$$ for $$t$$ years?

Accepted Answer

$$A = P e$$^{rt}$$

Question 6

What will be the amount in an account after 5 years if $$100 is $$invested at an annual nominal rate of $$8\%$$ compounded semiannually?

Accepted Answer

$$A = 100 \left(1 + \frac{0.08}{2}\$$right)^{2 	imes 5}$$

Question 7

What is the amount in an account after 5 years if $$100 is $$invested at an annual nominal rate of $$8\%$$ compounded continuously?

Accepted Answer

$$A = 100 e$$^{0.08 	imes 5}$$

Question 8

A savings account offers an annual interest rate of $$6\%$$ compounded monthly. If you deposit $$P = 500$$, what is the value of your account after 3 years?

Accepted Answer

$$A = 500 \left(1 + \frac{0.06}{12}\$$right)^{36}$$

Question 9

A chemical reaction decays according to $$N(t) = N_0$ e$$^{-kt}$$, where N_0$ is the initial amount and k$ is a positive constant. If N_0$ = 100$$ and $$k = 0.05$$, what is the amount after 10 hours?

Accepted Answer

$$N(10) = 100 e$$^{-0.5}$$

Question 10

A population of bacteria grows according to $$P(t) = 500 e$$^{0.03t}$$, where t$ is in hours. How many bacteria will there be after 24 hours?

Accepted Answer

$$P(24) = 500 e$$^{0.72}$$

Question 11

If $$e^x = 20$$, what is the value of $$x$$?

Accepted Answer

$$x = \ln 20$$

Question 12

A medication is eliminated from the bloodstream at a rate proportional to the amount present, modeled by $$C(t) = C_0$ e$$^{-kt}$$. If C_0$ = 50 mg $$and $$k = 0.2$$, what is the concentration after 8 hours?

Accepted Answer

$$C(8) = 50 e$$^{-1.6}$$

Question 13

A radioactive substance decays so that its mass after $$t$$ years is $$M(t) = M_0$ e$$^{-0.03t}$$. If the initial mass is 200$ grams, how long will it take for the mass to decrease to 100$ grams?

Accepted Answer

$$t = \frac{\ln 2}{0.03}$$

Question 14

A $$1000$$ principal is invested at $$5\%$$ annual interest compounded daily. What is the value after 2 years? (Assume 365 days in a year.)

Accepted Answer

$$A = 1000 \left(1 + \frac{0.05}{365}\$$right)^{730}$$

Question 15

A savings account compounds interest continuously at a rate of $$3\%$$ per year. If you deposit $$P = 2500$$, what is the value after 10 years?

Accepted Answer

$$A = 2500 e$$^{0.03 	imes 10}$$

Question 16

A company finds that the demand for its product is given by $$p(x) = 120e$$^{-0.02x}$$. What is the price per unit when 50 units are sold?

Accepted Answer

$$p(50) = 120e$$^{-1}$$

Question 17

If $$A = P e$$^{rt}$$, what is the value of r$ if A = 2000$, P = 1000$, and t = 10$?

Accepted Answer

$$r = \frac{\ln 2}{10}$$

Question 18

A population doubles every 12 years. If the population at time $$t is$$ $$P(t) = P_0$ e$$^{kt}$$, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{12}$$

Question 19

A $$500$$ principal is invested at $$4\%$$ annual interest compounded quarterly. What is the value after 6 years?

Accepted Answer

$$A = 500 \left(1 + \frac{0.04}{4}\$$right)^{24}$$

Exponential Functions and Compound Interest in Calculus

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