A company can earn additional profits of $500,000/year for 5 years by investing $2

Two quantities increase exponentially with growth constants k = 1.2 and k = 3.4, respectively. Which quantity doubles more rapidly?

A cell population grows exponentially beginning with one cell. Which takes longer: increasing from one to two cells or increasing from 15 million to 20 million cells?

Referring to his popular bookA Brief History of Time, the renowned physicist Stephen Hawking said, Someone told me that each equation I included in the book would halve its sales. Find a differential equation satisfied by S(n), the number of copies sold if the book has n equations

The PV of N dollars received at time T is (choose the correct answer): (a) The value at time T of N dollars invested today (b) The amount you would have to invest today in order to receive N dollars at time T

In 1 year, you will be paid $1. Will the PV increase or decrease if the interest rate goes up?

A quantity P obeys the exponential growth law P (t) = Cekt (t in years). Find the formula for P (t), assuming that the doubling time is 7 years and P (0) = 100.

Find all solutions to the differential equation y = 5y. Which solution satisfies the initial condition y(0) = 3.4?

Find the solution to y = 2y satisfying y(0) = 20.

Find the solution to y = 3y satisfying y(2) = 1000.

Find the function y = f (t) that satisfies the differential equation y = 0.7y and the initial condition y(0) = 10.

The decay constant of cobalt-60 is 0.13 year1. Find its half-life.

The half-life radium-226 is 1622 years. Find its decay constant.

One of the worlds smallest flowering plants, Wolffia globosa (Figure 13), has a doubling time of approximately 30 hours (h). Find the growth constant k and determine the initial population if the population grew to 1000 after 48 h.

A 10-kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope

The population of a city is P (t) = 2 e0.06t (in millions), where t is measured in years. Calculate the time it takes for the population to double, to triple, and to increase 7 fold.

What is the differential equation satisfied by P (t), the number of infected computer hosts in Example 4? Over which time interval would P (t) increase 100 fold?

The decay constant for a certain drug is k = 0.35 day1. Calculate the time it takes for the quantity present in the bloodstream to decrease by half, by one-third, and by one-tenth.

Light Intensity The intensity of light passing through an absorbing medium decreases exponentially with the distance traveled. Suppose the decay constant for a certain plastic block is k = 4 m1. How thick must the block be to reduce the intensity by a factor of one-third?

Assuming that population growth is approximately exponential, which of the following two sets of data is most likely to represent the population (in millions) of a city over a 5-year period? Year 2000 2001 2002 2003 2004 Set I 3.14 3.36 3.60 3.85 4.11 Set II 3.14 3.24 3.54 4.04 4.74

The atmospheric pressure P (h) (in kilopascals) at a height h meters above sea level satisfies a differential equation P = kP for some positive constant k. (a) Barometric measurements show that P (0) = 101.3 and P (30,900) = 1.013. What is the decay constant k? (b) Determine the atmospheric pressure at h = 500.

Degrees in Physics One study suggests that from 1955 to 1970, the number of bachelors degrees in physics awarded per year by U.S. universities grew exponentially, with growth constant k = 0.1. (a) If exponential growth continues, how long will it take for the number of degrees awarded per year to increase 14-fold? (b) If 2500 degrees were awarded in 1955, in which year were 10,000 degrees awarded?

The BeerLambert Law is used in spectroscopy to determine the molar absorptivity or the concentration c of a compound dissolved in a solution at low concentrations (Figure 14). The law states that the intensity I of light as it passes through the solution satisfies ln(I/I0) = cx, where I0 is the initial intensity and x is the distance traveled by the light. Show that I satisfies a differential equation dI /dx = kI for some constant k.

A sample of sheepskin parchment discovered by archaeologists had a C14-to-C12 ratio equal to 40% of that found in the atmosphere. Approximately how old is the parchment?

Chauvet Caves In 1994 three French speleologists (geologists specializing in caves) discovered a cave in southern France containing prehistoric cave paintings. A C14 analysis carried out by archeologist Helene Valladas showed the paintings to be between 29,700 and 32,400 years old, much older than any previously known human art. Given that the C14-to-C12 ratio of the atmosphere is R = 1012, what range of C14-to-C12 ratios did Valladas find in the charcoal specimens?

A paleontologist discovers remains of animals that appear to have died at the onset of the Holocene ice age, between 10,000 and 12,000 years ago. What range of C14-to-C12 ratio would the scientist expect to find in the animal remains?

Inversion of Sugar When cane sugar is dissolved in water, it converts to invert sugar over a period of several hours. The percentage f (t) of unconverted cane sugar at time t (in hours) satisfies f = 0.2f . What percentage of cane sugar remains after 5 hours (h)? After 10 h?

Continuing with Exercise 26, suppose that 50 g of sugar are dissolved in a container of water. After how many hours will 20 g of invert sugar be present?

Two bacteria colonies are cultivated in a laboratory. The first colony has a doubling time of 2 hours (h) and the second a doubling time of 3 h. Initially, the first colony contains 1000 bacteria and the second colony 3000 bacteria. At what time t will the sizes of the colonies be equal?

Moores Law In 1965 Gordon Moore predicted that the number N of transistors on a microchip would increase exponentially. (a) Does the table of data below confirm Moores prediction for the period from 1971 to 2000? If so, estimate the growth constant k. (b) Plot the data in the table. (c) Let N (t) be the number of transistors t years after 1971. Find an approximate formula N (t) Cekt , where t is the number of years after 1971. (d) Estimate the doubling time in Moores Law for the period from 1971 to 2000.(e) How many transistors will a chip contain in 2020 if Moores Law continues to hold? (f) Can Moore have expected his prediction to hold indefinitely? Processor Year No. Transistors 4004 1971 2250 8008 1972 2500 8080 1974 5000 8086 1978 29,000 286 1982 120,000 386 processor 1985 275,000 486 DX processor 1989 1,180,000 Pentium processor 1993 3,100,000 Pentium II processor 1997 7,500,000 Pentium III processor 1999 24,000,000 Pentium 4 processor 2000 42,000,000 Xeon processor 2008 1,900,000,000

Assume that in a certain country, the rate at which jobs are created is proportional to the number of people who already have jobs. If there are 15 million jobs at t = 0 and 15.1 million jobs 3 months later, how many jobs will there be after 2 years?

The only functions with a constant doubling time are the exponential functions y = P0ekt with k > 0. Show that the doubling time of linear function f (t) = at + b at time t0 is t0 + b/a (which increases with t0). Compute the doubling times of f (t) = 3t + 12 at t0 = 10 and t0 = 20.

Verify that the half-life of a quantity that decays exponentially with decay constant k is equal to (ln 2)/k.

Compute the balance after 10 years if $2000 is deposited in an account paying 9% interest and interest is compounded (a) quarterly, (b) monthly, and (c) continuously.

Suppose $500 is deposited into an account paying interest at a rate of 7%, continuously compounded. Find a formula for the value of the account at time t. What is the value of the account after 3 years?

A bank pays interest at a rate of 5%. What is the yearly multiplier if interest is compounded (a) three times a year? (b) continuously?

How long will it take for $4000 to double in value if it is deposited in an account bearing 7% interest, continuously compounded?

How much must one invest today in order to receive $20,000 after 5 years if interest is compounded continuously at the rate r = 9%?

An investment increases in value at a continuously compounded rate of 9%. How large must the initial investment be in order to build up a value of $50,000 over a 7-year period?

Compute the PV of $5000 received in 3 years if the interest rate is (a) 6% and (b) 11%. What is the PV in these two cases if the sum is instead received in 5 years?

Is it better to receive $1000 today or $1300 in 4 years? Consider r = 0.08 and r = 0.03.

Find the interest rate r if the PV of $8000 to be received in 1 year is $7300.

A company can earn additional profits of $500,000/year for 5 years by investing $2 million to upgrade its factory. Is the investment worthwhile if the interest rate is 6%? (Assume the savings are received as a lump sum at the end of each year.)

A new computer system costing $25,000 will reduce labor costs by $7000/year for 5 years. (a) Is it a good investment if r = 8%? (b) How much money will the company actually save?

After winning $25 million in the state lottery, Jessica learns that she will receive five yearly payments of $5 million beginning immediately. (a) What is the PV of Jessicas prize if r = 6%? (b) How much more would the prize be worth if the entire amount were paid today?

Use Eq. (3) to compute the PV of an income stream paying out R(t) = $5000/year continuously for 10 years, assuming r = 0.05

Find the PV of an investment that pays out continuously at a rate of $800/year for 5 years, assuming r = 0.08.

Find the PV of an income stream that pays out continuously at a rate R(t) = $5000e0.1t /year for 7 years, assuming r = 0.05.

A commercial property generates income at the rate R(t). Suppose that R(0) = $70,000/year and that R(t) increases at a continuously compounded rate of 5%. Find the PV of the income generated in the first 4 years if r = 6%.

Show that an investment that pays out R dollars per year continuously for T years has a PV of R(1 erT )/r.

Explain this statement: If T is very large, then the PV of the income stream described in Exercise 49 is approximately R/r.

Suppose that r = 0.06. Use the result of Exercise 50 to estimate the payout rate R needed to produce an income stream whose PV is $20,000, assuming that the stream continues for a large number of years.

Verify by differentiation:  tert dt = ert(1 + rt) r2 + C 5 Use Eq. (5) to compute the PV of an investment that pays out income continuously at a rate R(t) = (5000 + 1000t) dollars per year for 5 years, assuming r = 0.05

Use Eq. (5) to compute the PV of an investment that pays out income continuously at a rate R(t) = (5000 + 1000t)e0.02t dollars per year for 10 years, assuming r = 0.08.

Bankers Rule of 70 If you earn an interest rate of R percent, continuously compounded, your money doubles after approximately 70/R years. For example, at R = 5%, your money doubles after 70/5 or 14 years. Use the concept of doubling time to justify the Bankers Rule. (Note: Sometimes, the rule 72/R is used. It is less accurate but easier to apply because 72 is divisible by more numbers than 70.)

Drug Dosing Interval Let y(t) be the drug concentration (in micrograms per kilogram) in a patients body at time t. The initial concentration is y(0) = L. Additional doses that increase the concentration by an amount d are administered at regular time intervals of length T . In between doses, y(t) decays exponentiallythat is, y = ky. Find the value of T (in terms of k and d) for which the the concentration varies between L and L d as in Figure 15.

Show that y = Meaekt satisfies Eq. (6) for any constant a

To model mortality in a population of 200 laboratory rats, a scientist assumes that the number P (t) of rats alive at time t (in months) satisfies Eq. (6) with M = 204 and k = 0.15 month1 (Figure 16). Find P (t) [note that P (0) = 200] and determine the population after 20 months. 10 20 30 40 Rat population P(t) Time (months)

Isotopes for Dating Given that the age of the earth is approximately 4.5 billion years, which of the following would be most suitable for dating extremely old rocks: carbon-14 (half-life 5570 years), lead-210 (half-life 22.26 years), or potassium-49 (half-life 1.3 billion years)? Explain why.

Let P = P (t) be a quantity that obeys an exponential growth law with growth constant k. Show that P increases m-fold after an interval of (ln m)/k years.

Average Time of Decay Physicists use the radioactive decay law R = R0ekt to compute the average or mean time M until an atom decays. Let F (t) = R/R0 = ekt be the fraction of atoms that have survived to time t without decaying. (a) Find the inverse function t (F ). (b) By definition of t (F ), a fraction 1/N of atoms decays in the time interval  t  j N , t j 1 N  Use this to justify the approximation M 1 N N j=1 t  j N . Then argue, by passing to the limit as N , that M =  1 0 t (F ) dF. Strictly speaking, this is an improper integral because t (0) is infinite (it takes an infinite amount of time for all atoms to decay). Therefore, we define M as a limit M = lim c0  1 c t (F ) dF (c) Verify the formula  ln x dx = x ln x x by differentiation and use it to show that for c > 0, M = lim c0 1 k + 1 k (c ln c c) (d) Show that M = 1/k by evaluating the limit (use LHpitals Rule to compute limc0 c ln c). (e) What is the mean time to decay for radon (with a half-life of 3.825 days)?

Modify the proof of the relation e = lim n  1 + 1 n n given in the text to prove ex = lim n  1 + x n n . Hint: Express ln(1 + xn1) as an integral and estimate above and below by rectangles.

Prove that, for n > 0,  1 + 1 n n e  1 + 1 n n+1 Hint: Take logarithms and use Eq. (4)

A bank pays interest at the rate r, compounded M times yearly. The effective interest rate re is the rate at which interest, if compounded annually, would have to be paid to produce the same yearly return. (a) Find re if r = 9% compounded monthly. (b) Show that re = (1 + r/M)M 1 and that re = er 1 if interest is compounded continuously. (c) Find re if r = 11% compounded continuously. (d) Find the rate r that, compounded weekly, would yield an effective rate of 20%.