Homework Homework Assignment #9 Review Section 5.8 Page 365, Exercises: 1 – 49(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 1. A bacteria population P obeys the exponential growth law P(t) = 2,000e1.3t (t in hours). (a) How many bacteria are present initially? (b) At what time will there be 10,000 bacteria? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 5. The decay constant of Cobalt–60 is 0.13 years–1. What is its half-life? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 9. Find the solution to y ′ = 3y satisfying y(2) = 4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 13. Assuming that population growth is approximately exponential, which of the 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 Data I 3.14 3.36 3.60 3.85 4.11 Data II 3.24 3.54 4.04 4.74 Data II is most likely to represent the population of the city over the five-year period, as Data I is almost linear. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 17. A 10-kg quantity of radioactive isotope decays to 3-kg after 17 years. Find the decay constant of the isotope. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 21. The atmospheric pressure P(h) (in psi) at a height h (in miles) above sea level on earth satisfies a differential equation P′ = – kP for some positive constant k. (a) Measurements with a barometer show that P(0) = 14.7 and P(10) = 2.13. What is the decay constant k? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 21. (b) Determine the atmospheric pressure 15 miles above sea level. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 25. In 1965, Gordon Moore predicted that the number of transistors on a microchip would increase exponentially. (a) Does the table of data confirm Moore’s prediction? If so, estimate the growth constant. Year No. Trans, 1971 2,250 1972 2,500 1974 5,000 1978 29,000 1982 120,000 1985 275,000 1989 1,180,000 1993 3,100,000 1997 7,500,000 1999 24,000,000 2000 42,000,000 The data seems to support Moore’s prediction. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 25. (b) Plot the data in the table. Year No. Trans, 1971 2,250 1972 2,500 1974 5,000 1978 29,000 1982 120,000 1985 275,000 1989 1,180,000 1993 3,100,000 1997 7,500,000 1999 24,000,000 2000 42,000,000 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 25. (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. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 25. (d) Estimate the doubling time in Moore’s Law for the period 1971 – 2000. (e) If Moore’s Law holds to the end of the decade, how many transistors will a microchip hold in 2010? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 25. (e) Can Moore expect his prediction to hold indefinitely? Moore cannot expect his prediction to hold indefinitely, as a some point transistors will get as small as they can, one or more molecules, in size. That coupled with the finite length of the connecting conductors will limit the miniaturization we have seen over the last several decades. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 29. A certain quantity increases quadratically: P(t) =P0t2. (a) Starting at time t0 = 1, how long will it take for P to double in size? How long will it take starting at t0 = 2 or 3? (b) In general, starting at time t0, how long will it take for P to double in size? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 33. A bank pays interest at the rate of 5%. What is the yearly multiplier, if interest is compounded (a) annually? 1.05 (b) three times per year? (c) continuously? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 37. An investment increases in value at a continuously compounded rate of 9%. How large must the initial investment be to build up a value of $50,000 over a seven-year period? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 41. If a company invests $2,000,000 to upgrade a factory, it will earn additional profits of $500,000 per year. Is the investment worthwhile, assuming an interest rate of 6%? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 45. Use equation 3 to compute PV of an income stream paying out R(t) = $5,000/yr continuously for ten yr at r = 0.05. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 366 49. Show that PV of an investment that pays out R dollars/yr continuously for T years is R(1 – e–rt)/r where r is the interest rate. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework Homework Assignment #10 Review Sections 5.1 – 5.8 Page 369, Exercises: 1 – 97(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company