Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Scientists used carbon-14 dating of the soot from the oil lamps the artists used for lighting to date the paintings in the cave at Lascaux, France, to about 13–15,000 B.C. Because of the fragile nature of the paintings, a replica of the cave was built near the original as a tourist attraction. The modern materials used in the replica are resistant to deterioration from varying concentrations of CO 2, unlike the originals. Figure 8 shows a photo of one of the paintings in the replica cave.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 9 shows the decay of carbon 14 versus time.

Example, Page A quantity P obeys exponential growth law P(t) = Ce kt (t in years). Find the formula for P(t), assuming the doubling time is seven years and P(0) = 100. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page Find the decay constant of Radium–226, given that its half-life is 1,622 years. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page Measurements showed that a sample of sheepskin parchment discovered by archeologists had a C 14 to C 12 ratio equal to 40% of that found in the atmosphere. Approximately how old is the parchment? (C 14 – k = years -1 ) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Exponential functions are used to calculate interest earned on savings accounts, payments on loans, and other similar applications, as noted in Table 1.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Banks most commonly pay compound interest on savings accounts. This means that interest payments are made several times per year, usually quarterly. Since the interest paid at the end of one period, earns interest in following periods, we say the interest is compounded.

Example, Page Suppose $500 is deposited into an account paying interest of 7% compounded continuously. Find a formula for the value of the account at time t. What is the value of the account after 3 years? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 10 illustrates the limit for the function as n grows without bound. This limit is for the first function in Theorem 2.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Continuous compounding provides the ultimate in interest accumulation.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The present value (PV) of an amount of money to be received in the future is the smaller amount that, with continuously compounded interest, would equal the future amount. If one wins the California lottery, they may opt for either ten equal payments or a lump sum payment. The lump sum represents the PV of the ten equal payments.

Example Suppose Jose has the winning ticket for the Lotto, with the prize valued at $6.7-million. He has a choice of ten annual payments of $670,000 each or a lump sum payment. What would the lump sum payment be if the current interest rate is 9.5%? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page How much must be invested today to receive $20,000 after 5 years if interest is compounded continuously at 9%. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page An investment group purchased a building in 1998 for $17 million and sold it five years later for $26 million. What was the (continuously compounded) annual rate of return? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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