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Unformatted text preview: 420 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS Example 6 Suppose that $1000 is deposited in a bank that pays 5% compounded continuously. How much money will be in the account after 5 years, and what is the interest earned during this period?
SOLUTION Here we have A0 = 1000 and r = 0. 05. Thus, the amount of money in the account at any time t is given by A(t ) = 1000 e0.05t . The amount of money in the account after 5 years is A(5) = 1000 e0.05(5) = 1000 e0.25 ∼ $1284. 03, = and the interest earned during this period is $284. 03
Example 7 How long does it take for an investment to double at an interest rate r compounded continuously?
SOLUTION If an amount A0 is invested at an interest rate r compounded continuously, then the value of the investment at time t is A(t ) = A0 ert . To ﬁnd the “doubling time”, we solve 2A0 = A0 ert for t : ert = 2, rt = ln 2, t= ln 2 ∼ 0. 69 . = r r
0.69 0.08 For example, if the interest rate is 8% = 0. 08, then it will take the investment to double in value. = 8. 625 years for Remark A popular estimate for the “doubling time” of an investment at an interest rate r % is the rule of 72: 72 . doubling time = r For example, at an interest of 8%, it will take approximately investment to double in value. Here’s how this rule originated:
72 8 = 9 years for an 69 ∼ 72 0. 69 = . = r% r r Since we are only estimating, 72 is preferred to 69 because it has many more divisors. EXERCISES 7.6
c NOTE: Some of these exercises require a calculator or graphing utility.
1. Find the amount of interest earned by $500 compounded continuously for 10 years: (a) at 6%, (b) at 8%, (c) at 10%. 2. How long does it take for a sum of money to double if compounded continuously: (a) at 6%? (b) at 8%? (c) at 10%? 3. At what rate r of continuous compounding does a sum of money triple in 20 years? 4. At what rate r of continuous compounding does a sum of money double in 10 years? 5. A certain species of virulent bacteria is being grown in a culture. It is observed that the rate of growth of the bacterial population is proportional to the number present. If there were 1000 bacteria in the initial population and the number doubled after the ﬁrst 30 minutes, how many bacteria will be present after 2 hours? 6. In a bacteria growing experiment, a biologist observes that the number of bacteria in a certain culture triples every 7.6 EXPONENTIAL GROWTH AND DECAY 421 4 hours. After 12 hours, it is estimated that there are 1 million bacteria in the culture. (a) How many bacteria were present initially? (b) What is the doubling time for the bacteria population? 7. A population P of insects increases at a rate proportional to the current population. Suppose there are 10,000 insects initially and 20,000 insects 1 week later. (a) Find an expression for the number P (t ) of insects at any time t . (b) How many insects will there be in 1 year? In 1 year? 2 8. Determine the period in which y = Cekt changes by a factor of q. 9. The population of a certain country is increasing at the rate of 3. 5% per year. By what factor does it increase every 10 years? What percentage increase per year will double the population every 15 years? 10. According to the Bureau of the Census, the population of the United States in 1990 was approximately 249 million and in 2000, 281 million. Use this information to estimate the population in 1980. (The actual ﬁgure was about 227 million.) 11. Use the data of Exercise 10 to predict the population for 2010. Compare the prediction for 2001 with the actual (estimated) ﬁgure of 284.8 million. 12. Use the data of Exercise 10 to estimate how long it will take for the U.S population to double. 13. It is estimated that 1 of an acre is needed to provide 3 food for one person. It is further estimated that there are 10 billion square miles of arable land on the earth. Thus, a maximum of 30 billion people can be sustained according to current estimates. Use the data of Example 3 to ﬁnd when the maximum population will be reached. 14. Water is pumped into a tank to dilute a saline solution. The volume of the solution, call it V , is kept constant by continuous outﬂow. The amount of salt in the tank, call it s, depends on the amount of water that has been pumped in, call this x. Given that s ds =− , dx V ﬁnd the amount of water that must be pumped into the tank to eliminate 50% of the salt. Take V as 10,000 gallons. 15. A 200-liter tank initially full of water develops a leak at the bottom. Given that 20% of the water leaks out in the ﬁrst 5 minutes, ﬁnd the amount of water left in the tank t minutes after the leak develops if the water drains off at a rate that is proportional to the amount of water present. In Exercises 16–20, remember that the rate of decay of a radioactive substance is proportional to the amount of substance present. 16. What is the half-life of a radioactive substance if it takes 5 years for one-third of the substance to decay? 17. Two years ago there were 5 grams of a radioactive substance. Now there are 4 grams. How much will remain 3 years from now? 18. A year ago there were 4 grams of a radioactive substance. Now there are 3 grams. How much was there 10 years ago? 19. Suppose the half-life of a radioactive substance is n years. What percentage of the substance present at the start of a year will decay during the ensuing year? 20. A radioactive substance weighed n grams at time t = 0. Today, 5 years later, the substance weighs m grams. How much will it weigh 5 years from now? 21. The half-life of radium-226 is 1620 years. What percentage of a given amount of the radium will remain after 500 years? How long will it take for the original amount to be reduced by 75%? 22. Cobalt-60 is used extensively in medical technology. It has a half-life of 5.3 years. What percentage of a given amount of cobalt will remain after 8 years? If you have 100 grams of cobalt now, how much was there 3 years ago? 23. (The power of exponential growth) Imagine two racers competing on the x-axis (which has been calibrated in meters), a linear racer LIN [position function of the form x1 (t ) = kt + C ] and an exponential racer EXP [position function of the form x2 (t ) = ekt + C ]. Suppose that both racers start out simultaneously from the origin, LIN at one million meters per second, EXP at only one meter per second. In the early stages of the race, fast-starting LIN will move far ahead of EXP, but in time EXP will catch up to LIN, pass her, and leave her hopelessly behind. Show that this is true as follows: (a) Express the position of each racer as a function of time, measuring t in seconds. (b) Show that LIN’s lead over EXP starts to decline about 13.8 seconds into the race. (c) Show that LIN is still ahead of EXP some 15 seconds into the race but far behind 3 seconds later. (d) Show that, once EXP passes LIN, LIN can never catch up. 24. (The weakness of logarithmic growth) Having been soundly beaten in the race in Exercise 23, LIN ﬁnds an opponent she can beat, LOG, the logarithmic racer [position function x3 (t ) = k ln (t + 1) + C ]. Once again the racetrack is the xaxis calibrated in meters. Both racers start out at the origin, LOG at one million meters per second, LIN at only one meter per second. (LIN is tired from the previous race.) In this race LOG will shoot ahead and remain ahead for a long time, but eventually LIN will catch up to LOG, pass her, and leave her permanently behind. Show that this is true as follows: (a) Express the position of each racer as a function of time t , measuring t in seconds. (b) Show that LOG’s lead over LIN starts to decline 106 − 1 seconds into the race. (c) Show that LOG is still ahead of LIN 107 − 1 seconds into the race but behind LIN 108 − 1 seconds into the race. 422 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS (d) Show that, once LIN passes LOG, LOG can never catch up. 25. Atmospheric pressure p varies with altitude h according to the equation dp = kp, dh where k is a constant. 31. 32. Given that p is 15 pounds per square inch at sea level and 10 pounds per square inch at 10,000 feet, ﬁnd p at: (a) 5000 feet; (b) 15,000 feet. 26. According to the compound interest formula (7.6.3), if P dollars are deposited in an account now at an interest rate r compounded continuously, then the amount of money in the account t years from now will be Q = Pert . The quantity Q is sometimes called the future value of P (at the interest rate r ). Solving this equation for P , we get P = Qe−rt . In this formulation, the quantity P is called the present value of Q. Find the present value of $20,000 at 6% compounded continuously for 4 years. Find the interest rate r that is needed to have $6000 be the present value of $10,000 over an 8-year period. You are 45 years old and are looking forward to an annual pension of $50,000 per year at age 65. What is the presentday purchasing power (present value) of your pension if money can be invested over this period at a continuously compounded interest rate of: (a) 4%? (b) 6%? (c) 8%? The cost of the tuition, fees, room, and board at XYZ College is currently $16,000 per year. What would you expect to pay 3 years from now if the costs at XYZ are rising at the continuously compounded rate of: (a) 5%? (b) 8%? (c) 12%? A boat moving in still water is subject to a retardation proportional to its velocity. Show that the velocity t seconds 33. 27. 28. 34. after the power is shut off is given by the formula v = ce−kt where c is the velocity at the instant the power is shut off. A boat is drifting in still water at 4 miles per hour; 1 minute later, at 2 miles per hour. How far has the boat drifted in that 1 minute? (See Exercise 30.) During the process of inversion, the amount A of raw sugar present decreases at a rate proportional to A. During the ﬁrst 10 hours, 1000 pounds of raw sugar have been reduced to 800 pounds. How many pounds will remain after 10 more hours of inversion? The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12 C (a stable isotope), and carbon-14, denoted 14 C (a radioactive isotope). The ratio of the amount of 14 C to the amount of 12 C is essentially constant (approximately 1/10,000). When an organism dies, the amount of 12 C present remains unchanged, but the 14 C decays at a rate proportional to the amount present with a half-life of approximately 5700 years. This change in the amount of 14 C relative to the amount of 12 C makes it possible to estimate the time at which the organism lived. A fossil found in an archaeological dig was found to contain 25% of the original amount of 14 C. What is the approximate age of the fossil? The Dead Sea Scrolls are approximately 2000 years old. How much of the original 14 C remains in them? In Exercises 35–37, ﬁnd all the functions f that satisfy the equation for all real t . 35. f (t ) = t f (t ). HINT: Write f (t ) − t f (t ) = 0 and multiply 2 the equation by e−t /2 . 36. f (t ) = sin t f (t ). 37. f (t ) = cos t f (t ). 38. Let g be a function everywhere continuous and not identically zero. Show that if f (t ) = g (t ) f (t ) for all real t , then either f is identically zero, or f does not take on the value zero at any t . 29. 30. 7.7 THE INVERSE TRIGONOMETRIC FUNCTIONS
Since none of the trigonometric functions are one-to-one, none of them have inverses. What, then, are the inverse trigonometric functions? The Inverse Sine
The graph of y = sin x is shown in Figure 7.7.1. Since every horizontal line between −1 and 1 intersects the graph at inﬁnitely many points, the sine function does not have 1 an inverse. However, observe that if we restrict the domain to the interval [− 1 π , 2 π ] 2 (the solid portion of the graph in Figure 7.7.1), then y = sin x is one-to-one, and on that interval it takes on as a value every number in [− 1,1]. Thus, if x ∈ [− 1, 1], there is one and only one number in the interval [− 1 π , 1 π ] at which the sine function has 2 2 the value x. This number is called the inverse sine of x, or the angle whose sine is x, and is written sin−1 x. ...
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