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Precalculus Chapter 3 Practice Test Monday, November 6 Name 1. The mass, Q, of a sample of Tritium (a radioactive isotope of Hydrogen), decays at a rate of 5.626% per year, t. Write an equation to describe the decay of a 726 gram sample. Sketch a graph of the decay function. 2. Every year, a lake becomes more polluted, and 2% fewer organisms can live in it. If in 2010 there are one million organisms, write an equation relating O, the number of organisms, to time, t, in years since 2010. 2 3. Without a calculator or computer, match each exponential formula to one of the graphs I-VI. 10 1.2 10 1.5 20 1.2 t t t a) b) c) 30 0.85 30 0.95 30 1.05 t t t d) e) f) 4. In the year 2000, a total of 9.8 million passengers took a cruise vacation. The global cruise industry has been growing at approximately 8% per year for the last decade; assume that this growth rate continues. (a) Write a formula to approximate the number, N, of cruise passengers (in millions) t years after 2000. (b) How many cruise passengers are predicted in the year 2010? Approximately how many passengers went on a cruise in the year 1990? 3 5. The amount (in milligrams) of a drug in the body t hours after taking a pill is given by At 25 0.85 . t a) What is the initial dose given? b) What percent of the drug leaves the body each hour? c) What is the amount of drug left after 10 hours? d) After how many hours is there less than 1 milligram left in the body? 6. (a) Decide if the function is linear or exponential. (b) Find a possible formula for each function and sketch a graph on the same axes. 4 7. Give an equation for the exponential function described by f 3 and f 2 12 3 8 8. Give an equation for the exponential function shown below. 9. Describe the functions below as linear, exponential, or neither. Write possible equations for the linear or exponential functions. 5 The graphs of f x 1.1 , g x 1.2 and h x 1.25 are shown below. x x x 10. Briefly explain how you can match these formulas and graphs without a calculator. 11. The table below shows the populations of the planet Vulcan, which is growing exponentially, and of the planet Romulus, which is growing linearly. (a) Find a formula for the population (in millions) of each planet as a function of the number of years since 2010. (b) Use the formulas to predict the population of each planet in the year 2030. (c) Estimate the year in which the population of Vulcan reaches 50 million. (d) Estimate the year in which the population of Vulcan overtakes the population of Romulus. 6 12. Consider the exponential functions graphed in the figure below and the six constants a, b, c, d, p, q. (a) Which of these constants are definitely positive? (b) Which of these constants are definitely between 0 and 1? (c) Which of these constants could be between 0 and 1? (d) Which two of these constants are definitely equal? (e) Which one of the following pairs of constants could be equal? 7 13. Graph f x , a function defined for all real numbers and satisfying the condition. f x 3 as x 14. Graph the function to find horizontal asymptotes. f x 3x 2 2 15. The population of a colony of rabbits grows exponentially. The colony begins with 10 rabbits; five years later there are 340 rabbits. (a) Give a formula for the population of the colony of rabbits as a function of the time. (b) Use a graph to estimate how long it takes for the population of the colony to reach 1000 rabbits. 8 16. Let f be a piecewise-defined function given by 2 x , x0 f x 0, x0 1 1 x, x0 2 a) Graph f for 3 x 4 b) The domain for f x is all real numbers. What is its range? c) What are the intercepts of f ? d) What happens to f x as f x and x ? e) Over what intervals is f increasing? Decreasing? 9 17. A population grows from its initial level of 22,000 at a continuous growth rate of 7.1% per year. (a) Find a formula for P t , the population in year t. (b) By what percent does the population increase each year? 18. The same amount of money is deposited into two different bank accounts paying the same nominal rate, one compounded annually and the other compounded continuously. Which curve in the figure below corresponds to which compounding method? What is the initial deposit? 19. Find the effective annual yield and the continuous growth rate if Q 5500 e 0.19t . 10 20. A radioactive substance decays at a continuous rate of 14% per year, and 50 mg of the substance is present in the year 2000. a) Write a formula for the amount present, A (in mg), t years after 2000. b) How much will be present in the year 2010? c) Estimate when the quantity drops below 5 mg. 21. Without a calculator, match the functions y e x , y 2e x , and y 3e x to the graphs shown below. 22. Without a calculator, match the functions y e x , y e x , and y e x to the graphs shown below. 11 23. How long does it take an investment to double if it grows according to the formula V 537e 0.015t ? Assume t is in years. 24. If $5000 is deposited in an account paying a nominal interest rate of 4% per year, how much is in the account 10 years later if interest is compounded a) Annually? b) Continuously? 25. An investment decreases by 60% over a 12-year period. At what effective annual percent rate does it decrease? 12 Answer True or False to the following. Give an explanation for your answer. 26. Exponential functions are functions that increase or decrease at a constant percent rate. 27. The independent variable in an exponential function is always found in the exponent. t 2 28. If f t 3 , then f is a decreasing function. 5 29. A population that has 1000 members and decreases at 10% per year can be modeled as P 1000 0.10 . t 30. If a population increases by 50% each year, then in two years it increases by 100%. 13 Answer True or False to the following. Give an explanation for your answer. 31. In the equation Q ab t , the value of a tells us where the graph crosses the Q- axis. 32. If f x k as k , we say that the line y k is a horizontal asymptote. 33. Exponential graphs are always concave up. 34. There is no limit to the amount a twenty-year $10,000 investment at 5% interest can earn if the number of times the interest is compounded becomes greater and greater. 35. If you put $1000 into an account that earns 5.5% compounded continuously, then it takes about 18 years for the investment to grow to $2000.