8.2 Exercises. Section 8.2 Exponential Functions 783

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1 Section 8.2 Eponential Functions Eercises 1. The current population of Fortuna is 10,000 heart souls. It is known that the population is growing at a rate of 4% per ear. Assuming this rate remains constant, perform each of the following tasks. predict the population 40 ears from c. Use our calculator to sketch the graph of the population over the net 40 ears. 2. The population of the town of Imagination currentl numbers 12,000 people. It is known that the population is growing at a rate of 6% per ear. Assuming this rate remains constant, perform each of the following tasks. predict the population 30 ears from c. Use our calculator to sketch the graph of the population over the net 30 ears. 3. The population of the town of Despairia currentl numbers 1,000 individuals. It is known that the population is decaing at a rate of % per ear. Assuming this rate remains constant, perform each of the following tasks. predict the population 0 ears from c. Use our calculator to sketch the graph of the population over the net 0 ears. 4. The population of the town of Hopeless currentl numbers 2,000 individuals. It is known that the population is decaing at a rate of 6% per ear. Assuming this rate remains constant, perform each of the following tasks. predict the population 40 ears from c. Use our calculator to sketch the graph of the population over the net 40 ears. In Eercises -12, perform each of the following tasks for the given function. a. Find the -intercept of the graph of the function. Also, use our calculator to find two points on the graph to the right of the -ais, and two points to the lef b. Using our five points from (a) as a guide, set up a coordinate sstem on graph paper. Choose and label appropriate scales for each ais. Plot the five points, and an additional points ou feel are necessar to dis- 1 Coprighted material. See:

2 784 Chapter 8 Eponential and Logarithmic Functions cern the shape of the graph. c. Draw the horizontal asmptote with a dashed line, and label it with its equation. d. Sketch the graph of the function. e. Use interval notation to describe both the domain and range of the function f() = (2.) 6. f() = (0.1) 7. f() = (0.7) 8. f() = (1.1) f() = f() = f() = f() = In Eercises 13-20, the graph of an eponential function of the form f() = b + c is shown. The dashed red line is a horizontal asmptote. Determine the range of the function. Epress our answer in interval notation

3 Section 8.2 Eponential Functions f() = (/2) ; p = f() = 9 ; p = f() = ; p = f() = 9 ; p = f() = (6/) ; p = f() = (3/) ; p = 0 In Eercises 33-40, use our calculator to evaluate the function at the given value p. Round our answer to the nearest hundredth. 33. f() = 10 ; p = f() = 10 ; p = f() = (2/) ; p = f() = 2 ; p = 3/ f() = 10 ; p = f() = 7 ; p = 4/ f() = 10 ; p = 1/. 40. f() = (4/3) ; p = 1.1. In Eercises 21-32, compute f(p) at the given value p. 21. f() = (1/3) ; p = f() = (3/4) ; p = f() = ; p = 24. f() = (1/3) ; p = 4 2. f() = 4 ; p = f() = ; p = This eercise eplores the propert that eponential growth functions eventuall increase rapidl as increases. Let f() = 1.0. Use our graphing calculator to graph f on the intervals (a) [0, 10] and (b) [0, 100]. For (a), use Ymin = 0 and Yma = 10. For (b), use Ymin = 0 and Yma = 100. Make accurate copies of the images in our viewing window on our homework paper. What do ou observe when ou compare the two graphs?

4 786 Chapter 8 Eponential and Logarithmic Functions 8.2 Answers 1. a) P (t) = (1.04) t b) P (40) c). a) The -intercept is (0, 1). Evaluate the function at = 1, 2, 1, 2 to obtain the points (1, 2.), (2, 6.2), ( 1, 0.4), ( 2, 0.16) (other answers are possible). c) The horizontal asmptote is = 0. See the graph in part (. 3. a) P (t) = 1 000(0.9) t b) P (0) 1 14 c) 10 f()=(2.) =0 3 e) Domain = (, ), Range = (0, )

5 Section 8.2 Eponential Functions a) The -intercept is (0, 1). Evaluate the function at = 1, 2, 1, 2 to obtain the points (1, 0.7), (2, 0.6), ( 1, 1.34), ( 2, 1.78) (other answers are possible). 20 f()=3 +1 c) The horizontal asmptote is = 0. See the graph in part (. =1 3 f()=(0.7) =0 e) Domain = (, ), Range = (0, ) e) Domain = (, ), Range = (1, ) 11. a) The -intercept is (0, 2). Evaluate the function at = 1, 2, 1, 2 to obtain the points (1, 1), (2, 1), ( 1, 2.), ( 2, 2.7) (other answers are possible). c) The horizontal asmptote is = 3. See the graph in part (. 9. a) The -intercept is (0, 2). Evaluate the function at = 1, 2, 1, 2 to obtain the points (1, 4), (2, 10), ( 1, 1.34), ( 2, 1.11) (other answers are possible). f()=2 3 c) The horizontal asmptote is = 1. See the graph in part (. = 3 e) Domain = (, ), Range = ( 3, ) 13. ( 1, ) 1. (2, ) 17. (2, )

6 788 Chapter 8 Eponential and Logarithmic Functions 19. ( 2, ) a) The graph on the interval [0, 10] increases ver slowl. In fact, the graph looks almost linear. b) The graph on the interval [0, 100] increases slowl at first, but then increases ver rapidl on the second half of the interval.