Homework 3. (33-40) The graph of an exponential function is given. Match each graph to one of the following functions.

Transcription

1 Homework Section 4. (-40) The graph of an exponential function is given. Match each graph to one of the following functions. (a)y = x (b)y = x (c)y = x (d)y = x (e)y = x (f)y = x (g)y = x (h)y = x (46, 48, 0) Use transformations to graph each function. Determine the domain, range and horizontal asymptote of each ( ) function. x 46. f(x) = 4, 48. f(x) = x +, 0. f(x) = 2 x+ (6, 60) Begin with the graph of y = e x and use transformation to graph each functions. Determine the domain, range, and horizontal asymptote of each function. 6. f(x) = e x, 60. f(x) = 7 e 2x (66, ( 72, ) 78) Solve each equation. x 66. = 4 64, x+ = 27 x, 78. e x = e 2 x 82. If 2 x =, what does 4 x equal? 84. If x =, what does x equal? 94. Suppose that g(x) = x. (a) What is g( )? What point is on the graph of g? (b) If g(x) = 22, what is x? What point is on the graph of g?

2 ( ) 2 x 96. Suppose that F (x) =. (a) What is F ( )? What point is on the graph of F? (b) If F (x) = 24, what is x? What point is on the graph of F? (c) Find the zero of F. Section 4.4 (4, ) Change each exponential statement to an equivalent statement involving a logarithm. 4. x = 4.6,. e x = 8 (22, 24) Change each logarithmic statement to an equivalent statement involving exponential. 22. log 2 6 = x, 24. ln x = 4 (42, 44, 46) Find the domain of each function. ( ) ( ) x 42. g(x) = 8 + ln(2x + ), 44. g(x) = ln, 46. h(x) = log x x 60. Graph the function and its inverse on the same Cartesian plane. f(x) = 4 x ; f (x) = log 4 x. (6-70) The graph of a logarithmic function is given. Match each graph to one of the following functions. (a)y = log x (b)y = log ( x) (c)y = log x (d)y = log ( x) (e)y = log x (f)y = log (x ) (g)y = log ( x) (h)y = log x (72, 77, 82, 84, 86) Use the given function f to : (a) Find the domain of f. (b) Graph f.

3 (c) From the graph, determine the range and any asymptotes of f. (d) Find f. (e) Find the domain and the range of f. (f) Graph f. 72. f(x) = ln(x ), 77. f(x) = log(x 4) + 2, 82. f(x) = 2 log (x + ), 84. f(x) = e x + 2, 86. f(x) = x+ Section 4. (6, 7, 20, 22, 24, 26, 28) Use properties of logarithms to find the exact value of each expression. Do not use a calculator. 6. ln e log log log log 8 6 log log 8 log log 6+log 7 28.e log e 2 9 (0, 2) ( Write each) expression as a sum and/or difference of logarithms. Express powers as factors. [ x2 + ] x x + 0. log x 2 ; x >, 2. log (x ) 2 ; x > 2 (60, 62, 66) ( ) Write each ( expression ) as a single logarithm. x 60. log 2 + log x log(x 2 + x + 2) 2 log(x + ) log x + log (9x 2 ) log Express y as a function of x. The constant C is a positive number. ln y = 02x + ln C 98. Find the value of log 2 4 log 4 6 log 6 8. Section 4.6 (24, 28, 9, 49, 62, 68) Solve each equation. Express any irrational solution in exact form and as a decimal rounded to three decimal places. 24. log (x + ) = log (x ) 28. log 2 (x + ) + log 2 (x + 7) = 9. (log x) 2 (log x) = x = 4 x

4 x x+ + = x 4 x = 97. f(x) = log 2 (x + ) and g(x) = log 2 (x + ). (a) Solve f(x) =. What point is on the graph of f? (b) Solve g(x) = 4. What point is on the graph of g? (c) Solve f(x) = g(x). Do the graphs of f and g intersect? If so, where? (d) Solve (f + g)(x) = 7. (e) Solve (f g)(x) = f(x) = log (x + ) and g(x) = log (x ). (a) Solve f(x) = 2. What point is on the graph of f? (b) Solve g(x) =. What point is on the graph of g? (c) Solve f(x) = g(x). Do the graphs of f and g intersect? If so, where? (d) Solve (f + g)(x) =. (e) Solve (f g)(x) = 2. Section Find the present value to get $800 after 2 years at 7% compounded monthly. 22. Find the present value to get &800 after 2 2 years at 8% compounded continuously. 2. What rate of interest compounded annually is required to double an investment in 6 years? 6. (a) How long does it take for an investment to triple in value if it is invested at 6 % compounded monthly? (b) How long does it take if the interest is compounded continuously? Section The number N of bacteria present in a culture at time t(in hours) obeys the model N(t) = 000e 0.0t. (a) Determine the number of bacteria at t = 0 hours. (b) What is the growth rate of the bacteria? (c) What is the population after 4 hours? (d) When will the number of bacteria reach 700?

5 (e) When will the number of bacteria double? 9. (Radioactive Decay) The half-life of radium is 690 years. If 0 grams is present now, how much will be present in 0 years? Section. (, 4, 20, 22) Draw each angle in standard position.., 4. 20, 20. 2π, 22. 2π 4 (8, 44) Convert each angle in degrees to radians. Express your answer as a multiple of π. 8. 0, (4, 8) Convert each angle in radians to degrees. π 4. 2, 8. π 4 (76, 78, 8) Find the missing quantity. Round answers to three decimal places. s denotes the length of the arc of a circle of radius r subtended by the central angle θ. A denotes the area of the sector of a circle of radius r formed by the central angle θ. 76. r = 6 meters, s = 8 meters, θ =? 78. r = meters, θ = 20, s =? 8. r = 2 inches, θ = 0, A =? 90. Find the length s and area A. Round answers to three decimal places. 9. The minute hand of a clock is 6 inches long. How far the tip of the minute hand move in minutes? How far does it move in 2 minutes? Round answers to two decimal places. 92. A pendulum swings through an angle of 20 each second. If the pendulum is 40 inches long, how far does its tip move each second? Round answers to two decimal places. Section.2

6 6 (6, 20) P = (x, y) is the point on the unit circle that corresponds to a real number t. Find the exact values ( of the six) trigonometric ( functions of t. 6., 2 ) 6, 20., 2 (40, 42) Find the exact values of each expression. Do not use a calculator. 40. cos 90 8 sin 270, sin π 4 + tan π 4 (6, 64) Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say not defined. Do not use a calculator , 64. π Find the exact value of : sin 40 + sin 0 + sin sin If cos θ = 2, find sec θ. (07, 0, ) Find the value of each of the following, where f(x) = sin x, g(x) = cos x, h(x) = 2x, and x (f + g)(0 ( ) 4π 0. (f g) ( π ). (f g) 6

7 References 7 [] Precalculus, rd Edition, Michael Sullivan and Michael Sullivan, III.