Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

Exponential Functions from a Graph Find the exponential function f(x) = ax whose graph is given. 23.24E Exponential Functions from a Graph Match the exponential function with one of the graphs labeled I or II. 25. f(x) = 5x + 1 Exponential Functions from a Graph Match the exponential function with one of the graphs labeled I or II. 26. f(x) = 5x + 1 Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2. State the domain, range, and asymptote. 27. g(x) = 2x 3 28E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2. State the domain, range, and asymptote. 29. f(x) = 3x 30E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2. State the domain, range, and asymptote. 31. f(x) = 10x + 3 32E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2. State the domain, range, and asymptote. 33. y = 5x + 1 34E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2. State the domain, range, and asymptote. 35. y=2(13)x 36E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2. State the domain, range, and asymptote. 37. h(x) = 2x 4 + 1 38E 39E 40E 41E Comparing Exponential Functions In these exercises we compare the graphs of two exponential functions. 42. (a) Sketch the graphs of f(x) = 9x/2 and g(x) = 3x. (b) Use the Laws of Exponents to explain the relationship between these graphs.Comparing Exponential and Power Functions Compare the graphs of the power function f and exponential function g by evaluating both of them for x = 0, 1, 2, 3, 4, 6, 8, and 10. Then draw the graphs of f and g on the same set of axes. 43. f(x) = x3; g(x) = 3x Comparing Exponential and Power Functions Compare the graphs of the power function f and exponential function g by evaluating both of them for x = 0, 1, 2, 3, 4, 6, 8, and 10. Then draw the graphs of f and g on the same set of axes. 44. f(x) = x4; g(x) = 4x 45E 46E 47E Families of Functions Draw graphs of the given family of functions for c = 0.25, 0.5, 1, 2, 4. How are the graphs related? 48. f(x) = 2cx 49E 50E Difference Quotients These exercises involve a difference quotient for an exponential function. 51. If f(x) = 10x, show that f(x+h)f(x)h=10x(10h1h)52E Bacteria Growth A bacteria culture contains 1500 bacteria initially and doubles every hour. (a) Find a function N that models the number of bacteria after t hours. (b) Find the number of bacteria after 24 hours.Mouse Population A certain breed of mouse was introduced onto a small island with an initial population of 320 mice, and scientists estimate that the mouse population is doubling every year. (a) Find a function N that models the number of mice after t years. (b) Estimate the mouse population after 8 years.55E 56E Compound Interest If 10,000 is invested at an interest rate of 3% per year, compounded semiannually, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years Compound Interest If 2500 is invested at an interest rate of 2.5% per year, compounded daily, find the value of the investment after the given number of years. (a) 2 years (b) 3 years (c) 6 years Compound Interest If 500 is invested at an interest rate of 3.75% per year, compounded quarterly, find the value of the investment after the given number of years. (a) 1 year (b) 2 years (c) 10 years Compound Interest If 4000 is borrowed at a rate of 5.75% interest per year, compounded quarterly, find the amount due at the end of the given number of years. (a) 4 years (b) 6 years (c) 8 years Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. 61. Find the present value of 10,000 if interest is paid at a rate of 9% per year, compounded semiannually, for 3 years.Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. 62. Find the present value of 100,000 if interest is paid at a rate of 8% per year, compounded monthly, for 5 years.63E 64E 65E 66E The function f(x) = ex is called the __________ exponential function. The number e is approximately equal to __________In the formula A(t) = Pert for continuously compound interest, the letters P, r, and t stand for __________, __________, and __________, respectively, and A(t) stands for __________. So if 100 is invested at an interest rate of 6% compounded continuously, then the amount after 2 years is __________.Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. 3. h(x)=ex;h(1),h(),h(3),h(2)Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. 4. h(x)=e3x;h(13),h(1.5),h(1),h()5E 6E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graph of y = ex in Figure 1. State the domain, range, and asymptote. 7. g(x) = 2 + ex 8E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graph of y = ex in Figure 1. State the domain, range, and asymptote. 9. f(x) = ex 10E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graph of y = ex in Figure 1. State the domain, range, and asymptote. 11. y = ex 1 12E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graph of y = ex in Figure 1. State the domain, range, and asymptote. 13. f(x) = ex 2 14E Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graph of y = ex in Figure 1. State the domain, range, and asymptote. 15. h(x) = ex + 1 3 16E Hyperbolic Cosine Function The hyperbolic cosine function is defined by cosh(x)=ex+ex2 (a) Sketch the graphs of the functions y=12ex and y=12ex on the same axes, and use graphical addition (see Section 2.7) to sketch the graph of y = cosh(x). (b) Use the definition to show that cosh(x) = cosh(x).18E 19E 20E 21E 22E Medical Drugs When a certain medical drug is administered to a patient, the number of milligrams remaining in the patients bloodstream after t hours is modeled by D(t)=50e0.2t How many milligrams of the drug remain in the patients bloodstream after 3 hours?Radioactive Decay A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function m(t)=13e0.015t where m(t) is measured in kilograms. (a) Find the mass at time t = 0. (b) How much of the mass remains after 45 days?Sky Diving A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is 0.2. It can be shown that the downward velocity of the sky diver at time t is given by v(t)=180(1e0.2t) where t is measured in seconds (s) and v(t) is measured in feet per second (ft/s). (a) Find the initial velocity of the sky diver. (b) Find the velocity after 5 s and after 10 s. (c) Draw a graph of the velocity function v(t). (d) The maximum velocity of a falling object with wind resistance is called its terminal velocity. From the graph in part (c) find the terminal velocity of this sky diver.Mixtures and Concentrations A 50-gal barrel is filled completely with pure water. Salt water with a concentration of 0.3 lb/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time t is given by Q(t)=15(1e0.04t) where t is measured in minutes and Q(t) is measured in pounds. (a) How much salt is in the barrel after 5 min? (b) How much salt is in the barrel after 10 min? (c) Draw a graph of the function Q(t). (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as t becomes large. Is this what you would expect?Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: P(t)=d1+kect where c, d, and k are positive constants. For a certain fish population in a small pond d = 1200, k = 11, c = 0.2, and t is measured in years. The fish were introduced into the pond at time t = 0. (a) How many fish were originally put in the pond? (b) Find the population after 10, 20, and 30 years. (c) Evaluate P(t) for large values of t. What value does the population approach as t ? Does the graph shown confirm your calculations?Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model n(t)=56000.5+27.5e0.044t where r is measured in years. (a) Find the initial bird population. (b) Draw a graph of the function n(t). (c) What size does the population approach as time goes on?World Population The relative growth rate of world population has been decreasing steadily in recent years. On the basis of this, some population models predict that world population will eventually stabilize at a level that the planet can support. One such logistic model is P(t)=73.26.1+5.9e0.02t where t = 0 is the year 2000 and population is measured in billions. (a) What world population does this model predict for the year 2200? For 2300? (b) Sketch a graph of the function P for the years 2000 to 2500. (c) According to this model, what size does the world population seem to approach as time goes on?Tree Diameter For a certain type of tree the diameter D (in feet) depends on the trees age t (in years) according to the logistic growth model D(t)=5.41+2.9e0.01t Find the diameter of a 20-year-old tree.31E Compound Interest An investment of 7000 is deposited into an account in which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates. 32. t = 10 years Rate per years Amount 1% 2% 3% 4% 5% 6%Compound Interest If 2000 is invested at an interest rate of 3.5% per year, compounded continuously, find the value of the investment after the given number of years. (a) 2 years (b) 4 years (c) 12 years Compound Interest If 3500 is invested at an interest rate of 6.25% per year, compounded continuously, find the value of the investment after the given number of years. (a) 3 years (b) 6 years (c) 9 years Compound Interest If 600 is invested at an interest rate of 2.5% per year, find the amount of the investment at the end of 10 years for the following compounding methods. (a) Annually (b) Semiannually (c) Quarterly (d) Continuously 36E Compound Interest Which of the given interest rates and compounding periods would provide the best investment? (a) 212 per year, compounded semiannually (b) 214 per year, compounded monthly (c) 2% per year, compounded continuously 38E Investment A sum of 5000 is invested at an interest rate of 9% per year, compounded continuously. (a) Find the value A(t) of the investment after t years. (b) Draw a graph of A(t). (c) Use the graph of A(t) to determine when this investment will amount to 25,000.log x is the exponent to which the base 10 must be raised to get __________. So we can complete the following table for log x.The function f(x) = log9 x is the logarithm function with base __________. So f(9) = __________, f(1) = __________, f(19)= __________, f(81) = __________, and f(3) = __________.3E Match the logarithmic function with its graph. (a) f(x) = log2 x (b) f(x) = log2 (x) (c) f(x) = log2 x (d) f(x) = log2(x)The natural logarithmic function f(x) = ln x has the __________ asymptote x = _____.The logarithmic function f(x) = ln(x 1) has the __________ asymptote x = _____.Logarithmic and Exponential Forms Complete the table by finding the appropriate logarithmic or exponential form of the equation, as in Example 1. Logarithmic form Exponential form log88=1 __________ log864=2 __________ __________ 82/3 = 4 __________ 83 = 512 log8(18)=1 __________ __________ 82=164 8E 9E 10E 11E 12E 13E 14E 15E Exponential Form Express the equation in exponential form. 16. (a) ln(x + 1) = 2 (b) ln(x 1) = 4 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E Evaluating Logarithms Evaluate the expression. 28. (a) log2 32 (b) log8 817 (c) log6 1 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E Finding Logarithmic Functions Find the function of the form y = loga x whose graph is given. 53.Finding Logarithmic Functions Find the function of the form y = loga x whose graph is given. 54.Finding Logarithmic Functions Find the function of the form y = loga x whose graph is given. 55.Finding Logarithmic Functions Find the function of the form y = loga x whose graph is given. 56.57E Graphing Logarithmic Functions Match the logarithmic function with one of the graphs labeled I or II. 58. f(x) = ln(x 2)59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E Graphing Logarithmic Functions Graph the function, not by plotting points, but by starting from the graphs in Figures 4 and 9. State the domain, range, and asymptote. 71. y = | ln x |72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E 83E 84E 85E 86E Domain of a Composition Find the functions f g and g f and their domains. 87. f(x) = log2x, g(x) = x 2 88E 89E 90E 91E 92E 93E 94E 95E Absorption of Light A spectrophotometer measures the concentration of a sample dissolved in water by shining a light through it and recording the amount of light that emerges. In other words, if we know the amount of light that is absorbed, we can calculate the concentration of the sample. For a certain substance the concentration (in moles per liter, mol/L) is found by using the formula C=2500ln(II0) where I0 is the intensity of the incident light and I is the intensity of light that emerges. Find the concentration of the substance if the intensity I is 70% of I0.Carbon Dating The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it If D0 is the original amount of carbon-14 and D is the amount remaining, then the artifacts age A (in years) is given by A=8267ln(DD0) Find the age of an object if the amount D of carbon-14 that remains in the object is 73% of the original amount D0.Bacteria Colony A certain strain of bacteria divides every 3 hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by t=3log(N/50)log2 Find the time required for the colony to grow to a million bacteria.99E Charging a Battery The rate at which a battery charges is slower the closer the battery is to its maximum charge C0. The time (in hours) required to charge a fully discharged battery to a charge C is given by t=kln(1CC0) where k is a positive constant that depends on the battery. For a certain battery, k = 0.25. If this battery is fully discharged, how long will it take to charge to 90% of its maximum charge C0?Difficulty of a Task The difficulty in acquiring a target (such as using your mouse to click on an icon on your computer screen) depends on the distance to the target and the size of the target. According to Fittss Law, the index of difficulty (ID) is given by ID=log(2A/W)log2 where W is the width of the target and A is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 mm wide to clicking on one that is 10 mm wide. In each case, assume that the mouse is 100 mm from the icon.102E 103E 104E DISCUSS DISCOVER: The Number of Digits in an Integer Compare log 1000 to the number of digits in 1000. Do the same for 10,000. How many digits does any number between 1000 and 10,000 have? Between what two values must the common logarithm of such a number lie? Use your observations to explain why the number of digits in any positive integer x is logx+1. (The symbol n is the greatest integer function defined in Section 2.2.) How many digits does the number 2100 have?The logarithm of a product of two numbers is the same as the __________ of the logarithms of these numbers. So log5(25 125) = __________ + __________.The logarithm of a quotient of two numbers is the same as the __________ of the logarithms of these numbers. So log5(25125)=.The logarithm of a number raised to a power is the same as the __________ times the logarithm of the number. So log5(2510) = __________ __________.4E We can combine 2 log x + log y log z to get __________.6E True or False? 7. (a) log(A + B) is the same as log A + log B. (b) log AB is the same as log A + log B.(a) logAB is the same as log A log B. (b) logAB is the same as log A log B.Evaluating Logarithms Use the Laws of Logarithms to evaluate the expression. 9. log 50 + log 200 10E Evaluating Logarithms Use the Laws of Logarithms to evaluate the expression. 11. log2 60 log2 15 12E Evaluating Logarithms Use the Laws of Logarithms to evaluate the expression. 13. 14log381 14E 15E 16E 17E 18E 19E 20E 21E Evaluating Logarithms Use the Laws of Logarithms to evaluate the expression. 22. ln(lnee200)Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 23. log3 8x 24E 25E 26E 27E 28E Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 29. log2(xy)10 30E Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 31. log2(AB2)32E 33E Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 34. lnr3s Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 35. log5(3x2y3)36E 37E 38E 39E 40E Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 41. lnx4+2 42E 43E 44E 45E 46E 47E Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. 48. logxyz Combining Logarithmic Expressions Use the Laws of Logarithms to combine the expression. 49. log4 6 + 2 log4 7 50E 51E Combining Logarithmic Expressions Use the Laws of Logarithms to combine the expression. 52. 3ln2+2lnx12ln(x+4)Combining Logarithmic Expressions Use the Laws of Logarithms to combine the expression. 53. 4logx13log(x2+1)+2log(x1)54E Combining Logarithmic Expressions Use the Laws of Logarithms to combine the expression. 55. ln(a + b) + ln(a b) 2 ln c 56E 57E Combining Logarithmic Expressions Use the Laws of Logarithms to combine the expression. 58. loga b + c loga d r loga s 59E 60E 61E 62E 63E 64E 65E Change of Base Formula Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. 66. log12 2.5 67E 68E Change of Base Formula Use the Change of Base Formula to show that loge=1ln10 Change of Base Formula Simplify: (log2 5)(log5 7)71E Forgetting Use the Law of Forgetting (Example 4) to estimate a students score on a biology test two years after he got a score of 80 on a test covering the same material. Assume that c = 0.3 and t is measured in months.Wealth Distribution Vilfredo Pareto (18481923) observed that most of the wealth of a country is owned by a few members of the population. Paretos Principle is logP=logcklogW where W is the wealth level (how much money a person has) and P is the number of people in the population having that much money. (a) Solve the equation for P. (b) Assume that k = 2.1 and c = 8000, and that W is measured in millions of dollars. Use part (a) to find the number of people who have 2 million or more. How many people have 10 million or more?Biodiversity Some biologists model the number of species S in a fixed area A (such as an island) by the species-area relationship logS=logc+klogA where c and k are positive constants that depend on the type of species and habitat. (a) Solve the equation for S. (b) Use part (a) to show that if k = 3, then doubling the area increases the number of species eightfold.Magnitude of Stars The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by M=2.5log(BB0) where B is the actual brightness of the star and B0 is a constant. (a) Expand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. (c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero.DISCUSS: True or False? Discuss each equation, and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which any term is undefined.) (a) log(xy)=logxlogy (b) log2(x y) = log2 x log2 y (c) log5(ab2)=log5a2log5b (d) log 2z = z log 2 (e) (log P) (log Q) = log P + log Q (f) logalogb=logalogb (g) (log2 7)x = x log2 7 (h) loga aa = a (i) log(xy)=logxlogy (j) ln(1A)=lnA 77E PROVE: Shifting, Shrinking, and Stretching Graphs of Functions Let f(x) = x2. Show that f(2x) = 4f(x), and explain how this shows that shrinking the graph of f horizontally has the same effect as stretching it vertically. Then use the identities e2 + x = e2ex and ln(2x) = ln 2 + ln x to show that for g(x) = ex a horizontal shift is the same as a vertical stretch and for h(x) = ln x a horizontal shrinking is the same as a vertical shift.Lets solve the exponential equation 2ex = 50. (a) First, we isolate ex to get the equivalent equation __________. (b) Next, we take ln of each side to get the equivalent equation _______________. (c) Now we use a calculator to find x __________.Lets solve the logarithmic equation log3+log(x2)=logx (a) First, we combine the logarithms on the LHS to get the equivalent equation _______________. (b) Next, we use the fact that log is one-to-one to get the equivalent equation _______________. (c) Now we find x = __________.Exponential Equations Find the solution of the exponential equation, as in Example 1. 3. 5x 1 = 125 4E Exponential Equations Find the solution of the exponential equation, as in Example 1. 5. 52x 3 = 1 6E 7E 8E 9E 10E Exponential Equations (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. 11. 10x = 25 12E 13E 14E Exponential Equations (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. 15. 21 x = 3 16E 17E 18E Exponential Equations (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. 19. 300(1.025)12t = 1000 20E 21E 22E 23E 24E 25E 26E Exponential Equations (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. 27. 4(1 + 105x) = 9 28E 29E 30E 31E 32E 33E 34E

Page: [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]