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All Textbook Solutions for Calculus and Its Applications (11th Edition)
43ETwo cars start from the same point at the same time One travels north at 25 mph, and the other travels east at 60 mph. How fast is the distance between them increasing at the end of 1 hr? (Hint: D2=x2+y2. To find D after 1 hr, solve D2=252+602.)45. A ladder 26 ft long leans against a vertical wall. If the lower end is being moved away from the wall at the rate of 5 ft/sec, how fast is the height of the top changing (this will be a negative rate) when the lower end is 10 ft from the wall?
46EThe volume of a cantaloupe is approximated by V=43r3. The radius is growing at the rate of 0.7 cm/week, at a time when the radius is 7.5 cm. How fast is the volume changing at that moment?Differentiate implicitly to find.
48.
Differentiate implicitly to find.
49.
Differentiate implicitly to find.
50.
Differentiate implicitly to find.
51.
Differentiate implicitly to find.
52.
Differentiate implicitly to find dy/dx. (xy)3+(x+y)3=x5+y5Differentiate implicitly to find d2y/dx2 - xy+x2y=4Differentiate implicitly to find -
55.
Differentiate implicitly to find d2y/dx2 - x2y2=5Differentiate implicitly to find -
57.
Explain the usefulness of implicit differentiation.Look up the word implicit in a dictionary. Explain how that definition can be related to the concept of a function that is defined implicitlyGraph each of the following equations. Equations must be solved for y before they can be entered into most calculators. Graphicus does not require that equations be solved for y.
60.
Graph each of the following equations. Equations must be solved for y before they can be entered into most calculators. Graphicus does not require that equations be solved for y.
61.
Graph each of the following equations. Equations must be solved for y before they can be entered into most calculators. Graphicus does not require that equations be solved for y.
62.
Graph each of the following equations. Equations must be solved for y before they can be entered into most calculators. Graphicus does not require that equations be solved for y. x3=y2(2x)Graph each of the following equations. Equations must be solved for y before they can be entered into most calculators. Graphicus does not require that equations be solved for y.
64.
In Exercises 1-6, match each equation in column A with the most appropriate graph in column B Column B
Column A
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In Exercises 1-6, match each equation in column A with the most appropriate graph in column B Column B
Column A
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In Exercises 1-6, match each equation in column A with the most appropriate graph in column B Column B Column A P(t)=50e0.20tIn Exercises 1-6, match each equation in column A with the most appropriate graph in column B Column B Column A P(t)=lntIn Exercises 1-6, match each equation in column A with the most appropriate graph in column B Column B Column A P(t)=50(1e0.04t)6REClassify each statement as either true or false.
7. The base a in the exponential function given by must be greater than 1.
8RE9REClassify each statement as either true or false.
10. With exponential growth, the doubling time depends on the size of the original population.
Classify each statement as either true or false. A radioactive isotopes half-life can be used to determine the value of its decay constant.Classify each statement as either true or false. A radioactive isotopes half-life depends on how much of the substance is initially present.Classify each statement as either true or false. For any exponential function of the form f(x)=ax, it follows that f(x)=lnaax.Classify each statement as either true or false.
14. For any logarithmic function of the form it follows that.
Classify each statement as either true or false.
15. It is possible to amortize a loan faster by paying more than the minimum amount for each payment.
16. Find
a.
b.
c.
Differentiate each function. y=lnxDifferentiate each function.
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Differentiate each function.
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Differentiate each function. y=e2xDifferentiate each function. f(x)=lnxDifferentiate each function. f(x)=x4e3xDifferentiate each function. f(x)=lnxx3Differentiate each function.
24.
Differentiate each function.
25.
26REDifferentiate each function. F(x)=9x28REDifferentiate each function.
29.
Graph each function. f(x)=4xGraph each function.
31.
Given and, find each logarithm.
32.
Given and, find each logarithm.
33.
34REGiven loga2=1.8301 and loga7=5.0999, find each logarithm. loga3.5Given and, find each logarithm.
36.
Given and, find each logarithm.
37.
Find the function Q that satisfies dQ/dt=7Q, given that Q=25whent=0.39REBusiness: Interest compounded continuously. Suppose $8300 is invested in Noether Bond Fund, for which the interest rate is 3.8%, compounded continuously. How long will it take for the $8300 to double? Round to the nearest tenth of a year.41RE42. Business: Cost of Oreo Cookies. The average cost, C, of Oreo Cookies was $2.69/lb in 1990 and $5.14/lb in 2013. (Source: foodtimeline.org.) Assuming that the exponential growth model applies:
a. Find the exponential growth rate to three decimal places, and write the function that models the situation.
b. What will the cost of 1lb of Oreo Cookies be in 2020? In 2030?
43. Business: Franchise Growth. Fashionista Clothing is selling franchises throughout the United States and Canada. It is estimated that the number of franchises N will increase at the rate of 12% per year, that is,
where t is the time, in years.
a. Find the function that satisfies the equation, assuming that the number of franchises in 2014 is 60.
b. How many franchises will there be in 2018?
c. After how long will the number of franchises be 300? Round to the nearest tenth of a year.
44RELife Science: Decay Rate. The decay rate of a certain radioactive substance is 13% per year. What is its half-life? Round to the nearest tenth of a year.46RELife Science: Decay Rate. A certain radioactive isotope has a decay rate of 7% per day, that is, dAdt=0.07A, where A is the amount of isotope present after t days. a. Find a function that satisfies the equation if the amount of the isotope presents t=0 is 800g. b. After 20 days, how much of the 800 g will remain? Round to the nearest gram. c. After how long will half of the original amount remain?48RE49. Business: Present Value. Find the present value of $1,000,000 due 40 yr later at 4.2%, compounded continuously.
Business: Annuity. Patrice deposits $50 into a savings account every month at an annual interest rate of 4.7%, compounded monthly. a. How much will her account contain after 8 yr, assuming she makes no withdrawals? b. What is the rate of change in the value of Patrices account after 8 yr?Business: Car Loan. Glenda buys a used Subaru Outback for $13,000. She pays 25% down and finances the rest at an annual interest rate of 6.5%, compounded monthly for 5 yr. a. Find Glendas monthly car payment. b. If Glenda makes every payment for the life of the loan, find the total interest she will pay.52REBusiness: Credit Card. Vicki uses her credit card to buy $1200 in goods. Her credit card has an annual interest rate of 20.75%, compounded monthly over a 10-yr term. Assume that Vicki makes no additional purchases with this credit card. a. Find the amount of Vickis monthly payment. b. Assume that Vicki pays the amount found in part (a). In the first month, what will she pay in interest, and what will she pay toward the principal?54. Differentiate: .
55. Find the minimum value of.
56RE57REBusiness: shopping on the internet. Online sales of all types of consumer products increased at an exponential rate in recent years. Data in the following table show online retail sales in the United States, in billions of dollars. (Sources: Internetretailer.com; www.census.gov.) Years, t, after 2010 U.S. Online Retail Sales (in billions) 0 $167 1 194 2 231 3 262 4 291 a. Use regression to fit an exponential function y=abx to the data. Then convert that formula to an exponential function, base e, where t is the number of years after 2010, and determine the exponential growth rate. b. Estimate online sales in 2016 and in 2020. c. After what amount of time will online sales be $1 trillion? d. What is the doubling time for online sales?Differentiate. y=2e3xDifferentiate. y=(lnx)4Differentiate.
3.
Differentiate. f(x)=lnx7Differentiate.
5.
Differentiate. f(x)=3exlnxDifferentiate.
7.
8T9T10TGiven logb2=0.2560 and logb9=0.8114, find each of the following. logb4.5Given logb2=0.2560 and logb9=0.8114, find each of the following. logb313. Find the function that satisfies, if at.
14. The doubling time for a certain bacteria population is 3 hr. What is the growth rate? Round to the nearest tenth of a percent.
15. Business: interest compounded continuously. An investment is made at 6.931% per year, compounded continuously. What is the doubling time? Round to the nearest tenth of a year.
Business: Cost of Milk. The cost C of a gallon of milk was $3.22 in 2006. In 2013, it was $3.50. (Source: U.S. Department of Labor, Bureau of Labor Statistics.) Assuming that the exponential growth model applies: a. Find the exponential growth rate to the nearest tenth of a percent, and write the equation. b. Estimate the cost of a gallon of milk in 2016 and in 2020.17. Life science: drug dosage. A dose of a drug is injected into the body of a patient. The drug amount in the body decreases at the rate of 10% per hour, that is,
where A is the amount in the body and t is the time, in hours.
a. A dose of 3 cubic centimeters (cc) is administered. Assuming find the function that satisfies the equation.
b. How much of the initial dose 3 cc remains after 10 hr?
c. After how long does half of the original dose remain?
18. Life Science: decay rate. The decay rate of radium-226 is 4.209% per century. What is its half-life?
19. Life science: half-rate. The half-life of bohrium-267 is 17 sec. What is its decay rate? Express the rate as a percentage rounded to four decimal places.
Business: effect of advertising. Twin City Roasters introduced a new coffee in a trial run. The firm advertised the coffee on television and found that the percentage P(t) of people who bought the coffee after t ads had been run was P(t)=1001+24e0.28t. a. What percentage of people bought the coffee before seeing the ad (t=0)? b. What percentage bought the coffee after the ad had been run 1 time? 5 times? 10 times? 20 times? 30 times? c. Find the rate of change, P(t). d. Interpret the meaning of P(t). e. Sketch a graph of the function.21T22. Business: Amortized Loan. The Langways purchase a new home for $450,000. They pay 20% down and finance the rest through a 30-yr mortgage at an annual interest rate of 3.75%, compounded monthly.
a. How much is the Langways’ monthly mortgage payment?
b. Assuming that the Langways make the monthly payment found in part (a) for the life of the loan, how much will they pay in total?
c. How much interest will they pay?
23. Business: Car Loan. Giselle qualifies for a car loan at an annual interest rate of 5.75%, compounded monthly for 5 yr. She is willing to pay up to $400 per month in car payments. What is the maximum loan amount she can afford?
Differentiate: y=x(lnx)22xlnx+2x.Find the maximum and minimum values of f(x)=x4ex over [0,10].26T27T1ETEUse the exponential function to predict gross revenue, G, for weeks 9 through 13.3ETE5ETE7ETE8ETEGraph. y=5xGraph. y=4xGraph. y=23xGraph. y=34xGraph.
5.
Graph.
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7E8E9EGraph. y=1.13(0.81)xDifferentiate. f(x)=exDifferentiate.
12.
Differentiate.
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Differentiate. g(x)=e3xDifferentiate.
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Differentiate.
16.
Differentiate.
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Differentiate. F(x)=e4xDifferentiate. g(x)=3e5xDifferentiate.
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Differentiate.
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Differentiate. f(x)=3exDifferentiate.
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Differentiate.
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Differentiate.
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Differentiate. g(x)=45ex3Differentiate. F(x)=4e2xDifferentiate.
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Differentiate.
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Differentiate. f(x)=x52e6xDifferentiate.
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Differentiate.
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Differentiate. F(x)=e2xx4Differentiate. g(x)=e3xx6Differentiate. f(x)=(x22x+2)exDifferentiate.
36.
Differentiate.
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Differentiate. f(x)=exx5Differentiate.
39.
Differentiate.
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Differentiate. f(x)=ex2/2Differentiate.
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Differentiate. y=ex7Differentiate.
44.
Differentiate.
45.
Differentiate.
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Differentiate. y=ex+x3xex48EDifferentiate. y=1e3xDifferentiate. y=1exDifferentiate. y=1ekxDifferentiate. y=1emxDifferentiate. g(x)=(4x2+3x)ex27xDifferentiate.
54.
Graph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity.
55.
Graph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity.
56.
Graph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity. f(x)=e(1/3)xGraph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity. g(x)=e(1/2)xGraph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity.
59.
Graph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity. g(x)=13exGraph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity. F(x)=e(1/3)xGraph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity. G(x)=e(1/2)xGraph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity. f(x)=3ex, for x0Graph each function. Then determine any critical values, infection points, intervals over which the function is increasing or decreasing, and the concavity.
64. for
65E66E67E68E69E70Ea. 65-74. For each function given in Exercises 55-64, graph the function and its first and second derivatives using a graphing utility. F(x)=e(1/3)x72E73Ea. 65-74. For each function given in Exercises 55-64, graph the function and its first and second derivatives using a graphing utility.
74. for
Find the slope of the line tangent to the graph of f(x)=2e3x at the point (0,2).Find the slope of the line tangent to the graph of f(x)=ex at the point (0,1).77. Find an equation of the line tangent to the graph of at the point.
Find an equation of the line tangent to the graph of G(x)=ex at the point (0,1).For each of Exercises 77 and 78, graph the function and the tangent line using a graphing utility.
79. Find an equation of the line tangent to the graph of at the point.
For each of Exercises 77 and 78, graph the function and the tangent line using a graphing utility. Find an equation of the line tangent to the graph of G(x)=ex at the point (0,1).81. U.S. Travel Exports. U.S. travel exports (goods and services that international travelers buy while visiting the United States) are increasing exponentially. The value of such exports, t years after 2011, can be approximated by
,
where V is in billions of dollars. ( Source: WWW.census.gov/foreign-trade/data/index.html)
a. Estimate the value of U.S. travel exports in 2016 and 2018.
b. Estimate the growth rate for U.S. travel exports in 2016 and 2018.
Organic food. More Americans are buying organic fruit and vegetables and products made with organic ingredients. The amount A(t), in billions of dollars, sent on organic food and beverages t years after 1995 can be approximated by A(t)=2.43e0.18t a. Estimate the amount that Americans spent on organic food and beverages in 2009. b. Estimate the rate at which spending on organic food and beverages was growing in 2006.83. Marginal Cost. The total cost, in millions of dollars, for Cheevers, Inc., is given by
where t is the time in years since the start-up date.
Find each of the following.
a. The marginal cost,
b.
c. (Round to the nearest thousand.)
d. Find and .
Marginal cost. The total cost, in millions of dollars, for Marcotte Industries is given by C(t)=20040et, where t is the time in years since the start-up date. Find each of the following. a. The marginal cost C(t) b. C(1) c. C(5) (Round to the nearest thousand.) d. Find limxC(t) and limxC(t)85. Marginal demand. At a price of x dollars, the demand, in thousands of units, for a certain turntable is given by the demand function
.
a. How many turntables will be bought at a price of $250? Round to the nearest thousand.
b. Graph the demand function for .
c. Find the marginal demand, .
d. Interpret the meaning of the derivative.
86. Marginal supply. At a price of x dollars, the supply function for the turntable in Exercise 85 is given by
where q is in thousands of units.
a. How many turntables will be supplied at a price of $250? Round to the nearest thousand.
b. Graph the supply function for .
c. Find the marginal supply, .
d. Interpret the meaning of the derivative.
For Exercises 87-90, use the Tangent feature from the DRAW menu to find the rate of change in part (b).
87. Growth of a retirement fund. Maria deposits $20,000 in an IRA whose value increases by 5.6% every year. The value of the IRA after t years is modeled by
.
a. Use the model to estimate the value of Maria’s IRA after 7 yr.
b. What is the rate of change in the value of the IRA at the end of 7 yr?
c. When will Maria’s IRA have a value of $40,000?
For Exercises 87-90, use the Tangent feature from the DRAW menu to find the rate of change in part (b). Depreciation. Pelican Fabrics purchases a new video surveillance system. The value of the system is modeled by V(t)=17,500(0.92)t, where V is the value of the system, in dollars, t years after its purchase. a. Use the model to estimate the value of the system 5 yr after it was purchased. b. What is the rate of change in the value of the system at the end of 5 yr? c. When will the system be worth half of its original value?For Exercises 87-90, use the Tangent feature from the DRAW menu to find the rate of change in part (b). Depreciation. Perriots Restaurant purchased kitchen equipment on January 1, 2014. The value of the equipment decreases by 15% every year. On January 1, 2016, the value was $14,450. a. Find an exponential model for the value, V, of the equipment, in dollars, t years after January 1, 2016. V(t)=14,450(0.85)t b. What is the rate of change in the value of the equipment on January 1, 2016? c. What was the original value of the equipment on January 1, 2014? d. How many years after January 1, 2014 will the value of the equipment have decreased by half?For Exercises 87-90, use the Tangent feature from the DRAW menu to find the rate of change in part (b).
90. Stock prices. The value (price) of a share of stock in Barrington Gold was $90 on June 15, 2014, and was increasing by 3% every week.
a. Find an exponential model for the value, V, of a share of the stock, in dollars, t weeks after June 15, 2014.
b. What was the rate of change in the value of a share of the stock 6 weeks prior to June 15, 2014?
c. Use the model to estimate the value of a share of the stock 6 week prior to June 15, 2014.
d. How many weeks after June 15, 2014, will the stock’s share value have doubled?
Medication concentration. The concentration C, in parts per million, of a medication in the body t hours after ingestion is given by the function C(t)=10t2et. a. Find the concentration after 0 hr, 1hr, 2hr, 3hr, and 10hr. b. Sketch a graph of the function for 0t10. c. Find the rate of change of the concentration, C(t). d. Find the maximum value of the concentration and the time at which it occurs. e. Interpret the meaning of the derivatives.92. Ebbinghaus learning model. Suppose that you are given the task of learning 100% of knowledge. Human nature is such that we retain only a percentage P of knowledge t weeks after we have learned it. The Ebbinghaus learning model asserts that P is given by
where Q is the percentage that we would never forget and k is a constant that depends on the knowledge learned. Suppose that and .
a. Find the percentage retained after 0 weeks, 1 week, 2 weeks, 6 weeks, and 10 weeks.
b. Find
c. Sketch a graph of P.
d. Find the rate of change of with respect to time t.
e. Interpret the meaning of the derivative.
Differentiate. y=(e3x+1)594E95EDifferentiate.
96.
Differentiate. f(x)=ex/2x1Differentiate. f(x)=xex1+x2Differentiate. f(x)=exexex+exDifferentiate.
100.
101. Use the results from Exercises 85 and 86 to determine the equilibrium point (the point at which supply equals demand) and the rates at which supply and demand are changing at that point.
Exercises 102 and 103 each give an expression for e. Find the function values that are approximations for e. Round to five decimal places. For f(t)=(1+t)1/t, we have e=limt0f(t). Find f(1),f(0.5),f(0.2),f(0.1), and f(0.001).103E104EA student made the following error on test: ddxex=xex1. Identify the error and explain how to correct it.106E107E108EFor each of the functions in Exercises 109 – 112, graph
109.
For each of the functions in Exercises 109 – 112, graph
110.
For each of the functions in Exercises 109 – 112, graph
111.
For each of the functions in Exercises 109 – 112, graph
112.
113. Graph
Use the Table feature and very large values of x to confirm that e is approached as a limit.
114EWrite an equivalent equation.
1.
Write an equivalent equation.
2.
Write an equivalent equation. log273=13Write an equivalent equation.
4.
Write an equivalent equation. logaJ=KWrite an equivalent equation.
6.
Write an equivalent equation. logbV=wWrite an equivalent equation. log10h=pSolve for x. log749=xSolve for x. log5125=xSolve for x.
11.
Solve for x. logx64=3Solve for x. log3x=5Solve for x.
14.
Solve for x.
15.
Solve for x.
16.
Write an equivalent logarithmic equation. et=pWrite an equivalent logarithmic equation.
18.
Write an equivalent logarithmic equation.
19.
Write an equivalent logarithmic equation. 102=100Write an equivalent logarithmic equation. 102=0.01Write an equivalent logarithmic equation. 101=0.1Write an equivalent logarithmic equation.
23.
Write an equivalent logarithmic equation.
24.
Given logb3=1.099 and logb5=1.609, find each value. logb15Given and , find each value.
26.
Given logb3=1.099 and logb5=1.609, find each value. logbb3Given logb3=1.099 and logb5=1.609, find each value. logb15Given logb3=1.099 and logb5=1.609, find each value. logb75Given and , find each value.
30.
Given and , find each value. Do not use
31.
Given ln4=1.3863 and ln5=1.6094, find each value. Do not use ln20Given ln4=1.3863 and ln5=1.6094, find each value. Do not use ln15Given and , find each value. Do not use
34.
Given and , find each value. Do not use
35.
Given and , find each value. Do not use
36.
Given and , find each value. Do not use
37.
Given and , find each value. Do not use
38.
Given and , find each value. Do not use
39.
Given ln4=1.3863 and ln5=1.6094, find each value. Do not use ln14Given and , find each value. Do not use
41.
Given and , find each value. Do not use
42.
Find each logarithm. Round to six decimal places.
43.
Find each logarithm. Round to six decimal places. ln5894Find each logarithm. Round to six decimal places.
45.
Find each logarithm. Round to six decimal places.
46.
Find each logarithm. Round to six decimal places.
47.
Find each logarithm. Round to six decimal places. ln8100Solve for t.
49.
Solve for t. et=10Solve for t. e3t=900Solve for t. e2t=1000Solve for t. et=0.01Solve for t.
54.
Solve for t. e0.02t=0.06Solve for t.
56.
Differentiate y=9lnxDifferentiate y=8lnxDifferentiate y=7ln|x|Differentiate y=4ln|x|Differentiate y=x6lnx14x4Differentiate
62.
Differentiate f(x)=ln(9x)Differentiate
64.
Differentiate f(x)=ln|5x|Differentiate f(x)=ln|10x|Differentiate g(x)=x5ln(3x)Differentiate
68.
Differentiate g(x)=x4ln|6x|Differentiate
70.
Differentiate
71.
Differentiate y=lnxx4Differentiate y=ln|3x|x2