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All Textbook Solutions for Calculus and Its Applications (11th Edition)
35–68. Check the results of Exercises 1–34 using a calculator.
54.
35–68. Check the results of Exercises 1–34 using a calculator.
55.
Check the results of Exercises 134 using a calculator. g(x)=2x420x2+18Check the results of Exercises 134 using a calculator. G(x)=x+2335–68. Check the results of Exercises 1–34 using a calculator.
58.
Check the results of Exercises 134 using a calculator. f(x)=1x2/335–68. Check the results of Exercises 1–34 using a calculator.
60.
Check the results of Exercises 134 using a calculator. G(x)=8x2+135–68. Check the results of Exercises 1–34 using a calculator.
62.
Check the results of Exercises 134 using a calculator. g(x)=4xx2+1Check the results of Exercises 134 using a calculator. g(x)=x2x2+135–68. Check the results of Exercises 1–34 using a calculator.
65.
35–68. Check the results of Exercises 1–34 using a calculator.
66.
Check the results of Exercises 134 using a calculator. g(x)=x2+2x+535–68. Check the results of Exercises 1–34 using a calculator.
68.
For Exercises 6984, draw a graph to match the description given. Answers will vary. f(x) is increasing over (,2) and decreasing over (2,).For Exercises 6984, draw a graph to match the description given. Answers will vary. g(x) is decreasing over (,3) and increasing over (3,).For Exercises 69–84, draw a graph to match the description given. Answers will vary.
71. is decreasing over and and increasing over .
For Exercises 6984, draw a graph to match the description given. Answers will vary. F(x) is increasing over (,5) and (12,) and decreasing over (5,12).For Exercises 69–84, draw a graph to match the description given. Answers will vary.
73. has a positive derivative over and a negative derivative over
For Exercises 6984, draw a graph to match the description given. Answers will vary. f(x) has a negative derivative over (,1) and a positive derivative over (1,).For Exercises 69–84, draw a graph to match the description given. Answers will vary.
75. has a negative derivative over and and a positive derivative over and
For Exercises 6984, draw a graph to match the description given. Answers will vary. G(x) has a positive derivative over (,2) and (4,7) and a negative derivative over (2,4) and (7,).For Exercises 6984, draw a graph to match the description given. Answers will vary. f(x) has a positive derivative over (,3) and (3,9), a negative derivative over (9,), and a derivative equal to 0 at x=3.For Exercises 6984, draw a graph to match the description given. Answers will vary. g(x) has a negative derivative over (,5) and (5,8), a positive derivative over (8,), and a derivative equal to 0 at x=5.For Exercises 69–84, draw a graph to match the description given. Answers will vary.
79. has a negative derivative over and a positive derivative over and does not exist.
For Exercises 69–84, draw a graph to match the description given. Answers will vary.
80. has a positive derivative over and and a negative derivative over but neither nor exists.
For Exercises 69–84, draw a graph to match the description given. Answers will vary.
81. has a negative derivative over and and a positive derivative over
and but does not exist.
For Exercises 6984, draw a graph to match the description given. Answers will vary. g(x) has a positive derivative over (,3) and (0,3) a negative derivative over (3,0) and (3,) and a derivative equal to 0 at x=3 and x=3, but g(0) does not exist.83EFor Exercises 6984, draw a graph to match the description given. Answers will vary. K(x) is decreasing over (,), but the derivative does not exist at x=0 and x=2.Consider this graph. What makes an x-value a critical value? Which x-values above are critical values, and why?Consider this graph Using the graph and the intervals noted, explain how a function being increasing or decreasing relates to the first derivative.Employment. According to the U.S. Bureau of Labor statistics, the number of managerial and professional employees since 2006 is modeled by E(t)=107.833t3971.369t2+2657.917t+50,347.833, where t is the number of years since 2006 and E is thousands of employees. (Source: data.bls.gov.) Find the relative extrema of this function, and sketch the graph. Interpret the meaning of the relative extrema.Advertising. Brody Electronics estimates that it will sell N units of a new toy after spending a thousands of dollars on advertising where N (a)=a2+300a+6, 0a300. Find the relative extrema and sketch a graph of the function.Solar eclipse. On July 2, 2019, a total solar eclipse will occur over the South Pacific Ocean and parts of South America. (Source: eclipse.gsfc.nasa.gov.) The path of the eclipse is modeled by f(t)=0.00259t20.457t+36.237, Where f(t) is the latitude in degrees south of the equator at t minutes after the start of the total eclipse. What is the latitude closest to the equator, in degrees, at which the total eclipse will be visible?Temperature during an illness. The temperature of a person during an intestinal illness is given by T(t)=0.1t2+1.2t+98.6, 0t12, where T is the temperature (F) at t days. Find the relative extrema and sketch a graph of the function.In Exercises 9196, the graph of a derivative f is shown. Use the information in each graph to determine where f is increasing or decreasing and the x-values of any extrema. Then sketch a possible graph of f. Increasing on (1,), decreasing on (,1), relative minimum at x=1.In Exercises 91–96, the graph of a derivative is shown. Use the information in each graph to determine where f is increasing or decreasing and the x-values of any extrema. Then sketch a possible graph of .
92.
Increasing on , decreasing on , relative minimum at .
In Exercises 9196, the graph of a derivative f is shown. Use the information in each graph to determine where f is increasing or decreasing and the x-values of any extrema. Then sketch a possible graph of f.In Exercises 9196, the graph of a derivative f is shown. Use the information in each graph to determine where f is increasing or decreasing and the x-values of any extrema. Then sketch a possible graph of f.In Exercises 91–96, the graph of a derivative is shown. Use the information in each graph to determine where f is increasing or decreasing and the x-values of any extrema. Then sketch a possible graph of .
95.
In Exercises 9196, the graph of a derivative f is shown. Use the information in each graph to determine where f is increasing or decreasing and the x-values of any extrema. Then sketch a possible graph of f.97E98E99E100EUse a calculator’s absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist.
101.
Use a calculators absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist. f(x)=|2x5|Use a calculator’s absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist.
103.
Use a calculators absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist. f(x)=|x23x+2|Use a calculators absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist. f(x)=|9x2|Use a calculators absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist. f(x)=|x2+4x4|Use a calculator’s absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist.
107.
Use a calculators absolute-value feature to graph each function and determine relative extrema and intervals over which the function is increasing or decreasing. State any x-values the derivative does not exist. f(x)=|x42x2|109E110EDescribe a procedure that can be used to select an appropriate viewing window for the functions given in (a) Exercises 134 and (b) Exercises 97100.For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible.
1.
For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible.
2.
For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible.
3.
For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible. f(x)=x2xFor each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible.
5.
For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible.
6.
For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible. f(x)=x312x1For each function, find all relative extrema and classify each as a maximum. Use the second-derivative Test where possible.
8.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
9.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x312xSketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=2x33x236x+28Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
12.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
13.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
14.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x3+3x2Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
16.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=3x416x3+18x2 (Round results to three decimal places.)Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
18.
(Round results to three decimal places)
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
19.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=2x2x4Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x36x2+9x+1Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
22.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
23.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
24.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x36x2135xSketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
26.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
27.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
28.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
29.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x3+3x+1Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
31.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=20x33x5 (Round results to three decimal places.)Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
33.
(Round results to three decimal places.)
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x2(3x)2 (Round result to three decimal places.)Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=(x1)2/3Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=(x+1)2/3Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=(x3)1/31Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
38.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
39.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=3(x2)2/3+3Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x1x2Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=x4x2Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=8xx2+1Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
44.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down.
45.
Sketch the graph of each function. List the coordinates of any extrema or points of inflection. State where the function is increasing or decreasing and where its graph is concave up or concave down. f(x)=3x2+1For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f is decreasing and concave up on (,2), f is decreasing and concave down on (2,).For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f is increasing and concave up on (,4), f is increasing and concave down on (4,).For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary
49. is decreasing and concave down on f is decreasing and concave up on
For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f is increasing and concave down on (,1), f is increasing and concave up on (1,).For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f is concave up at (1,3), concave down at (8,7), and has an inflection point at (5,4).For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f is concave down at (1,5), concave up at (7,2), and has an inflection point at (4,1).For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f(3)=0,f(3)0,f(3)=8:f(9)=0, f(9)0,f(9)=6;f(2)=0, and f(2)=1For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary
54.
and
For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary f(0)=0,f(0),f(0)=5,f(2)=0,f(2)0, f(2)=2f(4)=0f(4)0 and f(4)=3For Exercises 47, 56, sketch a graph that possesses the characteristics listed Answers may vary
56.
and
Using the same set of axes, graph R, C, and P, the total-revenue, total-cost, and total-profit functions. R(x)=50x0.5x2,C(x)=4x+10Using the same set of axes, graph R, C, and P, the total-revenue, total-cost, and total-profit functions. R(x)=50x0.5x2,C(x)=10x+359. Small business. The percentage of the U.S. national income generated by nonfarm proprietors between 1970 and 2000 can be modeled by the function given by
where x is the number of years since 1970. (Source: Based on data from www.bls.gov.) Sketch the graph of this function for.
Labor force. The percentage of the U.S. civilian labor force aged 4554 between 1970 and 2000 can be modeled by the function f given by f(x)=0.025x20.71x+20.44, where x is the number of years after 1970. (Source: Based on data from www.bls.gov.) Sketch the graph of this function for 0x30.61. Rainfall in Reno. The average monthly rainfall in Reno, Nevada, is approximated by
where is the amount of rainfall in millimeters at t months since December. (Source: Based on data from weather.com.)
Find the point(s) of inflection for R. What is the significance of these point(s)?
Coughing velocity. A person coughs when a foreign object is in the windpipe. The velocity of the cough depends on the size of the object. Suppose a person has a windpipe with a 20-mm radius. If a foreign object has a radius r, in millimeters, then the velocity V, in millimeters per second, needed to remove the object by a cough is given by V(r)=k(20r2r3),0r20, where k is some positive constant. For what size object is the maximum velocity required to remove the object? Radius=403,or1313mmIn each of Exercises 63 and 64, determine which graph is the derivative of the other and explain why.In each of Exercises 63 and 64, determine which graph is the derivative of the other and explain why.65. Use calculus to prove that the relative minimum or maximum for any function f for which
occurs at .
66. Use calculus to prove that the point of inflection for any function g given by
occurs at .
For Exercises 6773, assume that f is differentiable over (,). Classify each of the following statements as either true or false. If a statement is false, explain why. If f has exactly two critical values at x=a and x=b, where ab, then there must exist exactly one point of inflection at x=c such that acb. In other words, exactly one point of inflection must exist between any two critical points.For Exercises 6773, assume that f is differentiable over (,). Classify each of the following statements as either true or false. If a statement is false, explain why. If f has exactly two critical values at x=a and x=b, where ab, then there must exist at least one point of inflection at x=c such that acb. In other words, at least one point of inflection must exist between any two critical points.For Exercises 67–73, assume that f is differentiable over. Classify each of the following statements as either true or false. If a statement is false, explain why.
69. A function can have no extrema but still have at least one point of inflection.
For Exercises 67–73, assume that f is differentiable over. Classify each of the following statements as either true or false. If a statement is false, explain why.
70. If has two points of inflection, then there is a critical value located between those points of inflection.
For Exercises 67–73, assume that f is differentiable over. Classify each of the following statements as either true or false. If a statement is false, explain why.
71. The function can have a point of inflection at a critical value.
For Exercises 67–73, assume that f is differentiable over. Classify each of the following statements as either true or false. If a statement is false, explain why.
72. The function can have a point of inflection at an extreme value.
For Exercises 6773, assume that f is differentiable over (,). Classify each of the following statements as either true or false. If a statement is false, explain why. A function f can have exactly one extreme value but no points of inflection.Hours of daylight. The number of hours of daylight in Chicago is represented in the graph below. On what dates is the number of hours of daylight changing most rapidly? How can you tell?Graph each function. Then estimate any relative extrema. Where appropriate, round to three decimal places. f(x)=4x6x2/3Graph each function. Then estimate any relative extrema. Where appropriate, round to three decimal places.
76.
Graph each function. Then estimate any relative extrema. Where appropriate, round to three decimal places.
77.
Graph each function. Then estimate any relative extrema. Where appropriate, round to three decimal places. f(x)=x2(x2)3Graph each function. Then estimate any relative extrema. Where appropriate, round to three decimal places.
79.
Graph each function. Then estimate any relative extrema. Where appropriate, round to three decimal places.
80.
81. Head of household education. The data in the following table relate the percentage of households headed by someone with a bachelor’s degree or higher to the number of years since 2000.
(Source: www.census.gov/prod/2013pubs/p20-569.pdf.)
Number of Years since 2000 Percentage of Households
0 66.0
1 75.2
3 78.3
7 84.0
9 88.5
10 89.2
11 89.9
a. Use regression (see Section R.6) to fit linear, cubic, and quartic functions to the data, where x is the number of years since 2000 and y is the percentage of households headed by someone with a bachelor’s degree or higher. Which function best fits the data?
b. What is the domain of?
c. Does f have any relative extrema? How can you tell?
Determine the vertical asymptote(s) of each function. If none exists, state that fact.
1.
Determine the vertical asymptote(s) of each function. If none exists, state that fact. f(x)=2x3x5Determine the vertical asymptote(s) of each function. If none exists, state that fact. f(x)=5xx225Determine the vertical asymptote(s) of each function. If none exists, state that fact.
4.
Determine the vertical asymptote(s) of each function. If none exists, state that fact.
5.
Determine the vertical asymptote(s) of each function. If none exists, state that fact.
6.
Determine the vertical asymptote(s) of each function. If none exists, state that fact. f(x)=x+2x2+6x+8Determine the vertical asymptote(s) of each function. If none exists, state that fact.
8.
Determine the vertical asymptote(s) of each function. If none exists, state that fact.
9.
Determine the vertical asymptote(s) of each function. If none exists, state that fact. f(x)=6x2+36Determine the horizontal asymptote of each function. If none exists, state that fact. f(x)=6x8x+3Determine the horizontal asymptote of each function. If none exists, state that fact. f(x)=3x26x2+xDetermine the horizontal asymptote of each function. If none exists, state that fact.
13.
Determine the horizontal asymptote of each function. If none exists, state that fact. f(x)=2x3x3x2Determine the horizontal asymptote of each function. If none exists, state that fact. f(x)=4+2xDetermine the horizontal asymptote of each function. If none exists, state that fact.
16.
Determine the horizontal asymptote of each function. If none exists, state that fact. f(x)=6x3+4x3x2xDetermine the horizontal asymptote of each function. If none exists, state that fact.
18.
Determine the horizontal asymptote of each function. If none exists, state that fact.
19.
Determine the horizontal asymptote of each function. If none exists, state that fact.
20.
Determine the horizontal asymptote of each function. If none exists, state that fact. f(x)=2x34x+14x3+2x3Determine the horizontal asymptote of each function. If none exists, state that fact.
22.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
23.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=5xSketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
25.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
26.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
27.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
28.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=2x+5Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
30.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
31.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=3x1xSketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x+2xSketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x+9xSketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=1x2Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=2x2Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=xx3Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
38.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
39.
40ESketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
41. (Hint: Simplify.)
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x1x21Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x1x+2Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
44.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
45.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x24x+3Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
47.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x+1x22x3Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=2x2x216Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x2+x22x22Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
51.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
52.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x2+1xSketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x3x21Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
55.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. f(x)=x29x357. Average cost. The total cost, in thousands of dollars, for Acme, Inc., to produce x pairs of rocket skates is given by
.
a. The average cost is given by. Find.
b. Graph the average cost.
c. Find the slant asymptote for the graph of, and interpret its significance.
Depreciation. Suppose that the value V of the inventory at Fidos Pet Supply, in hundreds of dollars, decreases (depreciates) after t months, where V(t)=5025t2(t+2)2. a. Find V(0),V(5),V(10), and V(70). b. Find the maximum value of the inventory over the interval [0,). c. Draw a graph of V. d. Does there seem to be a value below which V(t) will never fall? Explain.
59. Total cost and revenue. The total cost and total revenue, in dollars, from producing x couches are given by
.
a. Find the total-profit function, .
b. The average profit is given by. Find.
c. Find the slant asymptote for the graph of.
d. Graph the average profit.
Cost of pollution control. Cities and companies find that the cost of pollution control increases along with the percentage of pollutants being removed. Suppose the cost C, in dollars, of removing p% of the pollutants from a chemical spill is given by C(p)=48,000100p. a. Find C(0),C(20),C(80),andC(90). b. Find the domain of C. c. Draw a graph of C. d. d) Can the company or city afford to remove 100% of the pollutants due to this spill? Explain.61. Purchasing power. The purchasing power of the U.S. dollar t years after 1990 can be modeled by the function
.
(Source: Based on data from the Consumer Price Index.)
a. Find.
b. When was the purchasing power $0.50?
c. Find.
62. Medication in the bloodstream. After an injection, the amount of a medication A, in cubic centimeters (cc), in the bloodstream decreases with time t, in hours. Suppose that under certain conditions A is given by
where is the initial amount of the medication. Assume that an initial amount of 100 cc is injected.
a. Find.
b. Find the maximum amount of medication in the bloodstream over the interval
c. Graph the function.
d. d) According to this function, does the medication ever completely leave the bloodstream? Explain your answer.
64E65. Using graphs and limits, explain how three types of asymptotes are used when graphing rational functions.
Find each limit, if it exists. limx3x2+52x67E68E69E70E71E72E73E74EGraph each function using a graphing utility. f(x)=x3+2x23xx225Graph each function using a graphing utility.
76.
Graph the function given by f(x)=x232x4. a. Find any x-intercepts. b. Find the y-intercept if it exists. c. Find any asymptotes.78. Graph the function given by
.
a. Estimate and using the graph and input–output tables as needed to refine your estimates.
b. What appears to be the domain of the function? Explain.
c. Find.
79ERefer to the graph on p. 221. The function is given by f(x)=x21x2+x6. a. Inspect the graph and estimate the coordinates of any extrema. b. Find f and use it to determine the critical values. (Hint: you will need the quadratic formula.) Round the x-values to the nearest hundredth. c. Graph f in the window [0,0.2,0.16,0.17]. Use trace or maximum to confirm your results from part (b). d. Graph f in the window [9.8,10,0.9519,0.95195]. Use trace or minimum to confirm your results from part (b). e. How close were your estimates of part (a)? Would you have been able to identify the relative minimum point without calculus?81EIn Exercises 81–86, determine a rational function that meets the given conditions, and sketch its graph.
82. The function has a vertical asymptote at, a horizontal asymptote at, and.
In Exercises 8186, determine a rational function that meets the given conditions, and sketch its graph. The function g has vertical asymptotes at x=1 and x=1, a horizontal asymptote at y=1, and g(0)=2.In Exercises 8186, determine a rational function that meets the given conditions, and sketch its graph. The function g has vertical asymptotes at x=2 and x=0, a horizontal asymptote at y=3, and g(1)=4.85EIn Exercises 8186, determine a rational function that meets the given conditions, and sketch its graph. The function h has vertical asymptotes at x=12 and x=12, a horizontal asymptote at y=0, and h(0)=3.1. Fuel economy. According to the U.S. Department of Energy, a vehicle’s fuel economy, in miles per gallon (mpg), decreases rapidly for speeds over 60 mph.
(Sources: U.S. Dept. of Energy; a study by West, B.H., McGill, R.N., Hodgson, J.W., Sluder, S.S., and Smith, D.E., Oak Ridge National Laboratory, 1999; www.mpgforspeed.com, 2014.)
a. Estimate the speed at which the absolute maximum gasoline mileage is obtained.
b. Estimate the speed at which the absolute minimum gasoline mileage is obtained.
c. What is the mileage obtained at 70 mph?
Fuel economy. Using the graph in Exercise 1, estimate the absolute maximum and the absolute minimum fuel economy over the interval [30,70].Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
3.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=4+xx2;[0,2]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
5.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
6.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
7.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x3x2x+3;[1,0]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=2x+4;[1,1]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
10.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
11.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=23x;[10,10]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=5;[1,1]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x26x3;[1,5]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. g(x)=24;[4,13]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
16.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
17.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x33x2;[0,5]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=1+6x3x2;[0,4]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
20.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
21.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=3x22x3;[5,1]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=1x3;[8,8]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
24.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x36x2+10;[0,4]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
26.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x3x4;[1,1]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x42x3;[2,2]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x42x2+5;[2,2]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
30.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
31.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
32.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
33.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
34.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x2x2+1;[2,2]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur.
36.
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=(x+1)1/3;[2,26]Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x)=x3;[8,64]Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator. f(x)=5+xx2;[0,2]39–48. Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator.
40.
Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator. f(x)=2x+4;[1,1]39–48. Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator.
42.
Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator. f(x)=1+6x3x2;[0,4]Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator. f(x)=1x3;[8,8]39–48. Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator.
45.
Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator. f(x)=x2x2+1;[2,2]Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator. f(x)=(x+1)1/3;[2,26]39–48. Check Exercises 3, 5, 9, 13, 19, 23, 33, 35, 37, and 38 with a graphing calculator.
48.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
49.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
50.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
51.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=2x220x+340Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
53.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=x43x3;(0,)Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
55.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=x(25x)Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=13x35x;[3,3]Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
58.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=0.001x2+4.8x60Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
60.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=x3+x2+5x1;(0,)Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
62.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=15x212x3;[0,30]Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=4x212x3;[0,8]Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
65.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
66.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=x2+432x;(0,)Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=x2+250x;(0,)Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=2x4+x;[1,1]Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=2x4x;[1,1]Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
71.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
72.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=(x1)3Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=(x+1)3Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=2x3;[1,1]Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=95x;[10,10]