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All Textbook Solutions for Calculus and Its Applications (11th Edition)
See Exercise 61. a. Use the graph to approximate all the x-values (years since 1940) for which the cancer rate is 50 per 100,000. b. Use the graph to approximate all the x-values (years since 1940) for which the cancer rate is 70 per 100,000. c. Use the formula to approximate the lung and bronchus cancer rate in 2010.63. Is it possible for the domain and the range of a function to be the same set? Why or why not?
Is there an infinite number of functions for which 3 is not in the domain? Explain why or why not?65. Determine the range of each of the functions in Exercises 33, 35, 39, 40, and 47.
66. Determine the range of each of the functions in Exercises 34, 36, 48, 51, and 54.
Graph. x=5Graph. x=3Graph.
3.
Graph.
4.
Graph.
5.
Graph.
6.
Graph.
7.
Graph. y=3.75Graph. List the slope and y-intercept.
9.
Graph. List the slope and y-intercept. y=3xGraph. List the slope and y-intercept.
11.
Graph. List the slope and y-intercept.
12.
Graph. List the slope and y-intercept. y=3x4Graph. List the slope and y-intercept.
14.
Graph. List the slope and y-intercept. g(x)=x2.5Graph. List the slope and y-intercept. g(x)=x+3Graph. List the slope and y-intercept. y=7Graph. List the slope and y-intercept.
18.
Find the slope and y-intercept. y4x=1Find the slope and y-intercept. y3x=6Find the slope and y-intercept. 2x+y3=0Find the slope and y-intercept.
22.
Find the slope and y-intercept. 3x3y+6=0Find the slope and y-intercept.
24.
Find the slope and y-intercept. x=3y+7Find the slope and y-intercept. x=4y+3Find an equation of the line:
27. with containing.
Find an equation of the line: with m=5, containing (2,3).Find an equation of the line:
29. with containing.
Find an equation of the line: with m=3, containing (5,2).Find an equation of the line: with slope 5, containing (5,0).Find an equation of the line:
32. with slope 2, containing .
Find an equation of the line: with y-intercept (0,6) and slope 12.Find an equation of the line: with y-intercept (0,7) and slope 43.Find an equation of the line: with slope 0, containing (4,8).Find an equation of the line:
36. with slope 0, containing .
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
37. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
38. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
39. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact. (2,3) and (1,4)Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
41. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact. (4,2) and (4,10)Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
43. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact. (45,3) and (12,25)Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact. (2,3) and (1,3)Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
46. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
47. and
Find the slope of the line containing the given pair of points If a slope is undefined, state that fact.
48. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
49. and
Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact.
50. and
51-60. Find an equation of the line containing the pair of points in each of Exercises 37– 46.
51. and
Find an equation of the line containing the pair of points in each of Exercises 37 46. (2,1) and (6,3)51-60. Find an equation of the line containing the pair of points in each of Exercises 37– 46.
53. and
Find an equation of the line containing the pair of points in each of Exercises 37 46. (2,3) and (1,4)51-60. Find an equation of the line containing the pair of points in each of Exercises 37– 46.
55. and
51-60. Find an equation of the line containing the pair of points in each of Exercises 37– 46.
56. and
Find an equation of the line containing the pair of points in each of Exercises 37 46. (316,12) and (58,34)Find an equation of the line containing the pair of points in each of Exercises 37 46. (45,3) and (12,25)Find an equation of the line containing the pair of points in each of Exercises 37 46. (2,3) and (1,3)51-60. Find an equation of the line containing the pair of points in each of Exercises 37– 46.
60. and
61. Find the slope (or grade) of the treadmill.
Find the slope of the skateboard ramp.Find the slope (or head) of the river. Express the answer as a percentage.Highway tolls. It has been suggested that since heavier vehicles are responsible for more wear and tear on highways, drivers should pay tolls in direct proportion to the weight of their vehicles. Suppose a Toyota Camry weighting 3350 lb was charged $2.70 for traveling an 80-mile stretch of highway a. Find an equation of variation that expresses the amount of the toll T as a function of the vehicle's weight w. b. What would the toll be if a 3700-lb Jeep Cherokee drove the same stretch of highway?Inkjet cartridges. A registrar's office finds that the number of inkjet cartridges, I, required each year for its copiers and printers vanes directly with the number of students enrolled, s. a. Find an equation of variation that expresses I as a function of s, if the office requires 16 cartridges when 2800 students enroll. b. How many cartridges would be required if 3100 students enrolled?Profit-and-loss analysis. Boxowitz, Inc., a computer firm, is planning to sell a new graphing calculator. For the first year, the fixed costs for setting up the new production line are $100,000. The variable costs for each calculator are $20. The sales department projects that 150,000 calculators will be sold during the first year at a price of $45 each. a. Find and graph C(x), the total cost of producing x calculators. b. Using the same axes as in part (a), find and graph R(x), the total revenue from the sale of x calculators. c. Using the same axes as m part (a), find and graph P(x), the total profit from the production and sale of x calculators d. What profit or loss will the firm realize if the expected sale of 150,000 calculators occurs? e. How many calculators must the firm sell in order to break even?Profit-and-loss analysis. Red Tide is planning a new line of skis. For the first year, the fixed costs for setting up production are $45,000. The variable costs for producing each pair of skis are estimated at $80, and the selling price will be $450 per pair. It is projected that 3000 pairs will sell the first year. a. Find and graph C(x), the total cost of producing x pairs of skis. b. Find and graph R(x) the total revenue from the sale of x pairs of skis Use the same axes as in part (a). c. Using the same axes as in part (a), find and graph P(x), the total profit from the production and sale of x pairs of skis. d. What profit or loss will the company realize if the expected sale of 3000 pairs occurs? e. How many pairs must the company sell in order to break even?68. Profit-and-loss analysis. Jamal decides to mow lawns to earn money. The initial cost of his lawnmower is $250. Gasoline and maintenance costs are $4 per lawn.
a. Formulate a function for the total cost of mowing x lawns.
b. Jamal determines that the total-profit function for the lawn-mowing business is given by .
Find a function for the total revenue from mowing x lawns. How much does Jamal charge per lawn?
c. How many lawns must Jamal mow before he begins making a profit?
Straight-line depreciation. Quick Copy buys an office machine for $5200 on January 1 of a given year. The machine is expected to last for 8 yr, at the end of which time its salvage value will be $1100. If the company figures the decline in value to be the same each year, then the straight line depreciation value, V(t), after t years, 0t8, is given by V(t)=Ct(CSN), where C is the original cost of the item, N is the number of years of expected life, and S is the salvage value, V(t), is also called the book value. a. Find the linear function for the straight-line depreciation of the machine. b. Find the book value of the machine after 0 yr, 1 yr, 2 yr, 3 yr, 4 yr, 7 yr, and 8 yr.Straight-line depreciation. (See Exercise 69) Hanna's Photography spends $40 per square foot on improvements to a 25,000- ft2 office space. Under IRS guidelines for straight-line depreciation, these improvements will depreciate completely-that is, have zero salvage value-after 39 yr. Find the depreciated value of the improvements after 10 yr.Straight-line depreciation. The Video Game Wizard buys a new computer system for $60,000 and projects that its book value will be $2000 after 5 yr Using straight-line depreciation find the book value after 3 yr.72. Straight-line depreciation. Tyline Electric uses the function to find the book value, in dollars, of a photocopier t years after its purchase.
a. a) What do the numbers – 700 and 3500 signify?
b. How long will it take the copier to depreciate completely?
c. What is the domain of B? Explain.
Stair requirements. A North Carolina state law requires that stairs have minimum treads of 9 in and maximum risers of 8.25 in (Source: North Carolina Office of the State Fire Marshal.) According to this law what is the maximum grade of stairs in North Carolina? About 91% (don't round up, for legal reasons!)74. Health insurance premiums. Find the average rate of change in the annual premium for a family's health insurance.
(Source: The Kaiser Family Foundation; Health Research and Education Trust)
75. Health insurance premiums. Find the average rate of change m the annual premium for a single person.
(Source: The Kaiser Family Foundation; Health Research and Education Trust.)
76. Two-year college tuitions. Find the average rate of change of the tuition and fees at public two-year colleges.
(Source: U.S. National Center for Education Statistics, Digest of Education Statistics, annual)
Organic food sales. Find the average rate of change of organic food sales in the United States. (Source: Big Green Opportunity Report, 2013)Energy conservation. The R-factor of home insulation is directly proportional to its thickness T. a. Find an equation of variation if R=12.51 when T=3 in. b. What is the R-factor for insulation that is 6 in. thick?Nerve impulse speed. Impulses in nerve fibers travel at a speed of 293 ft/sec. The distance D, in feet traveled in t sec is given by D=293t. How long would it take an impulse to travel from the brain to the toes of a person who is 6 ft tall?80. Muscle weight. The weight M of a person's muscles is directly proportional to the person’s body weight W.
Muscle weight is directly proportional to body weight.
a. a) It is known that a person weighing 200 lb has 80 lb of muscles. Find an equation of variation expressing M as a function of W.
b. Express the variation constant as a percentage, and interpret the resulting equation.
c. What is the muscle weight of a person weighing 120 lb?
81. Brain weight. The weight B of a person's brain is directly proportional to the person's body weight W.
a. It is known that a person weighing 120 lb has a brain that weighs 3 lb. Find an equation of variation expressing B as a function of W.
b. Express the variation constant as a percentage, and interpret the resulting equation.
c. What is the weight of the brain of a person weighing 160 lb?
Stopping distance on glare ice. The stopping distance (at some fixed speed) of regular tires on glare ice is a linear function of the air temperature F, D(F)=2F+115, (a) D(0)=115ft,D(20)=75ft,D(10)=135ft,D(32)=179ft where D(F) is the stopping distance, in feet when the air temperature is F, in degrees Fahrenheit. a. Find D(0),D(20),D(10) and D(32). b. Explain why the domain should be restricted to the interval [57.5,32].83. Reaction time. While driving a car, you see a child suddenly crossing the street. Your brain registers the emergency and sends a signal to your foot to hit the brake. The car travels a reaction distance D, in feet, during this time, where D is a function of the speed r, in miles per hour that the car is traveling when you see the child. That reaction distance is a linear function given by
.
a. Find and.
b. Graph.
c. What is the domain of the function? Explain.
Estimating heights. An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones. The humerus is the bone from the elbow to the shoulder. Let x = the length of the humerus, in centimeters Then the height, in centimeters, of a male with a humerus of length x its given by M(x)=2.89x+70.64 The height, in centimeters, of a female with a humerus of length x is given by F(x)=2.75x+71.48 A 26-cm humerus was uncovered in some ruins. a. If we assume it was from a male, how tall was he? b. If we assume it was from a female, how tall was she?85. Percentage of young adults using social networking sites. In 2006, the percentage of 18 to 29 year olds who used social networking sites was 49%. In 2013, that percentage had risen to 89%. (Source: pewresearch.org.)
a. Use the year as the x-coordinate and the percentage as they-coordinate. Find the equation of the line that contains the data points.
b. Use the equation in part (a) to estimate the percentage of young adults using social networking sites in 2014.
c. Use the equation m part (a) to estimate the year in which the percentage of young adults using social networking sites will reach 100%
d. Explain why a linear equation cannot be used for years after the year found m part (c).
Manatee population. In January 2005, 3143 manatees were counted in an aerial survey of Florida in January 2011, 4834 manatees were counted. (Source Florida Fish and Wildlife Conservation Commission.) a. Using the year as the x-coordinate and the number of manatees as the y-coordinate, find an equation of the line that contains the two data points. b. Use the equation in part (a) to estimate the number of manatees counted m January 2010. c. c) The actual number counted in January 2010 was 5067. Does the equation found in part (a) give an accurate representation of the number of manatees counted each year? Why or why not?87. Urban population. The population of Woodland is P. After growing 2%, the new population is N.
a. Assuming that N is directly proportional to P, find an equation of variation.
b. Find N when.
c. Find P when.
88. Median age of women at first marriage. In general, people in our society are marrying at a later age. The median age, of women at first marriage can be approximated by
,
where t is the number of years after 1950. Thus, is the median age of women at first marriage in 1950, is the median age in 2000, and so on.
a. a) Find, and.
b. What was the median age of women at first marriage in 2008?
c. Graph.
89. Suppose, and all lie on the same line. Find y.
Describe one situation in which you would use the slope-intercept equation rather than the point-slope equation.Business: daily sales. Match each sentence below with the most appropriate of the following graphs (I, II, III, or IV) a. After January 1, daily sales continued to rise, but at a slower rate. b. After January 1, sales decreased faster than they ever grew. c. The rate of growth in daily sales doubled after January 1. d. After January 1, daily sales decreased at half the rate that they grew in December.Business: depreciation. A large crane is being depreciated according to the model V(t)=90060t, where V(t) is in thousands of dollars and t is the number of years since 2005 If the crane is to be depreciated until its value is $0, what is the domain of this model?Graph some of the total-revenue, total-cost, and total-profit functions in this exercise set using the same set of axes. Identify regions of profit and loss. Answers may vary.Graph each pair of equations on one set of axes. y=14x2 and y=14x2Graph each pair of equations on one set of axes.
2. and
Graph each pair of equations on one set of axes.
3. and
Graph each pair of equations on one set of axes.
4. and
Graph each pair of equations on one set of axes. y=3x2 and y=3x3+2Graph each pair of equations on one set of axes. y=2x2 and y=2x2+1Graph each pair of equations on one set of axes. y=|x| and y=|x3|Graph each pair of equations on one set of axes.
8. and
Graph each pair of equations on one set of axes. y=x3 and y=x3+1Graph each pair of equations on one set of axes.
10. and
Graph each pair of equations on one set of axes.
11. and
Graph each pair of equations on one set of axes.
12. and
For each of the following, state whether the graph of the function is a parabola. If the graph is a parabola find its vertex. f(x)=x2+4x7For each of the following, state whether the graph of the function is a parabola. If the graph is a parabola find its vertex.
14.
For each of the following, state whether the graph of the function is a parabola. If the graph is a parabola find its vertex. g(x)=2x44x23For each of the following, state whether the graph of the function is a parabola. If the graph is a parabola find its vertex. g(x)=3x26xGraph.
17.
Graph. y=x24x+3Graph. y=x2+2x1Graph.
20.
Graph. f(x)=3x26x+4Graph.
22.
Graph.
23.
Graph. g(x)=2x23x+7Graph. y=3xGraph.
26.
Graph. y=2xGraph. y=3xGraph.
29.
Graph.
30.
Graph.
31.
Graph. y=1|x|Graph.
33.
Graph.
34.
Graph.
35.
Graph.
36.
Graph.
37.
Graph. g(x)=(1.5)xGraph.
39.
Graph. g(x)=(0.25)xGraph. f(x)=3.(1.1)xGraph. g(x)=2.(1.45)xGraph. f(x)=4.(0.8)xGraph.
44.
Solve. x22x=2Solve. x22x+1=5Solve.
47.
Solve. x2+4x=3Solve. 4x2=4x+1Solve.
50.
Solve. 3y2+8y+2=0Solve. 2p25p=1Solve. x+7+9x=0 (Hint: Multiply both sides by x).Solve. 11w=1w2Rewrite each of the following as an equivalent expression using radical notation.
55.
56ERewrite each of the following as an equivalent expression using radical notation. y2/3Rewrite each of the following as an equivalent expression using radical notation. t2/5Rewrite each of the following as an equivalent expression using radical notation. t2/5Rewrite each of the following as an equivalent expression using radical notation. y2/3Rewrite each of the following as an equivalent expression using radical notation. b1/3Rewrite each of the following as an equivalent expression using radical notation.
62.
Rewrite each of the following as an equivalent expression using radical notation. (x23)1/2Rewrite each of the following as an equivalent expression using radical notation. (y2+7)1/4Rewrite each of the following as an equivalent expression with rational exponents. x3Rewrite each of the following as an equivalent expression with rational exponents. x5Rewrite each of the following as an equivalent expression with rational exponents. a35Rewrite each of the following as an equivalent expression with rational exponents. b24,b0Rewrite each of the following as an equivalent expression with rational exponents. x124,x0Rewrite each of the following as an equivalent expression with rational exponents.
70.
Rewrite each of the following as an equivalent expression with rational exponents. 1t5Rewrite each of the following as an equivalent expression with rational exponents.
72.
Rewrite each of the following as an equivalent expression with rational exponents.
71.
Rewrite each of the following as an equivalent expression with rational exponents.
74.
Simplify.
75.
Simplify.
76.
Simplify.
77.
Simplify. 82/3Determine the domain of each function. f(x)=x225x5Determine the domain of each function. f(x)=x24x+2Determine the domain of each function. f(x)=x3x25x+6Determine the domain of each function. f(x)=x4+7x2+6x+5Determine the domain of each function. f(x)=5x+4Determine the domain of each function. f(x)=2x6Determine the domain of each function.
85.
Determine the domain of each function.
86.
Find the equilibrium point for each pair of demand and supply functions. Demand: q=100010x; Supply: q=250+5xFind the equilibrium point for each pair of demand and supply functions. Demand: q=880030x; Supply: q=7000+15xFind the equilibrium point for each pair of demand and supply functions. Demand: q=5x; Supply: q=x5Find the equilibrium point for each pair of demand and supply functions. Demand: q=4x; Supply: q=x4Find the equilibrium point for each pair of demand and supply functions. Demand: q=(x3)2; Supply: q=x2+2x+1 (assume x3)Find the equilibrium point for each pair of demand and supply functions.
92. Demand: Supply: (assume)
Find the equilibrium point for each pair of demand and supply functions. Demand: q=5x; Supply: q=x+7Find the equilibrium point for each pair of demand and supply functions. Demand: q=7x; Supply: q=x+1295. Price of admission. The number of tickets sold for a fund-raiser is inversely proportional to the price of a ticket, p. If 175 tickets can be sold for $20 each, how many tickets will be sold if the price is $25 each?
Demand. The quantity sold x of a high-definition TV is inversely proportional to the price p. If 85,000 high definition TVs sold for $1200 each, how many will be sold if the price is $850 each?97. Radar Range. The function given by
can be used to approximate the maximum range, , in miles, of ARSR-3 surveillance radar with a peak power of x watts.
a. Determine the maximum radar range when the peak power is 40,000 watts, 50,000 watts, and 60,000 watts.
b. Graph the function.
98. Home Range. Refer to Example 14. The home range, m hectares, of an omnivorous mammal of mass w grams is given by
.
(Source: Harestad, A S., and Bunnel, F. L, "Home Range and Body Weight-A Reevaluation; “Ecology, Vol. 60, No. 2 (April, 1979), pp 389-402.) Complete the table of approximate function values and graph the function.
w 0 1000 2000 3000 4000 5000 6000 7000
0 34.0 64.2 93.3 121.6 149.3 176.5 203.4
99. Life Science: pollution control. Pollution control has become an important concern worldwide. The function
describes the average pollution, in micrograms per cubic meter of air, t years after 2000. (Source: epa.gov/airtrends/pm.html)
a. Predict the pollution in 2015, 2020, and 2025.
b. Graph the function over the interval .
Surface area and mass. The surface area of a person whose mass is 75 kg can be approximated by f(h)=0.144h1/2, where f(h) is measured in square meters and h is the person's height in centimeters (Source US Oncology.) a. Find the approximate surface area of a person whose mass is 75 kg and whose height is 180 cm. b. Find the approximate surface area of a person whose mass is 75 kg and whose height is 170 cm. c. Graph the function f for 0h200.Zipf's Law. According to Zipfs Law, the number of cities N with a population greater than S is inversely proportional to S. In 2012, there were 285 U.S. cities with a population greater than 100,000. Estimate the number of US cities with a population between 350,000 and 500 000; between 300,000 and 600,000.At most, how many y-intercepts can a function have? Why?What is the difference between a rational function and a polynomial function?Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places. f(x)=x3xUse the zero feature or the intersect feature to approximate the zeros of each function to three decimal places. f(x)=2x3x214x10Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places. f(x)=12(|x4|+|x7|)4Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places.
107.
Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places.
108.
Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places. f(x)=|x+1|+|x2|5Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places. f(x)=|x+1|+|x+2|Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places.
111.
Use the zero feature or the intersect feature to approximate the zeros of each function to three decimal places.
112.
113. Find the equilibrium point for the following demand and supply functions where q is the quantity in thousands of units, and x is the price per unit, in dollars,
Demand:
Supply:
For the scatter plots and graphs in Exercises 1 9, determine which, if any, of the following functions might be the best model for the data Linear, f(x)=mx+b Quadratic, f(x)=ax2+bx+c,a0 Quadratic f(x)=ax2+bx+c,a0 Polynomial, neither quadratic nor linear Exponential, f(x)=a0ax,0aora1 or a1For the scatter plots and graphs in Exercises 1 9, determine which, if any, of the following functions might be the best model for the data Linear, f(x)=mx+b Quadratic, f(x)=ax2+bx+c,a0 Quadratic f(x)=ax2+bx+c,a0 Polynomial, neither quadratic nor linear Exponential, f(x)=a0ax,0aora1 or a1For the scatter plots and graphs in Exercises 1 9, determine which, if any, of the following functions might be the best model for the data Linear, f(x)=mx+b Quadratic, f(x)=ax2+bx+c,a0 Quadratic f(x)=ax2+bx+c,a0 Polynomial, neither quadratic nor linear Exponential, f(x)=a0ax,0aora1 or a1For the scatter plots and graphs in Exercises 1– 9, determine which, if any, of the following functions might be the best model for the data
Linear,
Quadratic,
Quadratic
Polynomial, neither quadratic nor linear
Exponential, or
4.
(Source: U. S. Census Bureau, Foreign Trade Statistics)
For the scatter plots and graphs in Exercises 1– 9, determine which, if any, of the following functions might be the best model for the data
Linear,
Quadratic,
Quadratic
Polynomial, neither quadratic nor linear
Exponential, or
5.
(Source: U.S. Census Bureau.)
For the scatter plots and graphs in Exercises 1– 9, determine which, if any, of the following functions might be the best model for the data
Linear,
Quadratic,
Quadratic
Polynomial, neither quadratic nor linear
Exponential, or
6.
(Source: The Compendium of Professional Basketball, by Robert D. Bradley, Xaler Press, 2010; personal communication with author, 2013.)
For the scatter plots and graphs in Exercises 1 9, determine which, if any, of the following functions might be the best model for the data Linear, f(x)=mx+b Quadratic, f(x)=ax2+bx+c,a0 Quadratic f(x)=ax2+bx+c,a0 Polynomial, neither quadratic nor linear Exponential, f(x)=a0ax,0aora1 or a1For the scatter plots and graphs in Exercises 1 9, determine which, if any, of the following functions might be the best model for the data Linear, f(x)=mx+b Quadratic, f(x)=ax2+bx+c,a0 Quadratic f(x)=ax2+bx+c,a0 Polynomial, neither quadratic nor linear Exponential, f(x)=a0ax,0aora1 or a1For the scatter plots and graphs in Exercises 1 9, determine which, if any, of the following functions might be the best model for the data Linear, f(x)=mx+b Quadratic, f(x)=ax2+bx+c,a0 Quadratic f(x)=ax2+bx+c,a0 Polynomial, neither quadratic nor linear Exponential, f(x)=a0ax,0aora1 or a1 Cable video customers. a. For the data shown in Exercise 8, find a linear function that fits the data using the values given for 2006 and 2012. Let x represent the number or years since 2006 b. Use the linear function to estimate the number of cable video customers in 2018.Average salary in NBA. Use the data from the bar graph in Exercise 6. a. Find a linear function that fits the data using the average salaries given for the years 2006 and 2013. Use 0 for 2006 and 7 for 2013. b. Use the linear function to predict average salaries in 2018 and 2025. c. In what year will the average salary reach 7.0 million?Absorption of an asthma medication. Use the data from Exercise 3. a. Find a quadratic function that fits the data using the data points (0, 0), (2, 200), and (3, 167). b. Use the function to estimate the amount of albuterol in the bloodstream 4 hr after inhalation. c. Does it make sense to use this function for t = 6? Why or why not?Median income. Use the data given in Exercise 5. a. Find a quadratic function that fits the data using the data points (21,14483),(41,50168), and (61,49615). b. Use the function to estimate the median income of Americans who were 55 in 2012.Braking Distance. y=0.144x24.63x+60 Find a quadratic function that fits the following data. Travel Speed (mph) Braking Distance (ft) 20 25 40 105 60 300 (Source: New Jersey Department of Law and Public Safety.) b. Use the function to estimate the braking distance of a car traveling at 50 mph. c. Does it make sense to use this function when speeds are less than 15 mph? Why or why not?14. Daytime Accidents.
Find a quadratic function that fits the following data.
Travel Speed (in km/h) Number of Daytime Accidents (for every 200 million km driven)
60 100
80 130
100 200
(Source: New Jersey Department of Law and Public Safety.)
b. Use the function to estimate the number of daytime accidents that occur at 50 km/h for every 200 million km driven.
High blood pressure in women. a. Choose two points from the following data and find a linear function that fits the data. Age of Female Percentage of Females with High Blood Pressure 30 1.4 40 8.5 50 19.1 60 31.9 70 53.0 (Source: Based on data from Health United States 2005, CDC/NCHS.) b. Graph the scatter plot and the function on the same set of axes. c. Use the function to estimate the percentage of 55-yr-old women with high blood pressure.16. High blood pressure in men.
a. Choose two points from the following data and find a linear function that fits the data.
Age of Female Percentage of Males with High Blood Pressure
30 7.3
40 12.1
50 20.4
60 24.8
70 34.9
(Source: Based on data from Health United States 2005, CDC/NCHS.)
b. Graph the scatter plot and the function on the same set of axes.
c. Use the function to estimate the percent of 55-yr-old men with high blood pressure.
In Exercises 17 20, give the numbers in the exponential function to three decimal places. Population. a. Find the exponential function that best fits the following data. Years since 1970 Population of Texas (in millions) 0 11.2 10 14.2 20 17.0 30 20.9 43 26.4 (Source: www.census.gov) b. Graph the scatter plot and the function on the same set of axes. c. Use the function to estimate the population of Texas in 2020.In Exercises 17– 20, give the numbers in the exponential function to three decimal places.
18. Population.
a. Find the exponential function that best fits the following data.
Years since 1970 Population of Detroit (in millions)
0 1.5
10 1.2
20 1.0
30 0.95
40 0.71
(Source: www.census.gov)
b. Graph the scatter plot and the function on the same set of axes.
c. Use the function to estimate the population of Detroit in 2020.
In Exercises 17– 20, give the numbers in the exponential function to three decimal places.
19. Buying Power.
a. Find the exponential function that best fits the following data.
Years since 1980 Equivalent buying power of $100 (in 1980 dollars)
0 $100.00
10 $158.62
20 $208.98
30 $264.63
33 $282.85
(Source: http://data.bls.gov/cgi-bin/cpicalc.pl.)
b. Graph the scatterplot and the function on the same set of axes.
c. Use the function to estimate the equivalent buying power of S100 (in 1980 dollars) in 2020.
d. What other models also fit this data? Which model best predicts the equivalent buying power of $100 (in 1980 dollars) for future years? Why?
In Exercises 17– 20, give the numbers in the exponential function to three decimal places.
20. Stock Prices.
a. Find the exponential function that best fits the following data.
Years since 2010 Price of one share of starbucks stock at beginning of January
0 $20.59
1 $30.21
2 $46.61
3 $55.39
4 $76.17
(Source: yahoo.finance Nasdaq)
b. Graph the scatterplot and the function on the same set of axes.
c. Use the function to estimate the price of one share of Starbucks stock in 2018.
d. What other models also fit this data? Which model best predicts the price of one share of Starbucks stock in future years? Why?
21. Under what conditions might it make better sense to use a linear function rather than a quadratic or cubic function that fits a few data points more closely?
For modeling the number of hours of daylight for the dates April 22 to August 22, which would be a better choice: a linear function or a quadratic function? Why?What restrictions should be placed on the domain of the quadratic function found in Exercise 11? Why are such restrictions needed?What restrictions should be placed on the domain of the quadratic function found in Exercise 12? Why are such restrictions needed?25. Business: cable video customers.
a. Use regression to fit a linear function to the data shown in Exercise 8.
b. Use the function to estimate the number of cable video customers in 2018.
c. Compare your answer for part (b) to that found in Exercise 9. Which is more accurate?
The answers are very close, giving essentially the same information
d. Fit an exponential function to the data and use 1t to estimate the number of cable video customers in 2018.
e. Is a linear or an exponential model more appropriate for this set of data? Why
Business: trade deficit with Japan. a. Use regression to fit a cubic function to the data in Exercise 4. Let x be the number of years after 2006. b. Use the function to estimate the trade deficit with Japan in 2017. c. Why might a linear function be a more logical choice than a cubic function for modeling this set of data?Classify each statement as either true or false. Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] If limx5f(x) exists, then f(5) must exist. [1.1]Classify each statement as either true or false. Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] If limx2f(x)=LthenL=f(2). [1.1]Classify each statement as either true or false.
Match each function in column A with the most appropriate rule to use for differentiating the function.
3. If f is continuous at then .
Classify each statement as either true or false.
Match each function in column A with the most appropriate rule to use for differentiating the function.
4. A function’s average rate of change over interval is the same as its instantaneous rate of change at .
Classify each statement as either true or false.
Match each function in column A with the most appropriate rule to use for differentiating the function.
5. A function’s derivative at a point, if it exists, can be found as the limit of a difference quotient.
Classify each statement as either true or false.
Match each function in column A with the most appropriate rule to use for differentiating the function.
6. For to exists, f must be continuous at .
Classify each statement as either true or false.
Match each function in column A with the most appropriate rule to use for differentiating the function.
7. If f is continuous at 5, then must exist.
Classify each statement as either true or false.
Match each function in column A with the most appropriate rule to use for differentiating the function.
8. The acceleration function is the derivative of the velocity function.
Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] Column B a. Extended Power Rule b. Product Rule c. Sum Rule d. Different Rule e. Power Rule f. Quotient Rule f(x)=x7Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] Column B a. Extended Power Rule b. Product Rule c. Sum Rule d. Different Rule e. Power Rule f. Quotient Rule g(x)=x+9Match each function in column A with the most appropriate rule to use for differentiating the function.
Column B
a. Extended Power Rule
b. Product Rule
c. Sum Rule
d. Different Rule
e. Power Rule
f. Quotient Rule
11.
Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] Column B a. Extended Power Rule b. Product Rule c. Sum Rule d. Different Rule e. Power Rule f. Quotient Rule G(x)=2x+13x4Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] Column B a. Extended Power Rule b. Product Rule c. Sum Rule d. Different Rule e. Power Rule f. Quotient Rule H(x)=f(x)g(x)Match each function in column A with the most appropriate rule to use for differentiating the function. [1.5,1.6] Column B a. Extended Power Rule b. Product Rule c. Sum Rule d. Different Rule e. Power Rule f. Quotient Rule f(x)=2x7For Exercises 15-17, consider limx7f(x),wheref(x)=x2+4x21x+7. Limit numerically. [1.1] a. Find the limit by completing the following input-output tables. x7 f(x) 7.1 10.1 7.01 10.01 7.001 10.001 x7+ f(x) 6.9 9.9 6.99 9.99 6.999 9.999 b. Find limx7f(x),limx7+f(x),andlimx7f(x), if each exists.For Exercises 15-17, consider limx7f(x),wheref(x)=x2+4x21x+7. Limit graphically. Find the limit by graphing the function. [1.1]For Exercises 15-17, consider
.
17. Limit algebraically. Find the limit algebraically. Show your work.
Find each limit, if it exists. If a limit does not exist, state that fact.
18.
Find each limit, if it exists. If a limit does not exist, state that fact. [1.1,1.2] limx1(4x3x2+7x)Find each limit, if it exists. If a limit does not exist, state that fact. [1.1,1.2] limx7x2+2x35x+7Find each limit, if it exists. If a limit does not exist, state that fact.
21.
For Exercises 22-30, consider the function g graphed below. Find limx1g(x). [1.1]For Exercises 22-30, consider the function g graphed below. Find g(1). [1.1]For Exercises 22-30, consider the function g graphed below.
24. Is g continuous at 1? Why or why not?
For Exercises 22-30, consider the function g graphed below.
25. Find .
For Exercises 22-30, consider the function g graphed below.
26. Find .
For Exercises 22-30, consider the function g graphed below. Is g continuous at 2? Why or why not? [1.2]For Exercises 22-30, consider the function g graphed below. Find the average rate of change between x=2andx=1. [1.3]For Exercises 22-30, consider the function g graphed below. Find g(4). [1.4]For Exercises 22-30, consider the function g graphed below. For which value(s) is g(x) not defined? Why? [1.4]For Exercises 31-34, consider the function f graphed below. For which value(s) is f(x) not defined? Why? [1.4]For Exercises 31-34, consider the function f graphed below. Find limx0f(x). [1.1]For Exercises 31-34, consider the function f graphed below. Find limx2f(x). [1.1]For Exercises 31-34, consider the function f graphed below.
34. Is f continuous at ?
35. For find the average rate of change as x changes from –1 to 2.
Find a simplified difference quotient for g(x)=3x2+2. [1.3]37. Find a simplify difference quotient for
.
38. Find an equation of the tangent line to the graph of at the point .
39. Find the point(s) on the graph of at which the tangent line is horizontal.
40. Find the point(s) on the graph of at which the tangent line has slope .
Find .
41.
Find dy/dx. y=8x3 [1.5]Find .
43.
Find dy/dx. y=15x2/5 [1.5]Find .
45.
Differentiate. f(x)=512x6+8x42x [1.5]Differentiate.
47.
Differentiate. y=x2+88x [1.6]Differentiate.
49.
Differentiate. f(x)=(x53)7 [1.7]Differentiate. f(x)=x2(4x+2)3/4 [1.7]52. For .
For y=342x710x3+13x2+28x2,findy. [1.8]54. Social science: growth rate. The population of Lawton grows from an initial size of 10,000 to a size P, given by
, where t is in years.
a. Find the growth rate.
b. Find the number of people in Lawton after 20 yr .
c. Find the growth rate at .
For Exercises 55-58, consider the growth of , the distance a jogger has run after t minute.
55. In what open interval(s) is the Jogger running at a constant speed?
For Exercises 55-58, consider the growth of , the distance a jogger has run after t minute.
56. In what open interval(s) is she stopped?
For Exercises 55-58, consider the growth of y=s(t), the distance a jogger has run after t minute. [1.8] In what open interval(s) is she accelerating?For Exercises 55-58, consider the growth of y=s(t), the distance a jogger has run after t minute. [1.8] In what open interval(s) is she decelerating?For s(t)=t+t4, with t in seconds and s(t) in feet, find each of the following. [1.8] a. v(t) b. a(t) c. The velocity and the acceleration when t=2secBusiness: average revenue, cost, and profit. Given revenue and cost function R(x)=40x and C(x)=5x+100, R(x) find each for the following. Assume C(x) are in dollars and x is the number of lamps produced. [1.6] a. The average cost, the average revenue, and the average profit when x lamps are produced and sold b. The rate at which average cost is changing when 9 lamps are produced Average cost is dropping at approximately 1.33 per lamp.Find ddx(fg)(x) and ddx(gf)(x), given f(x)=x2+5 and g(x)=12x. [1.7]62RE63RE