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All Textbook Solutions for Precalculus
In Problems 51-66, find the domain of each function. P( t )= t-4 3t-21In Problems 51-66, find the domain of each function. h( z )= z+3 z-2In Problems 5170, find the domain of each function. f(x)=5x4364AYU65AYU66AYU67AYU68AYU69AYU70AYU71AYU72AYU73AYU74AYU75AYU76AYU77AYU78AYU79AYU80AYU81AYU82AYU83AYU84AYU85AYU86AYU87AYU88AYU89AYU90AYU91AYU92AYU93AYU94AYU95AYUEffect of Gravity on Jupiter If a rock falls from a height of 20 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately H( x )=2013 x 2 a. What is the height of the rock when x=1 second? x=1.1 seconds? x=1.2 seconds? b. When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters? c. When does the rock strike the ground?Cost of Transatlantic Travel A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by C( x )=100+ x 10 + 36,000 x where x is the ground speed (airspeed wind). a. What is the cost per passenger for quiescent (no wind) conditions? b. What is the cost per passenger with a head wind of 50 miles per hour? c. What is the cost per passenger with a tail wind of 100 miles per hour? d. What is the cost per passenger with a head wind of 100 miles per hour?Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A( x )=4x 1 x 2 , where x represents the length, in feet, of half the base of the beam. See the figure. Determine the cross-sectional area of the beam if the length of half the base of the beam is as follows: a. One-third of a foot b. One-half of a foot c. Two-thirds of a footEconomics The participation rate is the number of people in the labor force divided by the civilian population (excludes military). Let L( x ) represent the size of the labor force in year X , and P( x ) represent the civilian population in year X . Determine a function that represents the participation rate R as a function of X .Crimes Suppose that V( x ) represents the number of violent crimes committed in year x and P( x ) represents the number of property crimes committed in year X . Determine a function T that represents the combined total of violent crimes and property crimes in year X .Health Care Suppose that P( x ) represents the percentage of income spent on health care in year x and I( x ) represents income in year x . Determine a function H that represents total health care expenditures in year x .Income Tax Suppose that I( x ) represents the income of an individual in year x before taxes and T( x ) represents the individuals tax bill in year x . Determine a function N that represents the individuals net income (income after taxes) in year x .Profit Function Suppose that the revenue R , in dollars, from selling x cell phones, in hundreds, is R( x )=1.2 x 2 +220x . The cost C , in dollars, of selling x cell phones, in hundreds, is C( x )=0.05 x 3 2 x 2 +65x+500 . a. Find the profit function, P( x )=R( x )C( x ) . b. Find the profit if x=15 hundred cell phones are sold. c. Interpret P( 15 ) .104AYU105AYU106AYUInvestigate when, historically, the use of the function notation y=f( x ) first appeared.108AYUThe intercepts of the equation x 2 +4 y 2 =16 are _______. (pp. 18-19)True or False The point ( 2,6 ) is on the graph of the equation x=2y2 . (pp. 7-11)3. A set of points in the xy -plane is the graph of a function if and only if every _______ line intersects the graph in at most one point.4. If the point ( 5,3 ) is a point on the graph of f , then f( )= _________.5. Find a so that the point ( 1,2 ) is on the graph of f( x )=a x 2 +4 .True or False Every graph represents a function.True or False The graph of a function y=f( x ) always crosses the y-axis .True or False The y-intercept of the graph of the function y=f( x ) , whose domain is all real numbers, is f( 0 ) .Use the given graph the function f to answer parts (a)-(n). (a) Find f( 0 ) and f( 6 ) . (b) Find f( 6 ) and f( 11 ) . (c) Is f( 3 ) positive or negative? (d) Is f( 4 ) positive or negative? (e) For what values of x is f( x )=0 ? (f) For what values of x is f( x )0 ? (g) What is the domain of f ? (h) What is the range of f ? (i) What are the x-intercepts ? (j) What is the y-intercept ? (k) How often does the line y= 1 2 intersect the graph? (l) How often does the line x=5 intersect the graph? (m) For what values of x does f( x )=3 ? (n) For what values of x does f( x )=2 ?Use the given graph the function f to answer parts (a)-(n). (a) Find f( 0 ) and f( 6 ) . (b) Find f( 2 ) and f( 2 ) . (c) Is f( 3 ) positive or negative? (d) Is f( 1 ) positive or negative? (e) For what values of x is f( x )=0 ? (f) For what values of x is f( x )0 ? (g) What is the domain of f ? (h) What is the range of f ? (i) What are the x-intercepts ? (j) What is the y-intercept ? (k) How often does the line y=1 intersect the graph? (l) How often does the line x=1 intersect the graph? (m) For what values of x does f( x )=3 ? (n) For what values of f( x )=2 ?In Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn Problems 13-24, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis , the y-axis , or the originIn problems 2530, answer the questions about each function. f(x)=3x2+x2 Is the point (1,2) on the graph of f? If x=2, what is f(x)? What point is on the graph of f? If f(x)=2, what is x? What point(s) are on the graph of f? What is the domain of f? List the x intercepts, if any, of the graph of f? List the y intercept, if there is one, of the graph of f?In Problems 25-30, answer the questions about the given function. f( x )=-3 x 2 +5x (a) Is the point ( 1,2 ) on the graph of f ? (b) If x=2 . what is f( x ) ? What point is on the graph of f ? (c) If f( x )=2 , what is x ? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts , if any, of the graph of f . (f) List the y-intercepts , if there is one, of the graph of f .In Problems 25-30, answer the questions about the given function. f( x )= x+2 x-6 (a) Is the point ( 3,14 ) on the graph of f ? (b) If x=4 . what is f( x ) ? What point is on the graph of f ? (c) If f( x )=2 , what is x ? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts , if any, of the graph of f . (f) List the y-intercepts , if there is one, of the graph of f .In Problems 25-30, answer the questions about the given function. f( x )= x 2 +2 x+4 (a) Is the point ( 1, 3 5 ) on the graph of f ? (b) If x=0 . what is f( x ) ? What point is on the graph of f ? (c) If f( x )= 1 2 , what is x ? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts , if any, of the graph of f . (f) List the y-intercepts , if there is one, of the graph of f .In problems 2530, answer the questions about each function. f(x)=12x4x2+1 Is the point (1,6) on the graph of f? If x=3, what is f(x)? What point is on the graph of f? If f(x)=1, what is x? What point(s) are on the graph of f? What is the domain of f? List the x intercepts, if any, of the graph of f? List the y intercept, if there is one, of the graph of f?In Problems 25-30, answer the questions about the given function. f( x )= 2x x2 (a) Is the point ( 1 2 , 2 3 ) on the graph of f ? (b) If x=4 . what is f( x ) ? What point is on the graph of f ? (c) If f( x )=1 , what is x ? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts ., if any, of the graph of f . (f) List the y-intercepts , if there is one, of the graph of f .The graphs of two functions, f and g , arc illustrated. Use the graphs to answer parts ( a )( f ) . (a) ( f+g )( 2 ) (b) ( f+g )( 4 ) (c) ( fg )( 6 ) (d) ( gf )( 6 ) (e) ( fg )( 2 ) (f) ( f g )( 4 )30AYU31AYU32AYU33AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYU41AYU42AYU43AYU44AYU45AYU46AYU‘Are You Prepared?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 1. The interval ( 2,5 ) can be written as the inequality ______. (pp. A76-A77)‘Are You Prepared?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 2. The slope of the line containing the points ( 2,3 ) and ( 3,8 ) is ______. (pp. 30-32)‘Are You Prepared?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 3. Test the equation y=5 x 2 1 for symmetry with respect to the x-axis , the y-axis , and the origin. (pp. 19-21)‘Are You Prepared?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 4. Write the point-slope form of the line with slope 5 containing the point ( 3,2 ) . (p. 33)‘Are You Prepared?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 5. The intercepts of the equation y= x 2 9 are ______. (pp. 18-19)6. A function f is _____ on an interval I if, for any choice of x 1 , and x 2 in I , with x 1 x 2 we have f( x 1 )f( x 2 ) .7. A(n) ______ function f is one for which f( x )=f( x ) for every x in the domain of f ; a(n) ______ function f is one for which f( x )=f( x ) for every x in the domain of f .8. True or False A function f is decreasing on an interval I if, for any choice of x 1 and x 2 in I , with x 1 x 2 , we have f( x 1 )f( x 2 ) .9. True or False A function f has a local maximum at c if there is an open interval I containing c such that for all x in I , f( x )f( c )10. True or False Even functions have graphs that are symmetric with respect to the origin.In Problems 13-24, use the graph of the function f given. 13. Is f increasing on the interval [ 8,2 ] ?In Problems 13-24, use the graph of the function f given. 14. Is f decreasing on the interval [ 8,4 ] ?In Problems 13-24, use the graph of the function f given. 15. Is f increasing on the interval [ 2,6 ] ?In Problems 13-24, use the graph of the function f given. 16. Is f decreasing on the interval [ 2,5 ] ?In Problems 13-24, use the graph of the function f given. 17. List the interval(s) on which f increasing.In Problems 13-24, use the graph of the function f given. 18. List the interval(s) on which f decreasing.In Problems 13-24, use the graph of the function f given. 19. Is there a local maximum at 2? If yes, what is it?In Problems 13-24, use the graph of the function f given. 20. Is there a local maximum at 5? If yes, what is it?In Problems 13-24, use the graph of the function f given. 21. List the number(s) at which f has a local maximum. What are the local maximum values?In Problems 13-24, use the graph of the function f given. 22. List the number(s) at which f has a local minimum. What are the local minimum values?In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 25.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 26.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 27.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 28.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 29.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 30.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 31.In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The domain and range c. The intervals on which the function is increasing, decreasing, or constant d. Whether the function is even, odd, or neither 32.In Problems 33-36, the graph of a function f is given. Use the graph to find: a. The numbers, if any, at which f has a local maximum. What are the local maximum values? b. The numbers, if any, at which f has a local minimum. What are the local minimum values? 33.In Problems 33-36, the graph of a function f is given. Use the graph to find: a. The numbers, if any, at which f has a local maximum. What are the local maximum values? b. The numbers, if any, at which f has a local minimum. What are the local minimum values? 34.In Problems 33-36, the graph of a function f is given. Use the graph to find: a. The numbers, if any, at which f has a local maximum. What are the local maximum values? b. The numbers, if any, at which f has a local minimum. What are the local minimum values? 35.In Problems 33-36, the graph of a function f is given. Use the graph to find: a. The numbers, if any, at which f has a local maximum. What are the local maximum values? b. The numbers, if any, at which f has a local minimum. What are the local minimum values? 36.In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 37. f( x )=4 x 3In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 38. f( x )=2 x 4 x 2In Problems 3748, determine algebraically whether each function is even, odd, or neither. g(x)=10x2In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 40. h( x )=3 x 3 +5In Problems 3748, determine algebraically whether each function is even, odd, or neither. F(x)=4x3In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 42. G( x )= xIn Problems 37-48, determine algebraically whether each function is even, odd, or neither. 43. f( x )=x+| x |In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 44. f( x )= 2 x 2 +1 3In Problems 3748, determine algebraically whether each function is even, odd, or neither. g(x)=1x2+8In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 46. h( x )= x x 2 1In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 47. h( x )= x 3 3 x 2 9In Problems 37-48, determine algebraically whether each function is even, odd, or neither. 48. F( x )= 2x | x |In Problem 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 49.In Problem 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 50.In Problem 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 51.In Problem 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 52.In Problem 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 53.In Problems 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 54.In Problems 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 55.In Problems 49-56, for each graph of a function y=f( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 56.In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 57. f( x )= x 3 3x+2[ 2,2 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 58. f( x )= x 3 3 x 2 +5[ 1,3 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 59. f( x )= x 5 x 3 [ 2,2 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 60. f( x )= x 4 x 2 [ 2,2 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 61. f( x )=0.2 x 3 0.6 x 2 +4x6[ 6,4 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 62. f( x )=0.4 x 3 +0.6 x 2 +3x2[ 4,5 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 63. f( x )=0.25 x 4 +0.3 x 3 0.9 x 2 +3[ 3,2 ]In Problems 57-64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 64. f( x )=0.4 x 4 0.5 x 3 +0.8 x 2 2[ 3,2 ]61AYU62AYU63AYU64AYU65AYU66AYU67AYU68AYU69AYU70AYU71AYU72AYU73AYU74AYU75AYU80. Medicine Concentration The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function C( t )=O.OOZ x 4 +0.039 f 3 0.285 i 2 +0.766i+0.085 (a) After how many hours will the concentration be highest? (b) (b) A woman nursing a child must wait until the concentration is below 0.5 before she can feed him. After taking the medication, how long must she wait before feeding her child?83. E. coli Growth A strain of E. coli Beu 397-recA441 is placed into a nutrient broth at 30 Celsius and allowed to grow. The data shown in the table are collected. The population is measured in grams and the time in hours. Since population P depends on time t, and each input corresponds to exactly one output, we can say that population is a function of time. Thus P(t) represents the population at time t. (a) Find the average rate of change of the population from 0 to 2.5 hours. (b) Find the average rate of change of the population from 4.5 to 6 hours. (c) What is happening to the average rate of change as time passes?National Debt The size of the total debt owed by the United States federal government continues to grow. In fact, according to the Department of the Treasury, the debt per person living in the United States is approximately (or over per U.S. household). The following data represent the U.S. debt for the years. Since the debt depends on the year and each input corresponds to exactly one output, the debt is a function of the year. So represents the debt for each year
Plot the points, and so on.
Draw a line segment from the point to. What does the slope of this line segment represent?
Find the average rate of change of the debt from to.
Find the average rate of change of the debt from to.
Find the average rate of change of the debt from to.
What appears to be happening to the average rate of change as time passes?
85. For the function f( x )= x 2 , compute the average rate of change: (a) From 0 to 1 (b) From 0 to 0.5 (c) From 0 to 0.1 (d) From 0 to 0.01 (e) From 0 to 0.001 (f) Use a graphing utility to graph each of the secant lines along with f. (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?86. For the function f( x )= x 2 , compute the average rate of change: (a) From 1 to 2 (b) From 1 to 1.5 (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with f. (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 87. f( x )=2x+5Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 88. f( x )=-3x+2Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 89. f( x )= x 2 +2xProblems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 90. f( x )=2 x 2 +xProblems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 91. f( x )=2 x 2 -3x+1Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 92. f( x )=- x 2 +3x-2Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 93. f( x )= 1 xProblems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points (x,f(x)) and (x+h,f(x+h)) on the graph of a function y=f( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h=0.5 , 0.1, and 0.01 at x=1 . What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x=1 with h=0.01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 94. f( x )= 1 x 295. Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: ( 0,3 ) and ( 3,0 ) ; a local maximum value of — 2 is at — 1; a local minimum value of —6 is at 2. Compare your graph with those of others. Comment on any differences.96. Redo Problem 95 with the following additional information: increasing on (,1] , [2,) ; decreasing on [1,2] . Again compare your graph with others and comment on any differences.97. How many x-intercept can a function defined on an interval have if it is increasing on that interval? Explain.98. Suppose that a friend of yours does not understand the idea of increasing and decreasing functions. Provide an explanation, complete with graphs, that clarifies the idea.99. Can a function be both even and odd? Explain.100. Using a graphing utility, graph y=5 on the interval [ 3,3 ] . Use MAXIMUM to find the local maximum values on [ 3,3 ] . Comment on the result provided by the calculator.101. A function f has a positive average rate of change on the interval [2,5] . Is f increasing on [2,5] ? Explain.102. Show that a constant function f( x )=b has an average rate of change of 0. Compute the average rate of change of y= 4 x 2 on the interval [ 2,2 ] . Explain how this can happen.Sketch the graph of y= x . (p. 22)Sketch the graph of y= 1 x . (pp. 22-23)List the intercepts of the equation y= x 3 8 . (pp. 18-19)The function f( x )= x 2 is decreasing on the interval ______.When functions are defined by more than one equation, they are called ______ functions.True or False The cube function is odd and is increasing on the interval ( , ) .True or False The cube root function is odd and is decreasing on the interval ( , ) .True or False The domain and the range of the reciprocal function are the set of all real numbers.In Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 11-18, match each graph to its function. A. Constant function B. Identity function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root functionIn Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )=xIn Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )= x 2In Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )= x 3In Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )= xIn Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )= 1 xIn Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )=| x |In Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )= x 3In Problems 19-26, sketch the graph of each function. Be sure to label three points on the graph. f( x )=3If f(x)={x2ifx04ifx=03x2ifx0 find : f(3) (b) f(0) (c) f(3)If f( x )={ 3xifx1 0ifx=1 2 x 2 +1ifx1 find: (a) f( 2 ) (b) f( 1 ) (c) f( 0 )If f(x)={2x+4if3x1x31if1x5 find : f(2) (b) f(0) (c) f(1) (d) f(3)If f( x )={ x 3 if2x1 3x+2if1x4 find: (a) f( 1 ) (b) f( 0 ) (c) f( 1 ) (d) f( 3 )29AYU30AYU31AYU32AYU33AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYUIn Problems 43-46, the graph of a piecewise-defined function is given. Write a definition for each function.In Problems 43-46, the graph of a piecewise-defined function is given. Write a definition for each function.In Problems 43-46, the graph of a piecewise-defined function is given. Write a definition for each function.In Problems 43-46, the graph of a piecewise-defined function is given. Write a definition for each function.If f( x )=int( 2x ) , find (a) f( 1.2 ) (b) f( 1.6 ) (c) f( 1.8 )If f( x )=int( x 2 ) , find (a) f( 1.2 ) (b) f( 1.6 ) (c) f( 1.8 )47AYU48AYU49AYU50AYUFederal Income Tax Two 2018 Tax Rate Schedules are given in the accompanying table. If x equals taxable income and y equals the tax due, construct a function y=f(x) for Scheduled X. 2018TaxRateScheduleXSingleScheduleY-1-MarriedFilingJointlyorQualifiedWidow(er)IfTaxableIncomeisOverButNotOverTheTaxisThisAmountPlusthis%OftheExcessOverIfTaxableIncomeisOverButNotOverTheTaxisThisAmountPlusthis%OftheExcessOver09,5250+100019,0500+1009,52538,700952.50+129,52519,05077,4001,905+1219,05038,70082,5004,453.50+2238,70077,400165,0008,907.00+2277,40082,500157,50014,089.50+2482,500165,000315,00028,179.00+24165,000157,500200,00032,089.50+32157,500315,000400,00064,179.00+32315,000200,000500,00045,689.50+35200,000400,000600,00091,379.00+35400,000500,000150,689.50+37500,000600,000161,379.00+37600,000Federal Income Tax Refer to the 2018 tax rate schedules. If x equals taxable income and y equals the tax due, construct a function y=f(x) for Scheduled Y1. 2018TaxRateScheduleXSingleScheduleY-1-MarriedFilingJointlyorQualifiedWidow(er)IfTaxableIncomeisOverButNotOverTheTaxisThisAmountPlusthis%OftheExcessOverIfTaxableIncomeisOverButNotOverTheTaxisThisAmountPlusthis%OftheExcessOver09,5250+100019,0500+1009,52538,700952.50+129,52519,05077,4001,905+1219,05038,70082,5004,453.50+2238,70077,400165,0008,907.00+2277,40082,500157,50014,089.50+2482,500165,000315,00028,179.00+24165,000157,500200,00032,089.50+32157,500315,000400,00064,179.00+32315,000200,000500,00045,689.50+35200,000400,000600,00091,379.00+35400,000500,000150,689.50+37500,000600,000161,379.00+37600,000Cost of Transporting Goods A trucking company transports goods between Chicago and New York, a distance of 960 miles. The company’s policy is to charge, for each pound, 0.50 per mile for the first 100 miles, 0.40 per mile for the next 300 miles, 0.25 per mile for the next 400 miles, and no charge for the remaining 160 miles. (a) Graph the relationship between the cost of transportation in dollars and mileage over the entire 960-mile route. (b) Find the cost as a function of mileage for hauls between 100 and 400 miles from Chicago. (c) Find the cost as a function of mileage for hauls between 400 and 800 miles from Chicago.Car Rental Costs An economy car rented in Florida from Enterprise® on a weekly basis costs 185 per week. Extra days cost 37 per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also, any part of a day used counts as a full day. Find the cost C of renting an economy car as a function of the number x of days used, where 7x14 . Graph this function. Mortgage Fees Fannie Mae charges a loan-level price adjustment (LLPA) on all mortgages, which represents a fee homebuyers seeking a loan must pay. The rate paid depends on the credit score of the borrower, the amount borrowed, and the loan-to-value (LTV) ratio. The LTV ratio is the ratio of amount borrowed to appraised value of the home. For example, a homebuyer who wishes to borrow with a credit score of 730 and an LTV ratio of will pay of or The table shows the LLPA for various credit scores and an LTV ratio of.
Construct a function where is the loan-level price adjustment (LLPA) and is the credit score of an individual who wishes to borrow with an LTV ratio.
What is the LLPA on a loan with an 80% LTV ratio for a borrower whose credit score is
What is the LLPA on a loan with an 80% LTV ratio for a borrower whose credit score is
Minimum Payments for Credit Cards Holders of a credit cards issued by banks, department stores, oil companies, and so on, receive bills each month that state minimum amounts that must be paid by a certain due date. The minimum due depends on the total amount owed. One such credit card company uses the following rules: For a bill of less than $10, the entire amount is due. For a bill of at least 10 but less than 500, the minimum due is $10. A minimum of $30 is due on a bill of at least $500 but less than $1000, a minimum of $50 is due on a bill of at least $1000 but less than $1500, and a minimum of $70 is due on bills of $1500 or more. Find the function f that describes the minimum payments due on a bill of x dollars. Graph f.Wind Chill The wind chill factor represents the air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is W={ t0v1.79 33 ( 10.45+10 v v )( 33t ) 22.04 1.79v20 331.5958( 33t )v20 where v represents the wind speed (in meters per second) and t represents the air temperature ( C ) . Compute the wind chill for the following: (a) An air temperature of 10 C and a wind speed of 1 meter per second (m/sec) (b) An air temperature of 10 C and a wind speed of 5 m/sec (c) An air temperature of 10 C and a wind speed of 15 m/sec (d) An air temperature of 10 C and a wind speed of 25 m/sec (e) Explain the physical meaning of the equation corresponding to 0v1.79 . (f) Explain the physical meaning of the equation corresponding to v20 .Wind Chill Redo Problem 61(a)-(d) for an air temperature of - 10C .Wind Chill First-class Mail In 2015 the U.S. Postal Service charged 0.98 postage for first-class mail retail flats (such as an 8.5 by 11 envelope) weighing up to 1 ounce, plus 0.22 for each additional ounce up to 13 ounces. First-class rates do not apply to flats weighing more than 13 ounces. Develop a model that relates C , the first-class postage charged, for a flat weighing x ounces. Graph the function. Source: United States Postal ServiceIn Problems 64-71, use a graphing utility. Exploration Graph y= x 2 . Then on the same screen graph y= x 2 +2 , followed by y= x 2 +4 , followed by y= x 2 2 . What pattern do you observe? Can you predict the graph of y= x 2 4 ? Of y= x 2 +5 ?In Problems 64-71, use a graphing utility. Exploration Graph y= x 2 . Then on the same screen graph y= ( x2 ) 2 , followed by y= ( x4 ) 2 , followed by y= ( x+2 ) 2 . What pattern do you observe? Can you predict the graph of y= ( x+4 ) 2 ? Of y= ( x5 ) 2 ?In Problems 64-71, use a graphing utility. Exploration Graph y=| x | . Then on the same screen graph y=2| x | , followed by y=4| x | , followed by y= 1 2 | x | . What pattern do you observe? Can you predict the graph of y= 1 4 | x | ? Of y=5| x | ?In Problems 64-71, use a graphing utility. Exploration Graph y= x 2 . Then on the same screen graph y= x 2 . Now try y=| x | and y=| x | . What do you conclude?In Problems 64-71, use a graphing utility. Exploration Graph y= x . Then on the same screen graph y= x . Now try y=2x+1andy=2( x )+1 . What do you conclude?In Problems 64-71, use a graphing utility. Exploration Graph y= x 3 . Then on the same screen graph y= ( x1 ) 3 +2 . Could you have predicted the result?In Problems 64-71, use a graphing utility. Exploration Graph y= x 2 ,y= x 4 ,andy= x 6 on the same screen. What do you notice is the same about each graph? What do you notice is different?In Problems 64-71, use a graphing utility. Exploration Graph y= x 3 ,y= x 5 ,andy= x 7 on the same screen. What do you notice is the same about each graph? What do you notice is different?Consider the equation y={ 1ifxisrational 0ifxisrational Is this a function? What is its domain? What is its range? What is its y-intercept , if any? What are its x-intercept , if any? Is it even, odd, or neither? How would you describe its graph?Define some functions that pass through ( 0,0 )and( 1,1 ) and are increasing for x0 . Begin your list with y= x ,y=x,andy= x 2 . Can you propose a general result about such functions?1AYU2AYU3AYU4AYU5AYU6AYUIn problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In problems 7-18, match each graph to one of the following functions: A. y= x 2 +2 B. y=- x 2 +2 C. y=| x |+2 D. y=-| x |+2 E. y= ( x-2 ) 2 F. y=- ( x+2 ) 2 G. y=| x-2 | H. y=-| x+2 | I. y=2 x 2 J. y=-2 x 2 K. y=2| x | L. y=-2| x |In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Shifted to the right 4 units.In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Shifted to the left 4 units.In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Shifted up 4 units.In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Shifted down 4 units.In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Reflected about the y-axis .In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Reflected about the x-axis .In Problems, write the function whose graph is the graph of but is:
Vertically stretched by a factor of.
In Problem 19-26, write the function whose graph is the graph of y= x 3 , but is: Horizontally stretched by a factor of 4.In Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. (1) Shift up 2 units (2) Reflect about the x-axis (3) Reflect about the y-axisIn Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. (1) Reflect about the x-axis (2) Shift right 3 units (3) Shift down 2 unitsIn Problems 2932, find the function that is finally graphed after each of the following transformations is applied to the graph of y=x in the order stated. (1)Vertical stretch by a factor of 3 (2)Shift up 4 units (3) Shift left 5 unitsIn Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. (1) Shift up 2 units (2) Reflect about the y-axis (3) Shift left 3 unitsIn Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. (1) Shift up 2 units (2) Reflect about the y-axis (3) Shift left 3 unitsIn Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. If ( 3,6 ) is a point on the graph of y=f( x ) , which of the following points must be on the graph of y=f( x ) ? a. ( 6,3 ) b. ( 6,-3 ) c. ( 3,-6 ) d. ( 3,6 )In Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. If ( 1,3 ) is a point on the graph of y=f( x ) , which of the following points must be on the graph of y=2f( x ) ? a. ( 1, 3 2 ) b. ( 2,3 ) c. ( 1,6 ) d. ( 1 2 ,3 )In Problem 27-30, find the function that is finally graphed after each of the following transformations is applied to the graph of y= x in the order stated. If ( 4,2 ) is a point on the graph of y=f( x ) , which of the following points must be on the graph of y=2f( x ) ? a. ( 4,1 ) b. ( 8,2 ) c. ( 2,2 ) d. ( 4,4 )35AYU36AYU37AYU38AYUIn Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, y= x 2 ) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. f( x )= x 2 -1In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, y= x 2 ) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. f( x )= x 2 +4In Problems graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function.