Test Bank for Calculus: Concepts and Contexts, Enhanced 4th Edition

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Test Bank forPrepared byWilliam TomhaveConcordia CollegeXueqi ZengConcordia CollegeStewart’sCalculusConcepts and ContextsFOURTH EDITION

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PrefaceThese test items were designed to accompanyCalculus:Concepts and Contexts, 4th Editionby James Stewart.As in the previous editions the test items offer both multiple choice and freeresponse formats and approach the Calculus from four viewpoints: Descriptive, Algebraic, Numericand Graphic. These items are designed to help instructors assess both student manipulative skillsand conceptual understanding. It is our hope that you will find enough variety in item difficulty,approach and application areas to allow you substantial flexibility in designing examinations andquizzes that meet the needs of you and your students.This project could not have been completed without the assistance of several associates.Wewould especially like to thank Jessie Lenarz for her work related to page layout and design and hercareful checking. We would like to thank Jeannine Lawless for her patience, support and encour-agement throughout this writing process. As in previous editions we express our sincere thanks toJames Stewart for providing us with the opportunity to be part of one of his projects. Finally, weextend our deepest gratitude to Lois and Wentong, two spouses who have been incredibly support-ive as we carried out the work involved in a project as time-intensive as this one.William K. TomhaveXueqi Zengiii

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ContentsPrefaceiii1Functions and Models11.1Four Ways to Represent a Function ..............................................................................11.2Mathematical Models: A Catalog of Essential Functions............................................ 111.3New Functions From Old Functions .............................................................................. 171.4Graphing Calculators and Computers ........................................................................... 241.5Exponential Functions..................................................................................................... 271.6Inverse Functions and Logarithms ................................................................................. 311.7Parametric Curves ........................................................................................................... 382Limits and Derivatives442.1The Tangent and Velocity Problems.............................................................................. 442.2The Limit of a Function.................................................................................................. 512.3Calculating Limits Using the Limit Laws ..................................................................... 582.4Continuity ........................................................................................................................ 652.5Limits Involving Infinity ................................................................................................. 702.6Tangents, Velocities, and Other Rates of Change ........................................................ 762.7Derivatives........................................................................................................................ 822.8The Derivative as a Function ......................................................................................... 872.9What Doesf′Say Aboutf? .......................................................................................... 963Differentiation Rules1043.1Derivatives of Polynomials and Exponential Functions ...............................................104iv

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CONTENTSv3.2The Product and Quotient Rules...................................................................................1123.3Derivatives of Trigonometric Functions.........................................................................1203.4The Chain Rule ...............................................................................................................1243.5Implicit Differentiation....................................................................................................1363.6Inverse Trigonometric Functions and Their Derivatives ..............................................1423.7Derivatives of Logarithmic Functions ............................................................................1453.8Rates of Change in the Natural and Social Sciences....................................................1503.9Linear Approximations and Differentials ......................................................................1584Applications of Differentiation1644.1Related Rates...................................................................................................................1644.2Maximum and Minimum Values ....................................................................................1704.3Derivatives and the Shapes of Curves ...........................................................................1774.4Graphing with CalculusandCalculators ......................................................................1944.5Indeterminate Forms and L’Hospital’s Rule .................................................................1974.6Optimization Problems ...................................................................................................2014.7Newton’s Method.............................................................................................................2124.8Antiderivatives .................................................................................................................2165Integrals2265.1Areas and Distances ........................................................................................................2265.2The Definite Integral .......................................................................................................2335.3Evaluating Definite Integrals ..........................................................................................2405.4The Fundamental Theorem of Calculus ........................................................................2475.5The Substitution Rule.....................................................................................................2525.6Integration by Parts ........................................................................................................2575.7Additional Techniques of Integration.............................................................................2615.8Integration Using Tables And Computer Algebra Systems .........................................2665.9Approximate Integration.................................................................................................2685.10Improper Integrals ...........................................................................................................274

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viCONTENTS6Applications of Integration2816.1More About Areas ...........................................................................................................2816.2Volumes ............................................................................................................................2866.3Volumes by Cylindrical Shells ........................................................................................2906.4Arc Length .......................................................................................................................2926.5Average Value of a Function ..........................................................................................2966.6Applications to Physics and Engineering ......................................................................3006.7Applications to Economics and Biology ........................................................................3076.8Probability........................................................................................................................3117Differential Equations3187.1Modeling with Differential Equations............................................................................3187.2Direction Fields and Euler’s Method.............................................................................3267.3Separable Equations ........................................................................................................3447.4Exponential Growth and Decay .....................................................................................3557.5The Logistic Equation.....................................................................................................3627.6Predator-Prey Systems....................................................................................................3728Infinite Sequences and Series3818.1Sequences..........................................................................................................................3818.2Series.................................................................................................................................3918.3The Integral and Comparison Tests; Estimating Sums ...............................................4008.4Other Convergence Tests ................................................................................................4108.5Power Series .....................................................................................................................4268.6Representations of Functions as Power Series...............................................................4338.7Taylor and Maclaurin Series ...........................................................................................4378.8Applications of Taylor Polynomials ...............................................................................4449Vectors and the Geometry of Space4509.1Three-Dimensional Coordinate Systems........................................................................450

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CONTENTSvii9.2Vectors ..............................................................................................................................4539.3The Dot Product .............................................................................................................4599.4The Cross Product ..........................................................................................................4639.5Equations of Lines and Planes .......................................................................................4689.6Functions and Surfaces....................................................................................................4769.7Cylindrical and Spherical Coordinates ..........................................................................49010 Vector Functions49910.1Vector Functions and Space Curves...............................................................................49910.2Derivatives and Integrals of Vector Functions ..............................................................50110.3Arc Length and Curvature .............................................................................................50510.4Motion in Space: Velocity and Acceleration .................................................................50910.5Parametric Surfaces.........................................................................................................51411 Partial Derivatives51811.1Functions of Several Variables........................................................................................51811.2Limits and Continuity .....................................................................................................53111.3Partial Derivatives ...........................................................................................................53511.4Tangent Planes and Linear Approximations.................................................................54211.5The Chain Rule ...............................................................................................................54811.6Directional Derivatives and the Gradient Vector .........................................................55311.7Maximum and Minimum Values ....................................................................................56411.8Lagrange Multipliers .......................................................................................................57012 Multiple Integrals57612.1Double Integrals over Rectangles ...................................................................................57612.2Iterated Integrals .............................................................................................................57912.3Double Integrals over General Regions..........................................................................58312.4Double Integrals in Polar Coordinates ..........................................................................59012.5Applications of Double Integrals ....................................................................................596

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viiiCONTENTS12.6Surface Area.....................................................................................................................60212.7Triple Integrals.................................................................................................................60612.8Triple Integrals in Cylindrical and Spherical Coordinates...........................................61212.9Change of Variables in Multiple Integrals.....................................................................61913 Vector Calculus62713.1Vector Fields ....................................................................................................................62713.2Line Integrals ...................................................................................................................63213.3The Fundamental Theorem for Line Integrals ..............................................................63913.4Green’s Theorem .............................................................................................................64413.5Curl And Divergence.......................................................................................................65013.6Surface Integrals ..............................................................................................................65413.7Stokes’ Theorem ..............................................................................................................65913.8The Divergence Theorem ................................................................................................662

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1 Functions and Models1.1 Four Ways to Represent a Function1. Find the smallest value in the domain of the functionf(x) =√2x−5.(A)2(B)52(C)5(D)25(E)−2(F)1(G)0(H)−5Answer: (B)2. Find the smallest value in the range of the functionf(x) = 3x2+ 24x+ 40.(A)−4(B)−5(C)−6(D)−7(E)−8(F)−16(G)−24(H)−40Answer: (E)3. The range of the functionf(x) =√20 + 8x−x2is a closed interval [a, b].Find its lengthb−a.(A)1(B)2(C)3(D)4(E)5(F)6(G)7(H)9Answer: (F)4. Find the smallest value in the range of the functionf(x) =|2x|+|2x+ 3|.(A)2(B)3(C)5(D)12(E)32(F)52(G)0(H)1Answer: (B)5. Find the largest value in the domain of the functionf(x) =√3−2x4 + 3x.(A)−32(B)−23(C)0(D)2(E)23(F)32(G)3(H)No largest valueAnswer: (F)1

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21 Functions and Models6. Find the range of the functionf(x) =⎧⎨⎩x2−4xifx≤2|x−4|ifx >2.(A)[0,∞)(B)(−∞,2](C)[−4,∞)(D)(−∞,0](E)[4,∞)(F)(−∞,4](G)[2,∞)(H)(−∞,−4]Answer: (C)7. Find the range of the functionf(x) =|x−1|+x−1.(A)[1,∞)(B)(1,∞)(C)[0,∞)(D)(0,∞)(E)[−1,∞)(F)(−1,∞)(G)[0,1](H)RAnswer: (C)8. The functionf(x) =√x−1xhas as its domain all values ofxsuch that(A)x >0(B)x≥1(C)x≤0(D)x≤1(E)0< x≤1(F)x≥1 orx <0(G)x≥ −1(H)−1≤x <0Answer: (F)9. Find the range of the functionf(x) = 3x+ 45−2x.(A)(−∞,−32)∪(−32,∞)(B)(−∞,45)∪(45,∞)(C)(−∞,35)∪(35,∞)(D)(−∞,−2)∪(−2,∞)(E)(−∞,2)∪(2,∞)(F)(−∞,3)∪(3,∞)(G)(−∞,4)∪(4,∞)(H)(−∞,−32)∪(−32,∞)Answer: (A)

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1.1 Four Ways to Represent a Function310. Which of the following are graphs of functions?IIIIIIIVV(A)I only(B)II only(C)III only(D)I and II only(E)I and III only(F)I, II, and IV only(G)II and V only(H)I, II, and III onlyAnswer: (F)11. Each of the functions in the table below is increasing, but each increases in a different way.Select the graph from those given below which best fits each function:t123456f(t)263441464849g(t)162432404856h(t)364453647793(A)(B)(C)Answer:f(t):(B)g(t):(A)h(t):(C)

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41 Functions and Models12. Each of the functions in the table below is decreasing, but each decreases in a different way.Select the graph from those given below which best fits each function:t123456f(t)989181695435g(t)807163575352h(t)786960514233(A)(B)(C)Answer:f(t):(B)g(t):(C)h(t):(A)13. Suppose a pet owner decides to wash her dog in the laundry tub. She fills the laundry tubwith warm water, puts the dog into the tub and shampoos it, removes the dog from the tubto towel it, then pulls the plug to drain the tub. Lettbe the time in minutes, beginning whenshe starts to fill the tub, and leth(t) be the water level in the tub at timet. If the total timefor filling and draining the tub and washing the dog was 40 minutes, sketch a possible graphofh(t).Answer:(One possible answer — answers will vary.)

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1.1 Four Ways to Represent a Function514. A homeowner mowed her lawn on June 1, cutting it to a uniform height of 3′′. She mowedthe lawn at one-week intervals after that until she left for a vacation on June 30. A local lawnservice put fertilizer on her lawn shortly after she mowed on June 15, causing the grass togrow more rapidly. She returned from her vacation on July 13 to find that the neighborhoodboy whom she had hired to mow the lawn while she was away had indeed mowed on June 22and on June 29, but had forgotten to mow on July 6. Sketch a possible graph of the heightof the grass as a function of time over the time period from June 1 through July 13.Answer:(One possible answer — answers will vary.)15. A professor left the college for a professional meeting, a trip that was expected to take 4 hours.The graph below shows the distanceD(t) that the professor has traveled from the college asa function of the timet, in hours. Refer to the graph and answer the questions which follow.(a) Describe what might have happened atD(0.5).(b) Describe what might have happened atD(1.0).(c) Describe what might have happened atD(1.2).(d) Describe what might have happened atD(2.5).(e) Describe what might have happened atD(3.5).(f) Describe what might have happened atD(3.75).

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61 Functions and Models(g) Describe what might have happened atD(4.0).(h) Describe what might have happened atD(5.25).Answer:(a) He was traveling to the meeting.(b) He returned to the college (maybe he forgot something.)(c) He left the college for the meeting again.(d) He stopped to rest.(e) He stopped for a second time after traveling at a relatively high rate of speed, perhapsat the request of a highway patrol officer.(f) He continued on his trip but at a substantially lower rate of speed.(g) He was traveling to the meeting.(h) He arrived at his destination.16. Letf(x) = 4−x2. Find(a) the domain off.(b) the range off.Answer:(a) (−∞,∞)(b) (−∞,4]17. Letf(x) =√2x+ 5. Find(a) the domain off.(b) the range off.Answer:(a)[−25,∞)(b) [0,∞)18. Letf(x) =√16−x2. Find(a) the domain off.(b) the range off.Answer:(a) [−4,4](b) [0,4]

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1.1 Four Ways to Represent a Function719. Letf(x) =√3−xx+ 2 . Find(a) the domain off.(b) the range off.Answer:(a) (−2,3](b) (0,∞)20. Express the areaAof a circle as a function of its circumferenceC.Answer:A=C24π21. Letf(x) =⎧⎨⎩x2+ 3ifx≤ −12 + 3x6ifx >−1Find(a) the domain off.(b) the range off.Answer:(a) (−∞,∞)(b)(−16,∞)22. A function has domain [−4,4] and a portion of its graph is shown.(a) Complete the graph offif it is known thatfis an even function.(b) Complete the graph offif it is known thatfis an odd function.

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81 Functions and ModelsAnswer:(a)(b)23. A function has domain [−4,4] and a portion of its graph is shown.(a) Complete the graph offif it is known thatfis an even function.(b) Complete the graph offif it is known thatfis an odd function.Answer:(a)(b)

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1.1 Four Ways to Represent a Function924. Given the graph ofy=f(x):Find all values ofxwhere:(a)fis increasing.(b)fis decreasing.Answer:(a) (x2, x4) and (x5, x6)(b) (x0, x2) and (x4, x5)25. An ice cream vendor is stopped on the side of a city street 100 feet from a perpendicularintersection of the street with another straight city street. A bicyclist is riding on the perpen-dicular street at a rate of 1320 feet/second. If the bicyclist continues to ride straight aheadat the same rate of speed, write a function for the distance,d, between the ice cream vendorand the bicyclist for timetbeginning when the bicyclist is in the intersection.Answer:d(t) =√1002+ (1320t)2.26. A tank used for portland cement consists of a cylinder mounted on top of a cone, with itsvertex pointing downward. The cylinder is 30 feet high, both the cylinder and the cone haveradii of 4 feet, and the cone is 6 feet high.(a) Determine the volume of cement contained in the tank as a function of the depthdofthe cement.(b) What is the domain of this function?

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101 Functions and ModelsAnswer:(a)V(d) =⎧⎪⎨⎪⎩4πd327if 0≤d≤616πd−64πif 6< d≤36(b)d∈[0,36]27. A parking lot light is mounted on top of a 20-foot tall lamppost.A personTfeet tall iswalking away from the lamppost along a straight path. Determine a function which expressesthe length of the person’s shadow in terms of the person’s distance from the lamppost.Answer:LetLbe the length of the person’s shadow andxbe the person’s distance fromthe lamppost. ThenL=T x(20−T) .28. A small model rocket is launched vertically upward on a calm day.The engine delivers itsthrust at a constant rate for 2 seconds, at which point the engine burns out.The rocketcontinues until it begins to fall from its maximum height of 600 feet.Six seconds into theflight a parachute is automatically deployed and the rocket descends at a constant rate of 30feet per second. Sketch a possible graph of the altitude,h(t), of the rocket at timetfor thefirst 10 seconds of the flight.Answer:Answers will vary. One possible graph

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1.2 Mathematical Models: A Catalog of Essential Functions111.2 Mathematical Models: A Catalog of Essential Functions1. Classify the functionf(x) =x2+πx.(A)Power function(B)Root function(C)Polynomial function(D)Rational function(E)Algebraic function(F)Trigonometric function(G)Exponential function(H)Logarithmic functionAnswer: (D)2. Classify the functionf(x) =π2+x2e.(A)Power function(B)Root function(C)Polynomial function(D)Rational function(E)Algebraic function(F)Trigonometric function(G)Exponential function(H)Logarithmic functionAnswer: (C)3. Classify the functionf(x) = sin (5)x2+ sin (3)x.(A)Power function(B)Root function(C)Polynomial function(D)Rational function(E)Algebraic function(F)Trigonometric function(G)Exponential function(H)Logarithmic functionAnswer: (C)

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121 Functions and Models4. The following time-of-day and temperature (◦F) were gathered during a gorgeous midsummerday in Fargo, North Dakota:Time of DayTemperature187417731673157214701370126811661063962859758Source: National Weather Service website; www.weather.gov(a) Make a scatter plot of these data.(b) Fit a linear model to the data.(c) Fit an exponential model to the data.(d) Fit a quadratic model to the data.(e) Use your equations to make a table showing the predicted temperature for each model,rounded to the nearest degree.(f) The actual temperature at 8:00p.m.(20 hours) was 70◦F. Which model was closest?Which model was second-closest?(g) All of the models give values that are too high for each of the times after 6:00p.m.Whatis one possible explanation for this?

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1.2 Mathematical Models: A Catalog of Essential Functions13Answer:(a)(b)y= 1.561x+ 47.68(c)y= 49.89802e1.023831(d)y=−0.09263x2+ 3.902611x+ 33.934(e) Linear: 79Exponential: 80Quadratic: 75(f) Closest:quadratic.Second-closest:lin-ear(g) Answers will vary, but one explanationis that the data only reflect the part ofthe day when the air is warming and donot take into account cooling that takesplace later in the day into evening. Theonly model that begins to reflect this isthe quadratic model.5. Consider the data below:t123456y2.41964152295510(a) Fit both an exponential curve and a third-degree polynomial to the data.(b) Which of the models appears to be a better fit? Defend your choice.Answer:(a)(b) A third degree polynomial, for example,y= 2.40t3, appears to be the better fit.

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141 Functions and Models6. The following table contains United States population data for the years 1981–1990, as wellas estimates based on the 1990 census.YearU. S. Population(millions)1981229.51982231.61983233.81984235.81985237.91986240.11987242.31988244.41989246.81990249.5YearU. S. Population(millions)1991252.21992255.01993257.81994260.31995262.81996265.21997267.81998270.21999272.72000275.1Source: U.S. Census Bureau website(a) Make a scatter plot for the data and use your scatter plot to determine a mathematicalmodel of the U.S. population.(b) Use your model to predict the U.S. population in 2003.Answer:(a)A linear model seems appropriate.Takingt= 0 in 1981, we obtain the modelP(t) =2.4455t+ 228.5.(b)P(22)≈282.3

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1.2 Mathematical Models: A Catalog of Essential Functions157. The following table contains United States population data for the years 1790–2000 at inter-vals of 10 years.YearYears since1790U.S. Population(millions)179003.91800105.21810207.21820309.618304012.918405017.118506023.218607031.418708039.818809050.2189010062.9YearYears since1790U.S. Population(millions)190011076.0191012092.01920130105.71930140122.81940150131.71950160150.71960170178.51970180202.51980190225.51990200248.72000210281.4(a) Make a scatter plot for the data and use your scatter plot to determine a mathematicalmodel of the U.S. population.(b) Use your model to predict the U.S. population in 2005.Answer:(a)Answers will vary, but a quadratic or cubic model is most appropriate.Linear model:P(t) = 1.28545t−40.47668; quadratic model:P(t) = 0.006666t2−0.1144t+ 5.9; cubic model:P(t) =(6.6365×10−6)t3+ 0.004575t2+ 0.057155t+ 3.7;exponential model:P(t) = 6.04852453×1.020407795t(b) Linear model:P(215)≈235.9; quadratic model:P(215)≈289.4; cubic model:P(215)≈293.4; exponential model:P(215)≈465.6

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161 Functions and Models8. Refer to your models from Problems 6 and 7 above. Why do the two data sets produce suchdifferent models?Answer:Problem 6 covers a much shorter time span, so its data exhibit local linearity, whileProblem 7 shows nonlinear population growth over a longer time span.9. The following are the winning times for the Olympic Men’s 110 Meter Hurdles:YearTime189617.6190015.4190416190616.2190815191215.1192014.8192415192814.8YearTime193214.6193614.2194813.9195213.7195613.5196013.8196413.6196813.3197213.24YearTime197613.3198013.39198413.2198812.98199213.12199612.95200013200412.91(a) Make a scatter plot of these data.(b) Fit a linear model to the data.(c) Fit an exponential model to the data.(d) Fit a quadratic model to the data.(e) Use your equations to make a table showing the predicted winning time for each modelfor the 2008 Olympics, rounded to the nearest hundredth of a second.(f) The actual time for the 2008 Olympics was 12.93 seconds.Which model was closest?Which model was second-closest?

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1.3 New Functions From Old Functions17Answer:(a)(b)y=−0.0320057x+ 76.595(c)y= 1053.09176(0.997791842x)(d)y= 0.000322x2−1.2872778x+ 1299.573(e) Linear: 12.33Exponential: 12.44Quadratic: 13.04(f) Closest:quadratic.Second-closest:ex-ponential1.3 New Functions From Old Functions1. Letf(x) =x2−3x+ 7, thenf(2x) is equal to(A)2x2−6x+ 7(B)4x2−6x+ 7(C)2x2−6x+ 14(D)4x2−3x+ 7(E)2x2+ 6x−7(F)4x2+ 6x−7(G)2x2−3x+ 7(H)4x2−6x+ 14Answer: (B)2. Letf(x) =√x2+ 4andg(x) =−√x2−4.Find the domain of (g◦f) (x).(A)(−∞,0](B)(2,∞)(C)(−∞,−2)(D)(−∞,2)∪(2,∞)(E)[−2,∞)(F)(−∞,−2](G)(−∞,−2]∪[2,∞)(H)RAnswer: (H)

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181 Functions and Models3. Letf(x) =√x−1andg(x) =√10−x2.Find the domain of (f◦g) (x).(A)[1,∞)(B)[−√10,√10](C)(−∞,−√10]∪[√10,∞)(D)(−∞,−3]∪[3,∞)(E)[−3,3 ](F)(−∞,−1]∪[1,∞)(G)(−∞,−1](H)[√10,∞)Answer: (E)4. Leth(x) = sin2x+ 3 sinx−4 andg(x) = sinx. Findf(x) so thath(x) = (f◦g)(x).(A)f(x) = (3x+ 2)2(B)f(x) =x+ 3(C)f(x) = 3x2−4(D)f(x) =x2−3x+ 4(E)f(x) = 3x2−4x(F)f(x) =x2+ 3x−4(G)f(x) =x2−4(H)f(x) = (x−4)2Answer: (F)5. Letf(x) = 3x−2 andg(x) = 2−3x.Find the value of (f◦g) (x) whenx= 3.(A)−23(B)−9(C)−6(D)−3(E)3(F)6(G)9(H)23Answer: (A)6. Letf(x) = 2−x3andg(x) = 3 +x. Find the value of (f◦g) (x) whenx=−5.(A)−510(B)−5(C)−2(D)0(E)5(F)10(G)127(H)130Answer: (F)7. Letf(x) =12xand (f◦g) (x) =x2. Findg(2).(A)0(B)1(C)2(D)4(E)8(F)16(G)32(H)64Answer: (E)

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1.3 New Functions From Old Functions198. Relative to the graph ofy=x2+ 2, the graph ofy= (x−2)2+ 2 is changed in what way?(A)Shifted 2 units upward(B)Compressed vertically by a factor of 2(C)Compressed horizontally by a factor of 2(D)Shifted 2 units to the left(E)Shifted 2 units to the right(F)Shifted 2 units downward(G)Stretched vertically by a factor of 2(H)Stretched horizontally by a factor of 2Answer: (E)9. Relative to the graph ofy=x2, the graph ofy=x2−2 is changed in what way?(A)Shifted 2 units downward(B)Stretched horizontally by a factor of 2(C)Shifted 2 units to the right(D)stretched vertically by a factor of 2(E)Compressed horizontally by a factor of 2(F)Compressed vertically by a factor of 2(G)Stretched vertically by a factor of 2(H)Stretched horizontally by a factor of 2Answer: (A)10. Relative to the graph ofy=x3, the graph ofy=12x3is changed in what way?(A)Compressed horizontally by a factor of 2(B)Shifted 2 units downward(C)Stretched vertically by a factor of 2(D)Stretched horizontally by a factor of 2(E)Shifted 2 units upward(F)Compressed vertically by a factor of 2(G)Shifted 2 units to the right(H)Shifted 2 units to the leftAnswer: (F)11. Relative to the graph ofy=x2+ 2 , the graph ofy= 4x2+ 2 is changed in what way?(A)Compressed vertically by a factor of 2(B)Stretched horizontally by a factor of 2(C)Compressed horizontally by a factor of 2(D)Shifted 2 units upward(E)Shifted 2 units to the right(F)stretched vertically by a factor of 2(G)Shifted 2 units to the left(H)Shifted 2 units downwardAnswer: (C)

Test Bank for Calculus: Concepts and Contexts, Enhanced 4th Edition - Page 28 preview image

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201 Functions and Models12. Relative to the graph ofy= sinx, the graph ofy= 3 sinxis changed in what way?(A)Compressed horizontally by a factor of 3(B)Shifted 3 units to the right(C)Compressed vertically by a factor of 3(D)Shifted 3 units upward(E)Shifted 3 units to the left(F)Stretched vertically by a factor of 3(G)Shifted 3 units downward(H)Stretched horizontally by a factor of 3Answer: (F)13. Relative to the graph ofy=ex, the graph ofy=ex+5is changed in what way?(A)Shifted 5 units upward(B)Shifted 5 units downward(C)Shifted 5 units to the right(D)Shifted 5 units to the left(E)Stretched horizontally by a factor of 5(F)Stretched vertically by a factor of 5(G)Compressed horizontally by a factor of 5(H)Compressed vertically by a factor of 5Answer: (D)14. Relative to the graph ofy= sinx, wherexis in radians, the graph ofy= sinx, wherexis indegrees, is changed in what way?(A)Compressed horizontally by a factor of180π(B)Stretched vertically by a factor of180π(C)Compressed horizontally by a factor of90π(D)Stretched horizontally by a factor of90π(E)Compressed vertically by a factor of90π(F)Stretched vertically by a factor of90π(G)Stretched horizontally by a factor of180π(H)Compressed vertically by a factor of180πAnswer: (G)15. Letf(x) = 8 +x2. Find each of the following:(a)f(2) +f(−2)(b)f(x+ 2)(c) [f(x)]2(d)f(x2)Answer:(a) 24(b)x2+ 4x+ 12(c) 64 + 16x2+x4(d) 8 +x416. Letf(x) =√2x+ 5. Find each of the following:(a)f(0) +f(−2)(b)f(x+ 2)(c) [f(x)]2(d)f(x2)

Test Bank for Calculus: Concepts and Contexts, Enhanced 4th Edition - Page 29 preview image

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1.3 New Functions From Old Functions21Answer:(a)f(0) +f(−2) =√5 +√1 =√5 + 1(b)f(x+ 2) =√2 (x+ 2) + 5 =√2x+ 9(c) [f(x)]2= 2x+ 5,x≥ −52(d)f(x2)=√2x2+ 517. Letf(x) =√16−x2. Find each of the following:(a)f(0) +f(−2)(b)f(x+ 2)(c) [f(x)]2(d)f(x2)Answer:(a)f(0) +f(−2) =√16 +√12 = 4 + 2√3≈7.46(b)f(x+ 2) =√16−(x+ 2)2=√16−(x2+ 4x+ 4) =√12−4x−x2,−6≤x≤2(c) [f(x)]2= 16−x2,−4≤x≤4(d)f(x2)=√16−(x2)2=√16−x4, 0≤x≤218. Letf(x) =√2x+ 3, x >−3. Find each of the following:(a)f(−1)−f(−2)(b)f(x2−3)(c)f(x2)−3(d) [f(x−3)]2Answer:(a)f(−1)−f(−2) =√22−√21 = 1− √2(b)f(x2−3)=√2(x2−3) + 3 =√2x2=√2|x|,x= 0(c)f(x2)−3 =√2x2+ 3−3(d) [f(x−3)]2=(√2(x−3) + 3)2= 2x,x >0

Test Bank for Calculus: Concepts and Contexts, Enhanced 4th Edition - Page 30 preview image

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221 Functions and Models19. Evaluate the difference quotientf(x)−f(a)x−aforf(x) =1x2.Answer:f(x)−f(a)x−a=1x2−1a2x−a=−a+xa2x220. Given the graph ofy=f(x):Sketch the graph of each of the following functions:(a)−f(x)(b)f(−x)(c)f(2x)(d) 2f(x)(e)−f(−x)(f)f(12x)(g)12f(x)(h)f(x+ 1)(i)f(x)−1(j) 1−f(x)Answer:(a)(b)(c)(d)

Test Bank for Calculus: Concepts and Contexts, Enhanced 4th Edition - Page 31 preview image

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1.3 New Functions From Old Functions23(e)(f)(g)(h)(i)(j)21. Use the graphs offandggiven below to estimate the values off(g(x)) forx=−3,−2,−1,0, 1, 2, and 3, and use these values to sketch a graph ofy=f(g(x)).Answer:x−3−2−10123f(g(x))−0.53.883.502.883.503.88−0.5

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