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

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    Test Bank for Prepared by William Tomhave Concordia College Xueqi Zeng Concordia College Stewart’s Calculus Concepts and Contexts FOURTH EDITION

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    Preface These test items were designed to accompany Calculus: Concepts and Contexts, 4th Edition by James Stewart. As in the previous editions the test items offer both multiple choice and free response formats and approach the Calculus from four viewpoints: Descriptive, Algebraic, Numeric and Graphic. These items are designed to help instructors assess both student manipulative skills and 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 and quizzes that meet the needs of you and your students. This project could not have been completed without the assistance of several associates. We would especially like to thank Jessie Lenarz for her work related to page layout and design and her careful 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 to James Stewart for providing us with the opportunity to be part of one of his projects. Finally, we extend 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. Tomhave Xueqi Zeng iii

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    Contents Preface iii 1 Functions and Models 1 1.1 Four Ways to Represent a Function .............................................................................. 1 1.2 Mathematical Models: A Catalog of Essential Functions............................................ 11 1.3 New Functions From Old Functions .............................................................................. 17 1.4 Graphing Calculators and Computers ........................................................................... 24 1.5 Exponential Functions..................................................................................................... 27 1.6 Inverse Functions and Logarithms ................................................................................. 31 1.7 Parametric Curves ........................................................................................................... 38 2 Limits and Derivatives 44 2.1 The Tangent and Velocity Problems.............................................................................. 44 2.2 The Limit of a Function.................................................................................................. 51 2.3 Calculating Limits Using the Limit Laws ..................................................................... 58 2.4 Continuity ........................................................................................................................ 65 2.5 Limits Involving Infinity ................................................................................................. 70 2.6 Tangents, Velocities, and Other Rates of Change ........................................................ 76 2.7 Derivatives........................................................................................................................ 82 2.8 The Derivative as a Function ......................................................................................... 87 2.9 What Does f ′ Say About f ? .......................................................................................... 96 3 Differentiation Rules 104 3.1 Derivatives of Polynomials and Exponential Functions ...............................................104 iv

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    CONTENTS v 3.2 The Product and Quotient Rules...................................................................................112 3.3 Derivatives of Trigonometric Functions.........................................................................120 3.4 The Chain Rule ...............................................................................................................124 3.5 Implicit Differentiation....................................................................................................136 3.6 Inverse Trigonometric Functions and Their Derivatives ..............................................142 3.7 Derivatives of Logarithmic Functions ............................................................................145 3.8 Rates of Change in the Natural and Social Sciences....................................................150 3.9 Linear Approximations and Differentials ......................................................................158 4 Applications of Differentiation 164 4.1 Related Rates...................................................................................................................164 4.2 Maximum and Minimum Values ....................................................................................170 4.3 Derivatives and the Shapes of Curves ...........................................................................177 4.4 Graphing with Calculus and Calculators ......................................................................194 4.5 Indeterminate Forms and L’Hospital’s Rule .................................................................197 4.6 Optimization Problems ...................................................................................................201 4.7 Newton’s Method.............................................................................................................212 4.8 Antiderivatives .................................................................................................................216 5 Integrals 226 5.1 Areas and Distances ........................................................................................................226 5.2 The Definite Integral .......................................................................................................233 5.3 Evaluating Definite Integrals ..........................................................................................240 5.4 The Fundamental Theorem of Calculus ........................................................................247 5.5 The Substitution Rule.....................................................................................................252 5.6 Integration by Parts ........................................................................................................257 5.7 Additional Techniques of Integration.............................................................................261 5.8 Integration Using Tables And Computer Algebra Systems .........................................266 5.9 Approximate Integration.................................................................................................268 5.10 Improper Integrals ...........................................................................................................274

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    vi CONTENTS 6 Applications of Integration 281 6.1 More About Areas ...........................................................................................................281 6.2 Volumes ............................................................................................................................286 6.3 Volumes by Cylindrical Shells ........................................................................................290 6.4 Arc Length .......................................................................................................................292 6.5 Average Value of a Function ..........................................................................................296 6.6 Applications to Physics and Engineering ......................................................................300 6.7 Applications to Economics and Biology ........................................................................307 6.8 Probability........................................................................................................................311 7 Differential Equations 318 7.1 Modeling with Differential Equations............................................................................318 7.2 Direction Fields and Euler’s Method.............................................................................326 7.3 Separable Equations ........................................................................................................344 7.4 Exponential Growth and Decay .....................................................................................355 7.5 The Logistic Equation.....................................................................................................362 7.6 Predator-Prey Systems....................................................................................................372 8 Infinite Sequences and Series 381 8.1 Sequences..........................................................................................................................381 8.2 Series.................................................................................................................................391 8.3 The Integral and Comparison Tests; Estimating Sums ...............................................400 8.4 Other Convergence Tests ................................................................................................410 8.5 Power Series .....................................................................................................................426 8.6 Representations of Functions as Power Series...............................................................433 8.7 Taylor and Maclaurin Series ...........................................................................................437 8.8 Applications of Taylor Polynomials ...............................................................................444 9 Vectors and the Geometry of Space 450 9.1 Three-Dimensional Coordinate Systems........................................................................450

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    CONTENTS vii 9.2 Vectors ..............................................................................................................................453 9.3 The Dot Product .............................................................................................................459 9.4 The Cross Product ..........................................................................................................463 9.5 Equations of Lines and Planes .......................................................................................468 9.6 Functions and Surfaces....................................................................................................476 9.7 Cylindrical and Spherical Coordinates ..........................................................................490 10 Vector Functions 499 10.1 Vector Functions and Space Curves...............................................................................499 10.2 Derivatives and Integrals of Vector Functions ..............................................................501 10.3 Arc Length and Curvature .............................................................................................505 10.4 Motion in Space: Velocity and Acceleration .................................................................509 10.5 Parametric Surfaces.........................................................................................................514 11 Partial Derivatives 518 11.1 Functions of Several Variables........................................................................................518 11.2 Limits and Continuity .....................................................................................................531 11.3 Partial Derivatives ...........................................................................................................535 11.4 Tangent Planes and Linear Approximations.................................................................542 11.5 The Chain Rule ...............................................................................................................548 11.6 Directional Derivatives and the Gradient Vector .........................................................553 11.7 Maximum and Minimum Values ....................................................................................564 11.8 Lagrange Multipliers .......................................................................................................570 12 Multiple Integrals 576 12.1 Double Integrals over Rectangles ...................................................................................576 12.2 Iterated Integrals .............................................................................................................579 12.3 Double Integrals over General Regions..........................................................................583 12.4 Double Integrals in Polar Coordinates ..........................................................................590 12.5 Applications of Double Integrals ....................................................................................596

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    viii CONTENTS 12.6 Surface Area.....................................................................................................................602 12.7 Triple Integrals.................................................................................................................606 12.8 Triple Integrals in Cylindrical and Spherical Coordinates...........................................612 12.9 Change of Variables in Multiple Integrals.....................................................................619 13 Vector Calculus 627 13.1 Vector Fields ....................................................................................................................627 13.2 Line Integrals ...................................................................................................................632 13.3 The Fundamental Theorem for Line Integrals ..............................................................639 13.4 Green’s Theorem .............................................................................................................644 13.5 Curl And Divergence.......................................................................................................650 13.6 Surface Integrals ..............................................................................................................654 13.7 Stokes’ Theorem ..............................................................................................................659 13.8 The Divergence Theorem ................................................................................................662

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    1 Functions and Models 1.1 Four Ways to Represent a Function 1. Find the smallest value in the domain of the function f ( x ) = √ 2 x − 5. (A) 2 (B) 5 2 (C) 5 (D) 2 5 (E) − 2 (F) 1 (G) 0 (H) − 5 Answer: (B) 2. Find the smallest value in the range of the function f ( x ) = 3 x 2 + 24 x + 40. (A) − 4 (B) − 5 (C) − 6 (D) − 7 (E) − 8 (F) − 16 (G) − 24 (H) − 40 Answer: (E) 3. The range of the function f ( x ) = √ 20 + 8 x − x 2 is a closed interval [ a, b ]. Find its length b − a . (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 (F) 6 (G) 7 (H) 9 Answer: (F) 4. Find the smallest value in the range of the function f ( x ) = | 2 x | + | 2 x + 3 | . (A) 2 (B) 3 (C) 5 (D) 1 2 (E) 3 2 (F) 5 2 (G) 0 (H) 1 Answer: (B) 5. Find the largest value in the domain of the function f ( x ) = √ 3 − 2 x 4 + 3 x . (A) − 3 2 (B) − 2 3 (C) 0 (D) 2 (E) 2 3 (F) 3 2 (G) 3 (H) No largest value Answer: (F) 1

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    2 1 Functions and Models 6. Find the range of the function f ( x ) = ⎧ ⎨ ⎩ x 2 − 4 x if x ≤ 2 | x − 4 | if x > 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 function f ( x ) = | x − 1 | + x − 1. (A) [1 , ∞ ) (B) (1 , ∞ ) (C) [0 , ∞ ) (D) (0 , ∞ ) (E) [ − 1 , ∞ ) (F) ( − 1 , ∞ ) (G) [0 , 1] (H) R Answer: (C) 8. The function f ( x ) = √ x − 1 x has as its domain all values of x such that (A) x > 0 (B) x ≥ 1 (C) x ≤ 0 (D) x ≤ 1 (E) 0 < x ≤ 1 (F) x ≥ 1 or x < 0 (G) x ≥ − 1 (H) − 1 ≤ x < 0 Answer: (F) 9. Find the range of the function f ( x ) = 3 x + 4 5 − 2 x . (A) ( −∞ , − 3 2 ) ∪ ( − 3 2 , ∞ ) (B) ( −∞ , 4 5 ) ∪ ( 4 5 , ∞ ) (C) ( −∞ , 3 5 ) ∪ ( 3 5 , ∞ ) (D) ( −∞ , − 2) ∪ ( − 2 , ∞ ) (E) ( −∞ , 2) ∪ (2 , ∞ ) (F) ( −∞ , 3) ∪ (3 , ∞ ) (G) ( −∞ , 4) ∪ (4 , ∞ ) (H) ( −∞ , − 3 2 ) ∪ ( − 3 2 , ∞ ) Answer: (A)

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    1.1 Four Ways to Represent a Function 3 10. Which of the following are graphs of functions? I II III IV V (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 only Answer: (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: t 1 2 3 4 5 6 f ( t ) 26 34 41 46 48 49 g ( t ) 16 24 32 40 48 56 h ( t ) 36 44 53 64 77 93 (A) (B) (C) Answer: f ( t ): (B) g ( t ): (A) h ( t ): (C)

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    4 1 Functions and Models 12. 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: t 1 2 3 4 5 6 f ( t ) 98 91 81 69 54 35 g ( t ) 80 71 63 57 53 52 h ( t ) 78 69 60 51 42 33 (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 tub with warm water, puts the dog into the tub and shampoos it, removes the dog from the tub to towel it, then pulls the plug to drain the tub. Let t be the time in minutes, beginning when she starts to fill the tub, and let h ( t ) be the water level in the tub at time t . If the total time for filling and draining the tub and washing the dog was 40 minutes, sketch a possible graph of h ( t ). Answer: (One possible answer — answers will vary.)

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    1.1 Four Ways to Represent a Function 5 14. A homeowner mowed her lawn on June 1, cutting it to a uniform height of 3 ′′ . She mowed the lawn at one-week intervals after that until she left for a vacation on June 30. A local lawn service put fertilizer on her lawn shortly after she mowed on June 15, causing the grass to grow more rapidly. She returned from her vacation on July 13 to find that the neighborhood boy whom she had hired to mow the lawn while she was away had indeed mowed on June 22 and on June 29, but had forgotten to mow on July 6. Sketch a possible graph of the height of 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 distance D ( t ) that the professor has traveled from the college as a function of the time t , in hours. Refer to the graph and answer the questions which follow. (a) Describe what might have happened at D (0 . 5). (b) Describe what might have happened at D (1 . 0). (c) Describe what might have happened at D (1 . 2). (d) Describe what might have happened at D (2 . 5). (e) Describe what might have happened at D (3 . 5). (f) Describe what might have happened at D (3 . 75).

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    6 1 Functions and Models (g) Describe what might have happened at D (4 . 0). (h) Describe what might have happened at D (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, perhaps at 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. Let f ( x ) = 4 − x 2 . Find (a) the domain of f . (b) the range of f . Answer: (a) ( −∞ , ∞ ) (b) ( −∞ , 4] 17. Let f ( x ) = √ 2 x + 5. Find (a) the domain of f . (b) the range of f . Answer: (a) [ − 2 5 , ∞ ) (b) [0 , ∞ ) 18. Let f ( x ) = √ 16 − x 2 . Find (a) the domain of f . (b) the range of f . Answer: (a) [ − 4 , 4] (b) [0 , 4]

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    1.1 Four Ways to Represent a Function 7 19. Let f ( x ) = √ 3 − x x + 2 . Find (a) the domain of f . (b) the range of f . Answer: (a) ( − 2 , 3] (b) (0 , ∞ ) 20. Express the area A of a circle as a function of its circumference C . Answer: A = C 2 4 π 21. Let f ( x ) = ⎧ ⎨ ⎩ x 2 + 3 if x ≤ − 1 2 + 3 x 6 if x > − 1 Find (a) the domain of f . (b) the range of f . Answer: (a) ( −∞ , ∞ ) (b) ( − 1 6 , ∞ ) 22. A function has domain [ − 4 , 4] and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is an even function. (b) Complete the graph of f if it is known that f is an odd function.

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    8 1 Functions and Models Answer: (a) (b) 23. A function has domain [ − 4 , 4] and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is an even function. (b) Complete the graph of f if it is known that f is an odd function. Answer: (a) (b)

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    1.1 Four Ways to Represent a Function 9 24. Given the graph of y = f ( x ): Find all values of x where: (a) f is increasing. (b) f is decreasing. Answer: (a) ( x 2 , x 4 ) and ( x 5 , x 6 ) (b) ( x 0 , x 2 ) and ( x 4 , x 5 ) 25. An ice cream vendor is stopped on the side of a city street 100 feet from a perpendicular intersection 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 ahead at the same rate of speed, write a function for the distance, d , between the ice cream vendor and the bicyclist for time t beginning when the bicyclist is in the intersection. Answer: d ( t ) = √ 100 2 + (1320 t ) 2 . 26. A tank used for portland cement consists of a cylinder mounted on top of a cone, with its vertex pointing downward. The cylinder is 30 feet high, both the cylinder and the cone have radii 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 depth d of the cement. (b) What is the domain of this function?

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    10 1 Functions and Models Answer: (a) V ( d ) = ⎧ ⎪ ⎨ ⎪ ⎩ 4 πd 3 27 if 0 ≤ d ≤ 6 16 πd − 64 π if 6 < d ≤ 36 (b) d ∈ [0 , 36] 27. A parking lot light is mounted on top of a 20-foot tall lamppost. A person T feet tall is walking away from the lamppost along a straight path. Determine a function which expresses the length of the person’s shadow in terms of the person’s distance from the lamppost. Answer: Let L be the length of the person’s shadow and x be the person’s distance from the lamppost. Then L = T x (20 − T ) . 28. A small model rocket is launched vertically upward on a calm day. The engine delivers its thrust at a constant rate for 2 seconds, at which point the engine burns out. The rocket continues until it begins to fall from its maximum height of 600 feet. Six seconds into the flight a parachute is automatically deployed and the rocket descends at a constant rate of 30 feet per second. Sketch a possible graph of the altitude, h ( t ), of the rocket at time t for the first 10 seconds of the flight. Answer: Answers will vary. One possible graph

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    1.2 Mathematical Models: A Catalog of Essential Functions 11 1.2 Mathematical Models: A Catalog of Essential Functions 1. Classify the function f ( x ) = x 2 + π x . (A) Power function (B) Root function (C) Polynomial function (D) Rational function (E) Algebraic function (F) Trigonometric function (G) Exponential function (H) Logarithmic function Answer: (D) 2. Classify the function f ( x ) = π 2 + x 2 e . (A) Power function (B) Root function (C) Polynomial function (D) Rational function (E) Algebraic function (F) Trigonometric function (G) Exponential function (H) Logarithmic function Answer: (C) 3. Classify the function f ( x ) = sin (5) x 2 + 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 function Answer: (C)

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    12 1 Functions and Models 4. The following time-of-day and temperature ( ◦ F ) were gathered during a gorgeous midsummer day in Fargo, North Dakota: Time of Day Temperature 18 74 17 73 16 73 15 72 14 70 13 70 12 68 11 66 10 63 9 62 8 59 7 58 Source: 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 8:00 p.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 6:00 p.m. What is one possible explanation for this?

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    1.2 Mathematical Models: A Catalog of Essential Functions 13 Answer: (a) (b) y = 1 . 561 x + 47 . 68 (c) y = 49 . 89802 e 1 . 023831 (d) y = − 0 . 09263 x 2 + 3 . 902611 x + 33 . 934 (e) Linear: 79 Exponential: 80 Quadratic: 75 (f) Closest: quadratic. Second-closest: lin- ear (g) Answers will vary, but one explanation is that the data only reflect the part of the day when the air is warming and do not take into account cooling that takes place later in the day into evening. The only model that begins to reflect this is the quadratic model. 5. Consider the data below: t 1 2 3 4 5 6 y 2 . 4 19 64 152 295 510 (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 . 40 t 3 , appears to be the better fit.

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    14 1 Functions and Models 6. The following table contains United States population data for the years 1981–1990, as well as estimates based on the 1990 census. Year U. S. Population (millions) 1981 229 . 5 1982 231 . 6 1983 233 . 8 1984 235 . 8 1985 237 . 9 1986 240 . 1 1987 242 . 3 1988 244 . 4 1989 246 . 8 1990 249 . 5 Year U. S. Population (millions) 1991 252 . 2 1992 255 . 0 1993 257 . 8 1994 260 . 3 1995 262 . 8 1996 265 . 2 1997 267 . 8 1998 270 . 2 1999 272 . 7 2000 275 . 1 Source: U.S. Census Bureau website (a) Make a scatter plot for the data and use your scatter plot to determine a mathematical model of the U.S. population. (b) Use your model to predict the U.S. population in 2003. Answer: (a) A linear model seems appropriate. Taking t = 0 in 1981, we obtain the model P ( t ) = 2 . 4455 t + 228 . 5. (b) P (22) ≈ 282 . 3

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    1.2 Mathematical Models: A Catalog of Essential Functions 15 7. The following table contains United States population data for the years 1790–2000 at inter- vals of 10 years. Year Years since 1790 U.S. Population (millions) 1790 0 3 . 9 1800 10 5 . 2 1810 20 7 . 2 1820 30 9 . 6 1830 40 12 . 9 1840 50 17 . 1 1850 60 23 . 2 1860 70 31 . 4 1870 80 39 . 8 1880 90 50 . 2 1890 100 62 . 9 Year Years since 1790 U.S. Population (millions) 1900 110 76 . 0 1910 120 92 . 0 1920 130 105 . 7 1930 140 122 . 8 1940 150 131 . 7 1950 160 150 . 7 1960 170 178 . 5 1970 180 202 . 5 1980 190 225 . 5 1990 200 248 . 7 2000 210 281 . 4 (a) Make a scatter plot for the data and use your scatter plot to determine a mathematical model 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 . 28545 t − 40 . 47668; quadratic model: P ( t ) = 0 . 006666 t 2 − 0 . 1144 t + 5 . 9; cubic model: P ( t ) = ( 6 . 6365 × 10 − 6 ) t 3 + 0 . 004575 t 2 + 0 . 057155 t + 3 . 7; exponential model: P ( t ) = 6 . 04852453 × 1 . 020407795 t (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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    16 1 Functions and Models 8. Refer to your models from Problems 6 and 7 above. Why do the two data sets produce such different models? Answer: Problem 6 covers a much shorter time span, so its data exhibit local linearity, while Problem 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: Year Time 1896 17.6 1900 15.4 1904 16 1906 16.2 1908 15 1912 15.1 1920 14.8 1924 15 1928 14.8 Year Time 1932 14.6 1936 14.2 1948 13.9 1952 13.7 1956 13.5 1960 13.8 1964 13.6 1968 13.3 1972 13.24 Year Time 1976 13.3 1980 13.39 1984 13.2 1988 12.98 1992 13.12 1996 12.95 2000 13 2004 12.91 (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 model for 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 Functions 17 Answer: (a) (b) y = − 0 . 0320057 x + 76 . 595 (c) y = 1053 . 09176(0 . 997791842 x ) (d) y = 0 . 000322 x 2 − 1 . 2872778 x + 1299 . 573 (e) Linear: 12 . 33 Exponential: 12 . 44 Quadratic: 13 . 04 (f) Closest: quadratic. Second-closest: ex- ponential 1.3 New Functions From Old Functions 1. Let f ( x ) = x 2 − 3 x + 7, then f (2 x ) is equal to (A) 2 x 2 − 6 x + 7 (B) 4 x 2 − 6 x + 7 (C) 2 x 2 − 6 x + 14 (D) 4 x 2 − 3 x + 7 (E) 2 x 2 + 6 x − 7 (F) 4 x 2 + 6 x − 7 (G) 2 x 2 − 3 x + 7 (H) 4 x 2 − 6 x + 14 Answer: (B) 2. Let f ( x ) = √ x 2 + 4 and g ( x ) = − √ x 2 − 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) R Answer: (H)

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    18 1 Functions and Models 3. Let f ( x ) = √ x − 1 and g ( x ) = √ 10 − x 2 . 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. Let h ( x ) = sin 2 x + 3 sin x − 4 and g ( x ) = sin x . Find f ( x ) so that h ( x ) = ( f ◦ g )( x ). (A) f ( x ) = (3 x + 2) 2 (B) f ( x ) = x + 3 (C) f ( x ) = 3 x 2 − 4 (D) f ( x ) = x 2 − 3 x + 4 (E) f ( x ) = 3 x 2 − 4 x (F) f ( x ) = x 2 + 3 x − 4 (G) f ( x ) = x 2 − 4 (H) f ( x ) = ( x − 4) 2 Answer: (F) 5. Let f ( x ) = 3 x − 2 and g ( x ) = 2 − 3 x . Find the value of ( f ◦ g ) ( x ) when x = 3. (A) − 23 (B) − 9 (C) − 6 (D) − 3 (E) 3 (F) 6 (G) 9 (H) 23 Answer: (A) 6. Let f ( x ) = 2 − x 3 and g ( x ) = 3 + x . Find the value of ( f ◦ g ) ( x ) when x = − 5. (A) − 510 (B) − 5 (C) − 2 (D) 0 (E) 5 (F) 10 (G) 127 (H) 130 Answer: (F) 7. Let f ( x ) = 1 2 x and ( f ◦ g ) ( x ) = x 2 . Find g (2). (A) 0 (B) 1 (C) 2 (D) 4 (E) 8 (F) 16 (G) 32 (H) 64 Answer: (E)

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    1.3 New Functions From Old Functions 19 8. Relative to the graph of y = x 2 + 2, the graph of y = ( 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 2 Answer: (E) 9. Relative to the graph of y = x 2 , the graph of y = x 2 − 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 2 Answer: (A) 10. Relative to the graph of y = x 3 , the graph of y = 1 2 x 3 is 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 left Answer: (F) 11. Relative to the graph of y = x 2 + 2 , the graph of y = 4 x 2 + 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 downward Answer: (C)

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    20 1 Functions and Models 12. Relative to the graph of y = sin x , the graph of y = 3 sin x is 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 3 Answer: (F) 13. Relative to the graph of y = e x , the graph of y = e x +5 is 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 5 Answer: (D) 14. Relative to the graph of y = sin x , where x is in radians, the graph of y = sin x , where x is in degrees, is changed in what way? (A) Compressed horizontally by a factor of 180 π (B) Stretched vertically by a factor of 180 π (C) Compressed horizontally by a factor of 90 π (D) Stretched horizontally by a factor of 90 π (E) Compressed vertically by a factor of 90 π (F) Stretched vertically by a factor of 90 π (G) Stretched horizontally by a factor of 180 π (H) Compressed vertically by a factor of 180 π Answer: (G) 15. Let f ( x ) = 8 + x 2 . Find each of the following: (a) f (2) + f ( − 2) (b) f ( x + 2) (c) [ f ( x )] 2 (d) f ( x 2 ) Answer: (a) 24 (b) x 2 + 4 x + 12 (c) 64 + 16 x 2 + x 4 (d) 8 + x 4 16. Let f ( x ) = √ 2 x + 5. Find each of the following: (a) f (0) + f ( − 2) (b) f ( x + 2) (c) [ f ( x )] 2 (d) f ( x 2 )

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    1.3 New Functions From Old Functions 21 Answer: (a) f (0) + f ( − 2) = √ 5 + √ 1 = √ 5 + 1 (b) f ( x + 2) = √ 2 ( x + 2) + 5 = √ 2 x + 9 (c) [ f ( x )] 2 = 2 x + 5, x ≥ − 5 2 (d) f ( x 2 ) = √ 2 x 2 + 5 17. Let f ( x ) = √ 16 − x 2 . Find each of the following: (a) f (0) + f ( − 2) (b) f ( x + 2) (c) [ f ( x )] 2 (d) f ( x 2 ) Answer: (a) f (0) + f ( − 2) = √ 16 + √ 12 = 4 + 2 √ 3 ≈ 7 . 46 (b) f ( x + 2) = √ 16 − ( x + 2) 2 = √ 16 − ( x 2 + 4 x + 4) = √ 12 − 4 x − x 2 , − 6 ≤ x ≤ 2 (c) [ f ( x )] 2 = 16 − x 2 , − 4 ≤ x ≤ 4 (d) f ( x 2 ) = √ 16 − ( x 2 ) 2 = √ 16 − x 4 , 0 ≤ x ≤ 2 18. Let f ( x ) = √ 2 x + 3 , x > − 3. Find each of the following: (a) f ( − 1) − f ( − 2) (b) f ( x 2 − 3 ) (c) f ( x 2 ) − 3 (d) [ f ( x − 3)] 2 Answer: (a) f ( − 1) − f ( − 2) = √ 2 2 − √ 2 1 = 1 − √ 2 (b) f ( x 2 − 3 ) = √ 2 ( x 2 − 3) + 3 = √ 2 x 2 = √ 2 | x | , x = 0 (c) f ( x 2 ) − 3 = √ 2 x 2 + 3 − 3 (d) [ f ( x − 3)] 2 = (√ 2 ( x − 3) + 3 ) 2 = 2 x , x > 0

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    22 1 Functions and Models 19. Evaluate the difference quotient f ( x ) − f ( a ) x − a for f ( x ) = 1 x 2 . Answer: f ( x ) − f ( a ) x − a = 1 x 2 − 1 a 2 x − a = − a + x a 2 x 2 20. Given the graph of y = f ( x ): Sketch the graph of each of the following functions: (a) − f ( x ) (b) f ( − x ) (c) f (2 x ) (d) 2 f ( x ) (e) − f ( − x ) (f) f ( 1 2 x ) (g) 1 2 f ( x ) (h) f ( x + 1) (i) f ( x ) − 1 (j) 1 − f ( x ) Answer: (a) (b) (c) (d)

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    1.3 New Functions From Old Functions 23 (e) (f) (g) (h) (i) (j) 21. Use the graphs of f and g given below to estimate the values of f ( g ( x )) for x = − 3, − 2, − 1, 0, 1, 2, and 3, and use these values to sketch a graph of y = f ( g ( x )). Answer: x − 3 − 2 − 1 0 1 2 3 f ( g ( x )) − 0 . 5 3 . 88 3 . 50 2 . 88 3 . 50 3 . 88 − 0 . 5

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