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In doing the calculation in Example 1, assumptions were made that are implicit but that should be stated. Further Laws of Vector Addition and Scalar Multiplication 1. Adding : a ! 0 ! a ! 2. Associative Law for Scalars: m1na ! 2 3. Distributive Law for Scalars: 1m n2a ! 1mn2a ! ma ! mna ! na ! 0 ! CHAPTER 6 It is important to be aware of all these properties when calculating, but the properties can be assumed without having to refer to them for each simplification. EXAMPLE 2 Selecting appropriate vector properties to create new vectors If x ! 3i ! 4j ! k ! , y ! j ! , and , determine each of the following in terms of , , and . a. x ! y ! i ! 5k ! j ! b. x ! k ! z ! i ! j ! 4k ! y ! c. x ! 2y ! 3z ! Solution a. x ! y ! 13i ! 3i ! 4j ! 3i ! 4j ! 3j ! j ! k ! 2 4k ! k ! 1j ! 5k ! 5k ! 2 b. x ! y ! 13i ! 3i ! 4j ! 3i ! 4j ! 5j ! j ! k ! 2 6k ! k ! 1j ! 5k ! 5k ! 2 c. x ! 2y ! 3z ! 13i ! 3i ! 4j ! 9j ! 4j ! k ! 23k ! k ! 2 2j ! 21j ! 5k ! 10k ! 2 3i ! 31i ! 3j ! j ! 12k ! 4k ! 2 As stated previously, it is not necessary to state the rules as we simplify, and furthermore, it is better to try to simplify without writing in every step. The rules that were developed in this section will prove useful as we move ahead. They are necessary for our understanding of linear combinations, which will be dealt with later in this chapter. NEL 305
IN SUMMARY Key Idea • Properties used to evaluate numerical expressions and simplify algebraic expressions also apply to vector addition and scalar multiplication. Need to Know • Commutative Property of Addition: • Associative Property of Addition: 1a ! a ! • Distributive Property of Addition: , • Adding 0 ! : a ! 0 ! a ! k1a ! b ! b ! b ! 2 b ! c ! 2 ka ! a ! a ! 1b ! kb ! c ! 2 kR • Associative Law for Scalars: m1na ! 2 • Distributive Law for Scalars: 1m n2a ! 1mn2a ! ma ! mna ! na ! Exercise 6.4 PART A 1. If * is an operation on a set, S, the element x, such that a * x a, is called the identity element for the operation *. a. For the addition of numbers, what is the identity element? b. For the multiplication of numbers, what is the identity element? c. For the addition of vectors, what is the identity element? d. For scalar multiplication, what is the identity element? 2. Illustrate the commutative law for two vectors that are perpendicular. C 3. Redraw the following three vectors and illustrate the associative law. a c b 4. With the use of a diagram, show that the distributive law, k1a ! b ! 2 ka ! kb ! , holds where k 6 0, kR. 306 6.4 PROPERTIES OF VECTORS NEL
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Chapter 1 INTRODUCTION TO CALCULUS
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y x 3 1 , if 3 6 t 6 0 t 6. A func
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What Is Calculus? Two simple geomet
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If a and b were radicals, squaring
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Exercise 1.1 PART A 1. Write the co
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Consider a curve y f 1x2 and a poi
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EXAMPLE 2 Tech Support For help gra
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The slope of the tangent at 13, 42
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INVESTIGATION 4 Tech Support For he
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4. Simplify each of the following d
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18. Copy the following figures. Dra
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A. Determine the average velocity o
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Instantaneous Velocity The velocity
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Solution a. C11002 10V100 1000 1
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Exercise 1.3 C PART A 1. The veloci
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f 1x2 f 1a2 15. Use the alternate
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. Use your results for part a to ap
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The graph shown on your calculator
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limits because, in each case, only
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11. For each function, sketch the g
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EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 Using
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INVESTIGATION Here is an alternate
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In this section, we learned the pro
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A 11. Jacques Charles (1746-1823) d
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A second condition for continuity a
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d. The function F is not continuous
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x 3, if x 3 12. g1x2 e 2 k, if
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Key Concepts Review We began our in
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c. What is the present rate of chan
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16. Evaluate the limit of each diff
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Chapter 2 DERIVATIVES Imagine a dri
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2 4 5. Expand, and collect like te
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Section 2.1—The Derivative Functi
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For the Parabola f1x2 x 2 The slop
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Note that f 1t2 Vt is defined for
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EXAMPLE 6 Reasoning about different
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Exercise 2.1 PART A 1. State the do
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T 15. Match each function in graphs
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EXAMPLE 1 Applying the constant fun
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We conclude this section with the s
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EXAMPLE 6 Connecting the derivative
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x 1 8. Determine the slope of the
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Section 2.3—The Product Rule In t
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EXAMPLE 3 Selecting an efficient st
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EXAMPLE 6 Selecting a strategy to d
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PART B dy 5. Determine the value of
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9. Determine the equation of the ta
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EXAMPLE 1 Applying the quotient rul
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Exercise 2.4 PART A 1. What are the
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Section 2.5—The Derivatives of Co
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with respect to x is equal to the p
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EXAMPLE 7 Combining derivative rule
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Exercise 2.5 PART A 1. Given f 1x2
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Technology Extension: Derivatives o
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Key Concepts Review Now that you ha
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9. For what values of x does the cu
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24. At a manufacturing plant, produ
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Chapter 3 DERIVATIVES AND THEIR APP
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4. Determine the area of each figur
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Section 3.1—Higher-Order Derivati
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Motion on a Straight Line An object
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d. The object moves in a positive d
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EXAMPLE 5 Analyzing motion under gr
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Exercise 3.1 C K PART A 1. Explain
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T 12. A dragster races down a 400 m
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H. Repeat part D for the following
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Only t 3 is in the given interval,
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IN SUMMARY Key Ideas • The maximu
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T PART B 4. Using the algorithm for
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Mid-Chapter Review 1. Determine the
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Section 3.3—Optimization Problems
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For extreme values, set V¿1x2 0.
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Exercise 3.3 C PART A 1. A piece of
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a. Determine the dimensions that sh
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The revenue function is R1x2 17 0
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The minimal cost is approximately $
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T C 8. The fuel cost per hour for r
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CAREER LINK WRAP-UP Investigate and
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10. For each of the following cost
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s1t2 t 26. Find the absolute maxi
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Chapter 4 CURVE SKETCHING If you ar
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5. Determine the derivative of each
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Section 4.1—Increasing and Decrea
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dy Since this is a polynomial funct
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IN SUMMARY Key Ideas • A function
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9. Each of the following graphs rep
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Solution dy First determine dx . dy
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Note that there is no value of x fo
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EXAMPLE 4 Graphing the derivative g
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4. Find the x- and y-intercepts of
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Section 4.3—Vertical and Horizont
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The behaviour of the graph can be i
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EXAMPLE 2 Expressing a polynomial f
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When lim f 1x2 k or lim f 1x2 k,
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For horizontal asymptotes, 3x f 1x2
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Now let's consider the straight lin
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Exercise 4.3 PART A 1. State the eq
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12. Use the features of each functi
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11. a. What does f ¿1x2 7 0 imply
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From these investigations, we can m
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Setting dy dx 0, we obtain 3 1x 2
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The second derivative of f 1x2 is f
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Exercise 4.4 K PART A 1. For each f
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Section 4.5—An Algorithm for Curv
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Therefore, by the second derivative
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Interval 1, 12 11, 0.82 10.8, 22 12
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K A 1x PART B 4. Use the algorithm
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Key Concepts Review In this chapter
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5. For each of the following, check
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17. If f 1x2 x3 8 , determine the
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Chapter 5 DERIVATIVES OF EXPONENTIA
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Radian Measure A radian is the meas
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5. Convert the following angles to
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Section 5.1—Derivatives of Expone
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From the investigation, you should
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EXAMPLE 4 Connecting the derivative
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A C PART B 6. Determine the equatio
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Section 5.2—The Derivative of the
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In general, for the exponential fun
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Solution a. January 1, 1900, is exa
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Section 5.3—Optimization Problems
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The average revenue to the company
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Exercise 5.3 PART A 1. Use graphing
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12. Find the maximum and minimum va
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9. The rapid growth in the number o
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INVESTIGATION 2 A.Using your graphi
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With practice, you will learn how t
Page 255 and 256: We evaluate f 1x2 at the critical n
Page 257 and 258: A T 6. Determine the absolute extre
Page 259 and 260: Derivatives of Composite Functions
Page 261 and 262: CAREER LINK WRAP-UP Investigate and
Page 263 and 264: Review Exercise 1. Differentiate ea
Page 265 and 266: 18. The position of a particle is g
Page 267 and 268: Chapter 5 Test dy 1. Determine the
Page 269 and 270: 8. Use algebraic methods to evaluat
Page 271 and 272: f 1x2 142e 5x1 24. For each functi
Page 273 and 274: Review of Prerequisite Skills In th
Page 275 and 276: CAREER LINK Investigate CHAPTER 6:
Page 277 and 278: What is a Vector? A vector is a mat
Page 279 and 280: EXAMPLE 1 B A E D C Connecting vect
Page 281 and 282: A K B 5. Given the vector AB ! as s
Page 283 and 284: Section 6.2—Vector Addition In th
Page 285 and 286: the overall magnitude would be equa
Page 287 and 288: Solution c b a a - b + c a - b c a
Page 289 and 290: W N S E The vectors are drawn so th
Page 291 and 292: Exercise 6.2 PART A 1. The vectors
Page 293 and 294: 10. a. In ! the ! example ! involvi
Page 295 and 296: 0 0 The previous example illustrate
Page 297 and 298: Solution a. 30˚ y x 2x-y 2x -y 30
Page 299 and 300: Solution Draw a sketch and determin
Page 301 and 302: 8. The two vectors a ! and b ! are
Page 303 and 304: Section 6.4—Properties of Vectors
Page 305: Demonstrating the distributive law
Page 309 and 310: Mid-Chapter Review 1. ABCD is a par
Page 311 and 312: Section 6.5—Vectors in R 2 and R
Page 313 and 314: Right-Handed System of Coordinates
Page 315 and 316: Solution a. A16, 0, 02 is a point o
Page 317 and 318: There is one further observation th
Page 319 and 320: C A 14. a. What is the equation of
Page 321 and 322: R 2 EXAMPLE 1 Representing vectors
Page 323 and 324: EXAMPLE 2 Representing the sum and
Page 325 and 326: EXAMPLE 5 Calculating the magnitude
Page 327 and 328: A 10. Parallelogram is determined b
Page 329 and 330: EXAMPLE 1 Representing vectors in R
Page 331 and 332: . Using components, a ! b ! c !
Page 333 and 334: IN SUMMARY Key Ideas • In R 3 , i
Page 335 and 336: Section 6.8—Linear Combinations a
Page 337 and 338: In R 2 , it is possible to take any
Page 339 and 340: Geometrically, the linear combinati
Page 341 and 342: IN SUMMARY Key Ideas • In R 2 , O
Page 343 and 344: CAREER LINK WRAP-UP Investigate and
Page 345 and 346: Review Exercise 1. Determine whethe
Page 347 and 348: 15. A rectangular prism measuring 3
Page 349 and 350: Chapter 6 Test 1. The vectors a ! ,
Page 351 and 352: Review of Prerequisite Skills In th
Page 353 and 354: Section 7.1—Vectors as Forces The
Page 355 and 356: of 6 cm, also pointing east. The ve
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is drawn approximately to scale, an
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The magnitude of the vertical compo
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! T1 sin 105° 1961sin 45°2 T ! 1
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49 34 30 cos CBA 1 cos CBA 2 120
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C T f 3 40 N O 35 N 45° 30 N f 2 2
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Solution We start by drawing positi
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v a v + w w 20 triangles, Solving t
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A T C 9. A small airplane has an ai
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Perhaps the most important observat
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If we look at the right-angled tria
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Solution Consider the following dia
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A T 10. What is the value of the do
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Expanding, we get b 2 1 2a 1 b 1
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. 0 a ! 0 2 @b ! 112 2 122 2 142
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One of the most important propertie
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4. a. From the set of vectors e11,
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Mid-Chapter Review 1. a. If 0 a ! 0
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Section 7.5—Scalar and Vector Pro
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EXAMPLE 1 Reasoning about the chara
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x EXAMPLE 3 z P (2, 1, 4) g b a O(0
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EXAMPLE 4 Calculating a specific di
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IN SUMMARY Key Idea • A projectio
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B C 12. In the diagram shown, ^ ABC
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a 3 b, k = 1 O b 3 a, k = -1 b a is
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If we carry out an identical proced
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EXAMPLE 2 Calculating cross product
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K A C T 4. Calculate the cross prod
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EXAMPLE 1 Using the dot product to
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. We start by constructing position
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EXAMPLE 4 Using the cross product t
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Page 419 and 420:
Review Exercise 1. Given that a !
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18. For the vectors m ! 1 V3, 2, 3
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Chapter 7 Test 1. Given the vectors
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Review of Prerequisite Skills In th
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CAREER LINK Investigate CHAPTER 8:
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Both of these vectors can be multip
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c. If the point Q121, 232 lies on t
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In this section, the vector and par
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C K A T 6. a. If the equation of a
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EXAMPLE 1 Representing the Cartesia
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It is not possible, in this case, t
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AP ! 1x 4, y 1222 1x 4, y 22.
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u cos 1 a 3 12 1 10 21 25 2 b u
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A T 10. For each pair of lines, det
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is its direction vector. If represe
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EXAMPLE 4 Representing the equation
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C A T 6. a. Determine parametric eq
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13. The Cartesian equation of a lin
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s t s(1, 2, 1) t(0, 2, 1), s, tR P
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After deriving vector and parametri
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EXAMPLE 4 Representing the equation
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A K T 7. a. Determine parameters co
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To derive the equation of this plan
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A number of observations can be mad
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When we considered lines in R 2 , w
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IN SUMMARY Key Idea • The Cartesi
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Section 8.6—Sketching Planes in R
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EXAMPLE 2 Representing the graph of
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EXAMPLE 5 Graphing planes whose Car
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IN SUMMARY Key Idea • A sketch of
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Page 481 and 482:
Review Exercise 1. Determine vector
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20. Calculate the acute angle that
Page 485 and 486:
Chapter 8 Test 1. a. Given the poin
Page 487 and 488:
Review of Prerequisite Skills In th
Page 489 and 490:
CAREER LINK Investigate CHAPTER 9:
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4 13 s2 211 4s2 12 8s2 8 0 1
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Method 2: Again, this result can be
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We can now select any two of the th
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IN SUMMARY Key Ideas • Line and p
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A T 11. a. Show that the lines and
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equations. Since the lines would be
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Solution 1: Interchange equations 1
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Substituting into the second equati
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1: Multiply equation 1 by k, and ad
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K C PART B 4. Solve each system of
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Section 9.3—The Intersection of T
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If we had solved the system using e
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then x 3s 1. Now it is a matter o
Page 517 and 518:
Exercise 9.3 C K PART A 1. A system
Page 519 and 520:
Mid-Chapter Review 1. Determine the
Page 521 and 522:
Section 9.4—The Intersection of T
Page 523 and 524:
1: Create two new equations, 4 and
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Thus, 7y 5t 3. Dividing by 7, we g
Page 527 and 528:
Inconsistent Systems for Three Equa
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Therefore, we can choose ! ! ! m 1
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Exercise 9.4 PART A 1. A student is
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9. Solve each system of equations u
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Section 9.5—The Distance from a P
Page 537 and 538:
EXAMPLE 1 Calculating the distance
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In triangle PQR, sin u d @ QP ! @
Page 541 and 542:
Exercise 9.5 PART A 1. Determine th
Page 543 and 544:
Section 9.6—The Distance from a P
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It is also possible to use the dist
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p 1 p 2 U V d L 1 : r = (-2, 1, 0)
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The required distance between the t
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K A T PART B 2. Determine the follo
Page 553 and 554:
Review Exercise 1. The lines 2x y
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14. You are given the lines , and r
Page 557 and 558:
Chapter 9 Test 1. a. Determine the
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P(1, -2, 4) Q P9 2x - 3y - 4z + 66
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29. A line that passes through the
Page 563 and 564:
Solution a. Differentiate both side
Page 565 and 566:
Exercise PART A 1. State the chain
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dA b. To determine differentiate A
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1 16 dh dt Therefore, at the momen
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11. Two cyclists depart at the same
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Solution a. y ln15x2 Using the cha
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x1x 42 0 x 4 or x 0 But x 0 is
Page 577 and 578:
The Derivatives of General Logarith
Page 579 and 580:
The Derivative of a Composite Funct
Page 581 and 582:
To solve this, we take the natural
Page 583 and 584:
Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
Page 587 and 588:
Properties of a Matrix in Row-Echel
Page 589 and 590:
Exercise PART A 1. Write an augment
Page 591 and 592:
12. Determine the equation of the p
Page 593 and 594:
1 0 0 18 4 (row 3) row 1 £ 0 1 0
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Finally, convert the entry in the s
Page 597 and 598:
Review of Technical Skills Appendix
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3 Evaluating a Function 1. Enter th
Page 601 and 602:
7 Making a Table of Differences To
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4. Cursor along the curve to any po
Page 605 and 606:
5. Display and analyze the results.
Page 607 and 608:
14 Graphing a Trigonometric Functio
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2. Enter the second equation. Press
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Use either the calculator keypad or
Page 613 and 614:
20 Graphing the Derivative of a Fun
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4. Create a function. Right-click t
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composition: the process of combini
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G geometric vectors: vectors that a
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ationalizing the denominator: the p
Page 623 and 624:
Answers Chapter 1 0 0 0 0 0 Review
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d. about 1 e. about 7 8 f. no tang
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d. 13. m 3; b 1 14. a 3, b 2, c
Page 629 and 630:
Review Exercise, pp. 56-59 1. a. 3
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19. a. y 7 b. y 5x 5 c. y 18x 9
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20. a. 34.3 m> s b. 39.2 m> s c. 54
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17. a. 100 bacteria b. 1200 bacteri
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11. a. b. 160x y 16 0 60x y 61
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3. 1 2x 4. a. x 2 15x 6 b. 6012x
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14. t 1 s; away 15. a. s1t2 kt 2
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14. a. triangle side length 0.96 cm
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Review Exercise, pp. 156-159 11. 20
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c. i. ii. iii. d. i. ii. iii. 5 4 3
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. local minimum: 11.41, 39.62, loca
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2. increasing: x 6 1 and x 7 2, dec
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to a zero of its derivative, the nu
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7. (-2, 10) 10 8 y 10. a. 8 y e. 4
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4. hole at x 2; large and negative
Page 659 and 660:
sin x b. 1 sin 2 tan x sec x x L
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12. a. no maximum or minimum value
Page 663 and 664:
Review Exercise, pp. 263-265 1. a.
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x 2. 5. a. 19 000 fish> year b. 23
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Section 6.1, pp. 279-281 6. a. b. c
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17. P G 4. Answers may vary. For ex
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11. a. AG ! a ! b ! c ! , b. AG
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d. OD ! 11, 1, 12 (0, 0, 1) (1, 0,
Page 675 and 676:
Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
Page 679 and 680:
. Since the equilibrant is directed
Page 681 and 682:
7. a. scalar projection: 0, vector
Page 683 and 684:
2. a. 3 b. 7 c. 43 d. 217 e. 5 f. 1
Page 685 and 686:
cos A 1 cos B 1 cos C 1 area of t
Page 687 and 688:
. r ! 11, 1, 02 t10, 1, 12, tR; x
Page 689 and 690:
2. 12, 3, 42 3. P must lie on plane
Page 691 and 692:
25. A plane has two parameters, bec
Page 693 and 694:
coincident with the plane, meaning
Page 695 and 696:
Mid-Chapter Review, pp. 518-519 1.
Page 697 and 698:
8. a. b. y 1 z 1 4 , 2 c. x 3t
Page 699 and 700:
10. vector equation: r ! 10, 0, 42
Page 701 and 702:
. 23. a. b. 2a 1 2 b a 6 7 , 2 7 ,
Page 703 and 704:
16. 17. 18. 19. 2 3 4 m>min 144 m>m
Page 705 and 706:
ln x 1 e. ln 2 4x 1 x ln 13x 2 2
Page 707 and 708:
Index A Absolute extrema, 130, 132,
Page 709:
normal, 461-469, 486, 512, 524 para
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