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If we substitute this into equation 2 , we obtain ! ! 0.7071@ T 2 @ 0.8660311.414221@ T2 @ 24 196 ! 1.9318 @T 2 @ 196 ! 196 @T 2 @ 1.9318 101.46 N ! ! Since @T 1 @ 1.4142 @T2 @ , ! @T 1 @ 1.4142 1101.462 143.48 N Therefore, the tensions in the ropes are 143.48 N and 101.46 N, as before. EXAMPLE 5 Reasoning about equilibrium in a system involving three forces a. Is it possible for three forces of 15 N, 18 N, and 38 N to keep a system in a state of equilibrium? b. Three forces having magnitudes 3 N, 5 N, and 7 N are in a state of equilibrium. Calculate the angle between the two smaller forces. Solution a. For a system to be in equilibrium, it is necessary that a triangle be formed having lengths proportional to 15, 18, and 38. Since 15 18 6 38, a triangle cannot be formed because the triangle inequality states that for a triangle to be formed, the sum of any two sides must be greater than or equal to the third side. Therefore, three forces of 15 N, 18 N, and 38 N cannot keep a system in a state of equilibrium. b. We start by drawing the triangle of forces and the related parallelogram. C D 3 N C 7 N 5 N 5 N 7 N 5 N A 3 N B A 3 N B Using ^ ABC, @AC ! @ 2 @AB ! @ 2 @BC ! @ 2 2 @AB ! @@BC ! @ cos CBA (Cosine law) 7 2 3 2 5 2 2132152 cos CBA NEL CHAPTER 7 361
49 34 30 cos CBA 1 cos CBA 2 120° cos 1 10.52CBA The angle that is required is DAB, the supplement of CBA. DAB 60°, and the angle between the 3 N and 5 N force is 60°. IN SUMMARY Key Ideas • Problems involving forces can be solved using strategies involving vectors. • When two or more forces are applied to an object, the net effect of the forces can be represented by the resultant vector determined by adding the vectors that represent each of the forces. • A system is in a state of equilibrium when the net effect of all the forces acting on an object causes no movement of the object. Need to Know ! ! • F ! ! ! is the resultant of and . ! ! • F ! F 1 F2 F ! ! 1 F 2 is the equilibrant of and . • If a ! 1F b ! 1 F2 2 c ! 0 ! , then a ! , b ! , and c ! F 1 F 2 are in a state of equilibrium. Exercise 7.1 PART A 1. a. Name some common household items that have approximate weights of 10 N, 50 N, and 100 N. b. What is your weight in newtons? 2. Three forces of 10 N, 20 N, and 30 N are in a state of equilibrium. a. Draw a sketch of these three forces. b. What is the angle between the equilibrant and each of the smaller forces? 3. Two forces of 10 N and 20 N are acting on an object. How should these forces be arranged to produce the largest possible resultant? 4. Explain in your own words why three forces must lie in the same plane if they are acting on an object in equilibrium. 362 7.1 VECTORS AS FORCES 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
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We evaluate f 1x2 at the critical n
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A T 6. Determine the absolute extre
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Derivatives of Composite Functions
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Review Exercise 1. Differentiate ea
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18. The position of a particle is g
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Chapter 5 Test dy 1. Determine the
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8. Use algebraic methods to evaluat
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f 1x2 142e 5x1 24. For each functi
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Review of Prerequisite Skills In th
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CAREER LINK Investigate CHAPTER 6:
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What is a Vector? A vector is a mat
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EXAMPLE 1 B A E D C Connecting vect
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A K B 5. Given the vector AB ! as s
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Section 6.2—Vector Addition In th
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the overall magnitude would be equa
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Solution c b a a - b + c a - b c a
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W N S E The vectors are drawn so th
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Exercise 6.2 PART A 1. The vectors
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10. a. In ! the ! example ! involvi
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0 0 The previous example illustrate
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Solution a. 30˚ y x 2x-y 2x -y 30
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Solution Draw a sketch and determin
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8. The two vectors a ! and b ! are
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Section 6.4—Properties of Vectors
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Demonstrating the distributive law
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IN SUMMARY Key Idea • Properties
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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
Page 357 and 358: is drawn approximately to scale, an
Page 359 and 360: The magnitude of the vertical compo
Page 361: ! T1 sin 105° 1961sin 45°2 T ! 1
Page 365 and 366: C T f 3 40 N O 35 N 45° 30 N f 2 2
Page 367 and 368: Solution We start by drawing positi
Page 369 and 370: v a v + w w 20 triangles, Solving t
Page 371 and 372: A T C 9. A small airplane has an ai
Page 373 and 374: Perhaps the most important observat
Page 375 and 376: If we look at the right-angled tria
Page 377 and 378: Solution Consider the following dia
Page 379 and 380: A T 10. What is the value of the do
Page 381 and 382: Expanding, we get b 2 1 2a 1 b 1
Page 383 and 384: . 0 a ! 0 2 @b ! 112 2 122 2 142
Page 385 and 386: One of the most important propertie
Page 387 and 388: 4. a. From the set of vectors e11,
Page 389 and 390: Mid-Chapter Review 1. a. If 0 a ! 0
Page 391 and 392: Section 7.5—Scalar and Vector Pro
Page 393 and 394: EXAMPLE 1 Reasoning about the chara
Page 395 and 396: x EXAMPLE 3 z P (2, 1, 4) g b a O(0
Page 397 and 398: EXAMPLE 4 Calculating a specific di
Page 399 and 400: IN SUMMARY Key Idea • A projectio
Page 401 and 402: B C 12. In the diagram shown, ^ ABC
Page 403 and 404: a 3 b, k = 1 O b 3 a, k = -1 b a is
Page 405 and 406: If we carry out an identical proced
Page 407 and 408: EXAMPLE 2 Calculating cross product
Page 409 and 410: K A C T 4. Calculate the cross prod
Page 411 and 412: 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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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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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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Review Exercise 1. Determine vector
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20. Calculate the acute angle that
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Chapter 8 Test 1. a. Given the poin
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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
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Exercise 9.3 C K PART A 1. A system
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Mid-Chapter Review 1. Determine the
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Section 9.4—The Intersection of T
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1: Create two new equations, 4 and
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Thus, 7y 5t 3. Dividing by 7, we g
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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
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EXAMPLE 1 Calculating the distance
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In triangle PQR, sin u d @ QP ! @
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Exercise 9.5 PART A 1. Determine th
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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
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Solution a. Differentiate both side
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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
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The Derivatives of General Logarith
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The Derivative of a Composite Funct
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To solve this, we take the natural
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Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
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Properties of a Matrix in Row-Echel
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Exercise PART A 1. Write an augment
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12. Determine the equation of the p
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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
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7 Making a Table of Differences To
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4. Cursor along the curve to any po
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5. Display and analyze the results.
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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
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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
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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
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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
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sin x b. 1 sin 2 tan x sec x x L
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12. a. no maximum or minimum value
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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,
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Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
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. Since the equilibrant is directed
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7. a. scalar projection: 0, vector
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2. a. 3 b. 7 c. 43 d. 217 e. 5 f. 1
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cos A 1 cos B 1 cos C 1 area of t
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. 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.
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8. a. b. y 1 z 1 4 , 2 c. x 3t
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10. vector equation: r ! 10, 0, 42
Page 701 and 702:
. 23. a. b. 2a 1 2 b a 6 7 , 2 7 ,
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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,
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normal, 461-469, 486, 512, 524 para
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