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x 11 6. Show that y 4 and x 0, y 1 3t, z 3 2t, 2 4 z 27 5 tR, are skew lines. 7. a. Determine the intersection of the lines x 5 1x 3, y 20, z 72 t12, 4, 52, tR, and y 2 z 4 2 3 . b. What conclusion can you make about these lines? 8. Determine the point of intersection between the lines x 1 2s, y 4 s, z 3s, sR, and x 3, y t 3, z 2t, tR. 9. Determine the point of intersection for each pair of lines. a. r ! and r ! 15, 1, 72 s12, 0, 52, sR, b. r ! 11, 1, 32 t14, 2, 12, tR and r ! 12, 1, 32 s15, 1, 62, sR, 18, 1, 92 t15, 1, 62, tR 10. You are given a pair of vector equations that both represent lines in R 3 . a. Explain all the possible ways that these lines could be related to each other. Support your explanation with diagrams. b. Explain how you could use the equations you are given to help you identify which of the situations you described in part a. you are dealing with. 11. a. Explain when a line and a plane can have an infinite number of points of intersection. b. Give an example of a pair of vector equations (one for a line and one for a plane) that have an infinite number of points of intersection. 12. Use elementary operations to solve each system of equations. a. 1 2x 3y 30 2 x 2y 13 b. 1 x 4y 3z 6 0 2 2x 8y 6z 11 0 c. 1 x 3y 2z 9 2 2x 5y z 3 3 3x 6y 2z 8 13. For the system of equations given in parts a. and b. of question 12, describe the corresponding geometrical representation. 14. L is the line of intersection of planes x y 1 and y z 3, and L 1 is the line of intersection of the planes y z 0 and x 1 2 . a. Determine the point of intersection of L and L 1 . b. Determine the angle between the lines of intersection. c. Determine the Cartesian equation of the plane that contains the point you found in part a. and the two lines of intersection. NEL CHAPTER 9 519
Section 9.4—The Intersection of Three Planes In the previous section, we discussed the intersection of two planes. In this section, we will extend these ideas and consider the intersection of three planes. Algebraically, the three planes are typically represented by a system of three linear equations in three unknowns. First we will consider consistent systems. Later in the section, we will consider inconsistent systems. Consistent Systems for Three Equations Representing Three Planes There are four cases that should be considered for the intersection of three planes. These four cases, which all result in one or more points of intersection between all three planes, are shown below. Case 1: P is the point of intersection of three planes. Case 2a: L is the line of intersection of three planes. p 1 p 2 p 3 p 1 p 2 p 3 P L Case 2b: L is the line of intersection of two coincident planes and a third plane not parallel to the coincident planes. Case 3: The plane of intersection of three coincident planes is the plane itself. p 1 , p 2 , p 3 p 3 L p 1 , p 2 520 9.4 THE INTERSECTION OF THREE PLANES 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
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Section 6.5—Vectors in R 2 and R
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Right-Handed System of Coordinates
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Solution a. A16, 0, 02 is a point o
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There is one further observation th
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C A 14. a. What is the equation of
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R 2 EXAMPLE 1 Representing vectors
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EXAMPLE 2 Representing the sum and
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EXAMPLE 5 Calculating the magnitude
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A 10. Parallelogram is determined b
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EXAMPLE 1 Representing vectors in R
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. Using components, a ! b ! c !
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IN SUMMARY Key Ideas • In R 3 , i
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Section 6.8—Linear Combinations a
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In R 2 , it is possible to take any
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Geometrically, the linear combinati
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IN SUMMARY Key Ideas • In R 2 , O
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Review Exercise 1. Determine whethe
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15. A rectangular prism measuring 3
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Chapter 6 Test 1. The vectors a ! ,
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Section 7.1—Vectors as Forces The
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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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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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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
Page 469 and 470: IN SUMMARY Key Idea • The Cartesi
Page 471 and 472: Section 8.6—Sketching Planes in R
Page 473 and 474: EXAMPLE 2 Representing the graph of
Page 475 and 476: EXAMPLE 5 Graphing planes whose Car
Page 477 and 478: IN SUMMARY Key Idea • A sketch of
Page 479 and 480: CAREER LINK WRAP-UP Investigate and
Page 481 and 482: Review Exercise 1. Determine vector
Page 483 and 484: 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:
Page 491 and 492: 4 13 s2 211 4s2 12 8s2 8 0 1
Page 493 and 494: Method 2: Again, this result can be
Page 495 and 496: We can now select any two of the th
Page 497 and 498: IN SUMMARY Key Ideas • Line and p
Page 499 and 500: A T 11. a. Show that the lines and
Page 501 and 502: equations. Since the lines would be
Page 503 and 504: Solution 1: Interchange equations 1
Page 505 and 506: Substituting into the second equati
Page 507 and 508: 1: Multiply equation 1 by k, and ad
Page 509 and 510: K C PART B 4. Solve each system of
Page 511 and 512: Section 9.3—The Intersection of T
Page 513 and 514: If we had solved the system using e
Page 515 and 516: 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: Mid-Chapter Review 1. Determine the
Page 523 and 524: 1: Create two new equations, 4 and
Page 525 and 526: Thus, 7y 5t 3. Dividing by 7, we g
Page 527 and 528: Inconsistent Systems for Three Equa
Page 529 and 530: Therefore, we can choose ! ! ! m 1
Page 531 and 532: Exercise 9.4 PART A 1. A student is
Page 533 and 534: 9. Solve each system of equations u
Page 535 and 536: Section 9.5—The Distance from a P
Page 537 and 538: EXAMPLE 1 Calculating the distance
Page 539 and 540: 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
Page 545 and 546: It is also possible to use the dist
Page 547 and 548: p 1 p 2 U V d L 1 : r = (-2, 1, 0)
Page 549 and 550: The required distance between the t
Page 551 and 552: K A T PART B 2. Determine the follo
Page 553 and 554: Review Exercise 1. The lines 2x y
Page 555 and 556: 14. You are given the lines , and r
Page 557 and 558: Chapter 9 Test 1. a. Determine the
Page 559 and 560: P(1, -2, 4) Q P9 2x - 3y - 4z + 66
Page 561 and 562: 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
Page 567 and 568: dA b. To determine differentiate A
Page 569 and 570: 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
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,
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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 !
Page 679 and 680:
. Since the equilibrant is directed
Page 681 and 682:
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
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. 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
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ln x 1 e. ln 2 4x 1 x ln 13x 2 2
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Index A Absolute extrema, 130, 132,
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normal, 461-469, 486, 512, 524 para
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