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Cumulative Review of Vectors 1. For the vectors a ! 12, 1, 22 and b ! 13, 4, 122, determine the following: a. the angle between the two vectors b. the scalar and vector projections of a ! on c. the scalar and vector projections of b ! b ! on a ! 2. a. Determine the line of intersection between p 1 : 4x 2y 6z 14 0 and p 2 : x y z 5 0. b. Determine the angle between the two planes. 3. If x ! and y ! are unit vectors, and the angle between them is 60°, determine the value of each of the following: a. b. c. 4. Expand and simplify each of the following, where i ! , j ! 012x ! and k ! y ! 2 # ! ! 02x! # ! 0x! # ! y 0 3y 0 1x 3y 20 , represent the standard basis vectors in a. 21i ! b. 213i ! 2j ! 4j ! 3k ! R 3 2 412i ! : 5k ! 4j ! 5k ! 2 ! ! 2 # ! 1i ! j ! 2 ! ! 12i 3k 2 2i # 13j 2k 2 5. Determine the angle that the vector a ! 14, 2, 32 makes with the positive x-axis, y-axis, and z-axis. 6. If a ! 11, 2, 32, b ! 11, 1, 22, and c ! 13, 4, 12, determine each of the following: a. a ! b ! c. the area of the parallelogram determined by and ! ! b. d. c ! a ! b ! ! # ! 2a 3b 1b a 2 7. Determine the coordinates of the unit vector that is perpendicular to a ! 11, 1, 12 and b ! 12, 2, 32. 8. a. Determine vector and parametric equations for the line that contains A12, 3, 12 and B11, 2, 32. b. Verify that C14, 13, 32 is on the line that contains A and B. 9. Show that the lines L 1 : r ! 12, 0, 92 t11, 5, 22, tR, and L 2 : x 3 y 5 are parallel and distinct. 5 z 10 2 10. Determine vector and parametric equations for the line that passes through (0, 0, 4) and is parallel to the line with parametric equations x 1, y 2 t, and z 3 t, tR. 11. Determine the value of c such that the plane with equation 2x 3y cz 8 0 is parallel to the line with equation x 1 2 y 2 3 z 1. NEL CHAPTERS 6–9 557
P(1, –2, 4) Q P9 2x – 3y – 4z + 66 = 0 25 N 60° 40 N x 2 12. Determine the intersection of the line y 5 z 3 with the plane 3 5 5x y 2z 2 0. 13. Sketch the following planes, and give two direction vectors for each. a. x 2y 2z 6 0 b. 2x 3y 0 c. 3x 2y z 0 14. If P11, 2, 42 is reflected in the plane with equation 2x 3y 4z 66 0, determine the coordinates of its image point, P¿. (Note that the plane 2x 3y 4z 66 0 is the right bisector of the line joining P11, 2, 42 with its image.) 15. Determine the equation of the line that passes through the point and intersects the line r ! A11, 0, 22 12, 3, 42 s11, 1, 22, sR, at a right angle. 16. a. Determine the equation of the plane that passes through the points A11, 2, 32, B12, 0, 02, and C11, 4, 02. b. Determine the distance from O10, 0, 02 to this plane. 17. Determine a Cartesian equation for each of the following planes: a. the plane through the point A11, 2, 52 with n ! 13, 5, 42 b. the plane through the point K14, 1, 22 and perpendicular to the line joining the points (2, 1, 8) and 11, 2, 42 c. the plane through the point 13, 1, 32 and perpendicular to the z-axis d. the plane through the points 13, 1, 22 and 11, 3, 12 and parallel to the y-axis 18. An airplane heads due north with a velocity of 400 km> h and encounters a wind of 100 km> h from the northeast. Determine the resultant velocity of the airplane. 19. a. Determine a vector equation for the plane with Cartesian equation 3x 2y z 6 0, and verify that your vector equation is correct. b. Using coordinate axes you construct yourself, sketch this plane. 20. a. A line with equation r ! 11, 0, 22 s12, 1, 22, sR, intersects the plane x 2y z 2 at an angle of u degrees. Determine this angle to the nearest degree. b. Show that the planes p 1 : 2x 3y z 1 0 and p 2 : 4x 3y 17z 0 are perpendicular. c. Show that the planes p 3 : 2x 3y 2z 1 0 and p 4 : 2x 3y 2z 3 0 are parallel but not coincident. 21. Two forces, 25 N and 40 N, have an angle of 60° between them. Determine the resultant and equilibrant of these two vectors. 558 CUMULATIVE REVIEW 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
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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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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
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 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
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: Chapter 9 Test 1. a. Determine the
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
Page 571 and 572: 11. Two cyclists depart at the same
Page 573 and 574: Solution a. y ln15x2 Using the cha
Page 575 and 576: 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
Page 585 and 586: 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
Page 595 and 596: Finally, convert the entry in the s
Page 597 and 598: Review of Technical Skills Appendix
Page 599 and 600: 3 Evaluating a Function 1. Enter th
Page 601 and 602: 7 Making a Table of Differences To
Page 603 and 604: 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
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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
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
Page 645 and 646:
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
Page 671 and 672:
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 !
Page 677 and 678:
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
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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,
Page 709:
normal, 461-469, 486, 512, 524 para
show all

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