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From equation 7 , 3z 3 z 1 If we then substitute into equation 6 , y 112 3 y 2 If y 2 and z 1, these values can now be substituted into equation 1 to obtain x 2 112 1, or x 4. Therefore, the solution to this system is 14, 2, 12. Check: These values should be substituted into each of the original equations and checked to see that they satisfy each equation. To solve this system of equations, we used elementary operations and ended up with a triangle of zeros in the lower left part: ax by cz d 0x ey fz g 0x 0y hz i The use of elementary operations to create the lower triangle of zeros is our objective when solving systems of equations. Large systems of equations are solved using computers and elementary operations to eliminate unknowns. This is by far the most efficient and cost-effective method for their solution. In the following example, we consider a system of equations with different possibilities for its solution. EXAMPLE 5 Connecting the value of a parameter to the nature of the intersection between two lines in R 2 Consider the following system of equations: 1 2 x ky 4 kx 4y 8 Determine the value(s) of k for which this system of equations has a. no solutions b. one solution c. an infinite number of solutions Solution Original System of Equations: 1 2 x ky 4 kx 4y 8 NEL CHAPTER 9 505
1: Multiply equation 1 by k, and add it to equation 2 to eliminate x from equation 2 . 1 3 x ky 4 0x k 2 y 4y 4k 8, k 1 Actual Solution to Problem: To solve the problem, it is only necessary to deal with the equation 0x k 2 y 4y 4k 8 to determine the necessary conditions on k. k 2 y 4y 4k 8 (Factor) y1k 2 42 41k 22 (Multiply by 1) y1k 2 42 41k 22 1k 221k 22y 41k 22 There are three different cases to consider. Case 1: k 2 If k 2, this results in the equation 12 2212 22y 412 22, or 0y 0. Since this equation is true for all real values of y, we will have an infinite number of solutions. Substituting k 2 into the original system of equations gives 2 1 2 x 2y 4 21x 2y2 2142 This system can then be reduced to just a single equation, x 2y 4, which, as we have seen, has an infinite number of solutions. Case 2: k 2 If k 2, this equation becomes 12 2212 22y 412 22, or 0y 16. There are no solutions to this equation. Substituting k 2 into the original system of equations gives 1 2 x 2y 4 21x 2y2 2142 This system can be reduced to the two equations, x 2y 4 and x 2y 4, which are two parallel lines that do not intersect. Thus, there are no solutions. Case 3: k ; 2 If k ; 2, we get an equation of the form ay b, a 0. This equation will always have a unique solution for y, which implies that the original system of equations will have exactly one solution, provided that k ; 2. 506 9.2 SYSTEMS OF EQUATIONS 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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Page 419 and 420:
Review Exercise 1. Given that a !
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18. For the vectors m ! 1 V3, 2, 3
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Chapter 7 Test 1. Given the vectors
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Page 427 and 428:
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
Page 455 and 456: s t s(1, 2, 1) t(0, 2, 1), s, tR P
Page 457 and 458: After deriving vector and parametri
Page 459 and 460: EXAMPLE 4 Representing the equation
Page 461 and 462: A K T 7. a. Determine parameters co
Page 463 and 464: To derive the equation of this plan
Page 465 and 466: A number of observations can be mad
Page 467 and 468: 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: Substituting into the second equati
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 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
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1 16 dh dt Therefore, at the momen
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11. Two cyclists depart at the same
Page 573 and 574:
Solution a. y ln15x2 Using the cha
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x1x 42 0 x 4 or x 0 But x 0 is
Page 577 and 578:
The Derivatives of General Logarith
Page 579 and 580:
The Derivative of a Composite Funct
Page 581 and 582:
To solve this, we take the natural
Page 583 and 584:
Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
Page 587 and 588:
Properties of a Matrix in Row-Echel
Page 589 and 590:
Exercise PART A 1. Write an augment
Page 591 and 592:
12. Determine the equation of the p
Page 593 and 594:
1 0 0 18 4 (row 3) row 1 £ 0 1 0
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Finally, convert the entry in the s
Page 597 and 598:
Review of Technical Skills Appendix
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
Page 609 and 610:
2. Enter the second equation. Press
Page 611 and 612:
Use either the calculator keypad or
Page 613 and 614:
20 Graphing the Derivative of a Fun
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4. Create a function. Right-click t
Page 617 and 618:
composition: the process of combini
Page 619 and 620:
G geometric vectors: vectors that a
Page 621 and 622:
ationalizing the denominator: the p
Page 623 and 624:
Answers Chapter 1 0 0 0 0 0 Review
Page 625 and 626:
d. about 1 e. about 7 8 f. no tang
Page 627 and 628:
d. 13. m 3; b 1 14. a 3, b 2, c
Page 629 and 630:
Review Exercise, pp. 56-59 1. a. 3
Page 631 and 632:
19. a. y 7 b. y 5x 5 c. y 18x 9
Page 633 and 634:
20. a. 34.3 m> s b. 39.2 m> s c. 54
Page 635 and 636:
17. a. 100 bacteria b. 1200 bacteri
Page 637 and 638:
11. a. b. 160x y 16 0 60x y 61
Page 639 and 640:
3. 1 2x 4. a. x 2 15x 6 b. 6012x
Page 641 and 642:
14. t 1 s; away 15. a. s1t2 kt 2
Page 643 and 644:
14. a. triangle side length 0.96 cm
Page 645 and 646:
Review Exercise, pp. 156-159 11. 20
Page 647 and 648:
c. i. ii. iii. d. i. ii. iii. 5 4 3
Page 649 and 650:
. local minimum: 11.41, 39.62, loca
Page 651 and 652:
2. increasing: x 6 1 and x 7 2, dec
Page 653 and 654:
to a zero of its derivative, the nu
Page 655 and 656:
7. (-2, 10) 10 8 y 10. a. 8 y e. 4
Page 657 and 658:
4. hole at x 2; large and negative
Page 659 and 660:
sin x b. 1 sin 2 tan x sec x x L
Page 661 and 662:
12. a. no maximum or minimum value
Page 663 and 664:
Review Exercise, pp. 263-265 1. a.
Page 665 and 666:
x 2. 5. a. 19 000 fish> year b. 23
Page 667 and 668:
Section 6.1, pp. 279-281 6. a. b. c
Page 669 and 670:
17. P G 4. Answers may vary. For ex
Page 671 and 672:
11. a. AG ! a ! b ! c ! , b. AG
Page 673 and 674:
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
Page 685 and 686:
cos A 1 cos B 1 cos C 1 area of t
Page 687 and 688:
. r ! 11, 1, 02 t10, 1, 12, tR; x
Page 689 and 690:
2. 12, 3, 42 3. P must lie on plane
Page 691 and 692:
25. A plane has two parameters, bec
Page 693 and 694:
coincident with the plane, meaning
Page 695 and 696:
Mid-Chapter Review, pp. 518-519 1.
Page 697 and 698:
8. a. b. y 1 z 1 4 , 2 c. x 3t
Page 699 and 700:
10. vector equation: r ! 10, 0, 42
Page 701 and 702:
. 23. a. b. 2a 1 2 b a 6 7 , 2 7 ,
Page 703 and 704:
16. 17. 18. 19. 2 3 4 m>min 144 m>m
Page 705 and 706:
ln x 1 e. ln 2 4x 1 x ln 13x 2 2
Page 707 and 708:
Index A Absolute extrema, 130, 132,
Page 709:
normal, 461-469, 486, 512, 524 para
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