Table Of Content

Calculus and Vectors 12 McGraw-Hill Ryerson Preface Chapter 1 Rates of Change Prerequisite Skills 1.1 Rates of Change and the Slope of a Curve 1.2 Rates of Change Using Equations 1.3 Limits 1.4 Limits and Continuity 1.5 Introduction to Derivatives Extension: Use a Computer Algebra System to Determine Derivatives Review Practice Test Task: The Water Skier: Where’s the Dock? Chapter 2 Derivatives Prerequisite Skills 2.1 Derivative of a Polynomial Function Extension: Problem Solving With a Computer Algebra System 2.2 The Product Rule 2.3 Velocity, Acceleration, and Second Derivatives 2.4 The Chain Rule 2.5 Derivatives of Quotients Extension: The Quotient Rule 2.6 Rate of Change Problems Review Practice Test Task: The Disappearing Lollipop Chapter 3 Curve Sketching Prerequisite Skills 3.1 Increasing and Decreasing Functions 3.2 Maxima and Minima 3.3 Concavity and the Second Derivative Test 3.4 Simple Rational Functions 3.5 Putting It All Together 3.6 Optimization Problems Review Practice Test Chapters 1 to 3 Review Task: An Intense Source of Light Chapter 4 Derivatives of Sinusoidal Functions Prerequisite Skills 4.1 Instantaneous Rates of Change of Sinusoidal Functions 4.2 Derivatives of the Sine and Cosine Functions 4.3 Differentiation Rules for Sinusoidal Functions 4.4 Applications of Sinusoidal Functions and Their Derivatives Review Practice Test Task: Double Ferris Wheel Chapter 5 Exponential and Logarithmic Functions Prerequisite Skills 5.1 Rates of Change and the Number e 5.2 The Natural Logarithm 5.3 Derivatives of Exponential Functions 5.4 Differentiation Rules for Exponential Functions 5.5 Making Connections: Exponential Models Review Practice Test Chapters 4 and 5 Review Task: Headache Relief? Be Careful! Chapter 6 Geometric Vectors Prerequisite Skills 6.1 Introduction to Vectors 6.2 Addition and Subtraction of Vectors 6.3 Multiplying a Vector by a Scalar 6.4 Applications of Vector Addition 6.5 Resolution of Vectors Into Rectangular Components Review Practice Test Task: Taxi Cab Vectors Chapter 7 Cartesian Vectors Prerequisite Skills 7.1 Cartesian Vectors 7.2 Dot Product 7.3 Applications of the Dot Product 7.4 Vectors in Three-Space 7.5 The Cross Product and Its Properties 7.6 Applications of the Dot Product and Cross Product Review Practice Test Task: The Cube Puzzle Chapter 8 Lines and Planes Prerequisite Skills 8.1 Equations of Lines in Two-Space and Three-Space 8.2 Equations of Planes 8.3 Properties of Planes 8.4 Intersections of Lines in Two-Space and Three-Space 8.5 Intersections of Lines and Planes 8.6 Intersections of Planes Extension: Solve Systems of Equations Using Matrices Review Practice Test Chapters 6 to 8 Review Task: Simulating 3-D Motion on a Television Screen Chapters 1 to 8 Course Review Prerequisite Skills Appendix Technology Appendix Answers Glossary Index Credits Eighth pages 1 Chapter Rates of Change Our world is in a constant state of change. Understanding the nature of change and the rate at which it takes place enables us to make important predictions and decisions. For example, climatologists monitoring a hurricane measure atmospheric pressure, humidity, wind patterns, and ocean temperatures. These variables affect the severity of the storm. Calculus plays a significant role in predicting the storm’s development as these variables change. Similarly, calculus is used to analyse change in many other fields, from the physical, social, and medical sciences to business and economics. By the end of this chapter, you will describe examples of real-world applications of rate of change of the function at x (cid:5) a and the rates of change, represented in a variety of ways f(a(cid:3)h)(cid:4)f(a) value of the limit lim describe connections between the average rate h→ 0 h of change of a function that is smooth over an compare, through investigation, the calculation interval and the slope of the corresponding of instantaneous rate of change at a point secant, and between the instantaneous rate of (a, f (a)) for polynomial functions, with and without change of a smooth function at a point and the f(a(cid:3)h)(cid:4)f(a) simplifying the expression before slope of the tangent at that point h substituting values of h that approach zero make connections, with or without graphing technology, between an approximate value of the generate, through investigation using technology, instantaneous rate of change at a given point on a table of values showing the instantaneous the graph of a smooth function and average rate rate of change of a polynomial function, f (x), of change over intervals containing the point for various values of x, graph the ordered pairs, recognize, through investigation with or without recognize that the graph represents a function technology, graphical and numerical examples of called the derivative, f′(x) or dy, and make limits, and explain the reasoning involved dx connections between the graphs of f (x) and f′(x) make connections, for a function that is smooth dy over the interval a (cid:2) x (cid:2) a (cid:3) h, between the or y and dx average rate of change of the function over this interval and the value of the expression determine the derivatives of polynomial f(a(cid:3)h)(cid:4)f(a) functions by simplifying the algebraic expression , and between the instantaneous f(x(cid:3)h)(cid:4)f(x) h and then taking the limit of the h simplified expression as h approaches zero 1 Eighth pages Prerequisite Skills First Differences Expanding Binomials 1. Complete the following table for the function 5. Use Pascal’s triangle to expand each binomial. y(cid:5) x2 (cid:3) 3x (cid:3) 5. a) (a (cid:3) b)2 b) (a (cid:3) b)3 c) (a (cid:4) b)3 a) What do you notice about the first d) (a (cid:3) b)4 e) (a (cid:4) b)5 f) (a (cid:3) b)5 differences? b) Does this tell you anything about the shape Factoring of the curve? 6. Factor. x y First Difference a) 2x2 (cid:4) x (cid:4) 1 b) 6x2 (cid:3) 17x (cid:3) 5 (cid:4)4 c) x3 (cid:4) 1 d) 2x4 (cid:3) 7x3 (cid:3) 3x2 (cid:4)3 e) x2 (cid:4) 2x (cid:4) 4 f) t3 (cid:3) 2t2 (cid:4) 3t (cid:4)2 Factoring Difference Powers (cid:4)1 7. Use the pattern in the first row to complete the 0 table for each difference of powers. 1 Difference of Factored Form 2 Powers a) an (cid:4) bn (a (cid:4) b)(an(cid:4)1 (cid:3) an(cid:4)2b (cid:3) an(cid:4)3b2 (cid:3) Slope of a Line … (cid:3) a2bn(cid:4)3 (cid:3) abn(cid:4)2 (cid:3) bn(cid:4)1) 2. Determine the slope of the line that passes b) a2 (cid:4) b2 through each pair of points. c) (a (cid:4) b)(a2 (cid:3) ab (cid:3) b2) a) ((cid:4)2, 3) and (4, 1) b) (3, (cid:4)7) and (0, (cid:4)1) d) a4 (cid:4) b4 c) (5, 1) and (0, 0) d) (0, 4) and ((cid:4)9, 4) e) a5 (cid:4) b5 f) (x (cid:3) h)n (cid:4) xn Slope-Intercept Form of the Equation of a Line 3. Rewrite each equation in slope-intercept form. Expanding Difference of Squares State the slope and y-intercept for each. 8. Expand and simplify each difference of squares. a) 2x (cid:4) 4y (cid:5) 7 b) 5x (cid:3) 3y (cid:4) 1 (cid:5) 0 a) ( x(cid:4) 2)( x(cid:3) 2) c) (cid:4)18x (cid:5) 9y (cid:3) 10 d) 5y (cid:5) 7x (cid:3) 2 b) ( x(cid:3)1(cid:4) x)( x(cid:3)1(cid:3) x) 4. Write the slope-intercept form of the equations c) ( x(cid:3)1(cid:4) x(cid:4)1)( x(cid:3)1(cid:3) x(cid:4)1) of lines that meet the following conditions. d) ( 3(x(cid:3)h)(cid:4) 3x)( 3(x(cid:3)h)(cid:3) 3x) a) The slope is 5 and the y-intercept is 3. b) The line passes through the points ((cid:4)5, 3) Simplifying Rational Expressions and (1, 1). 9. Simplify. c) The slope is (cid:4)2 and the point (4, 7) is on 1 1 1 1 a) (cid:4) b) (cid:4) the line. 2(cid:3)h 2 x(cid:3)h x d) The line passes through the points (3, 0) 1 1 and (2, (cid:4)1). (cid:4) 1 1 x(cid:3)h x c) (cid:4) d) (x(cid:3)h)2 x2 h 2 MHR • Calculus and Vectors • Chapter 1 Eighth pages Function Notation Representing Intervals 10. Determine the points ((cid:4)2, f ((cid:4)2)) and (3, f (3)) 14. An interval can be represented in several ways. for each given function. Complete the missing information in the a) f (x) (cid:5) 3x (cid:3) 12 following table. b) f (x) (cid:5) (cid:4)5x2 (cid:3) 2x (cid:3) 1 Interval Notation Inequality Number Line c) f (x) (cid:5) 2x3 (cid:4) 7x2 (cid:3) 3 ((cid:4)3, 5) 11. For each function, determine f (3 (cid:3) h) in (cid:4)3 (cid:2) x (cid:2) 5 simplified form. (cid:4)3 (cid:2) x (cid:6) 5 (cid:2)3 0 5 a) f (x) (cid:5) 6x (cid:4) 2 ((cid:4)3, 5] b) f (x) (cid:5) 3x2 (cid:3) 5x (cid:2)3 0 5 c) f (x) (cid:5) 2x3 (cid:4) 7x2 [(cid:4)3, ∞) f(2(cid:3)h)(cid:4)f(2) x (cid:6) 5 12. For each function, determine h x (cid:2) 5 in simplified form. ((cid:4)∞, ∞) (cid:92) a) f (x) (cid:5) 6x b) f (x) (cid:5) 2x3 Graphing Functions Using Technology 1 4 c) f(x)(cid:5) d) f(x)(cid:5)(cid:4) x x 15. Use a graphing calculator to graph each function. State the domain and range of each Domain of a Function using set notation. 13. State the domain of each function. a) y (cid:5) (cid:4)5x3 a) f (x) (cid:5) 3 (cid:4) 5x b) y(cid:5)8(cid:3)x b) y(cid:5) x 8(cid:4)x x2(cid:4)4 c) y(cid:5) c) Q(x) (cid:5) x4 (cid:4) x2 (cid:3) 4x d) y(cid:5) x x(cid:3)2 e) y(cid:5) x2 d) y (cid:5) 0.5x2 (cid:3) 1 x2(cid:3)x(cid:4)6 f) D(x)(cid:5) x (cid:3) 9(cid:4)x P R O B L E M R E Alicia is considering a career as either a demographer or T a climatologist. Demographers study changes in human P A populations with respect to births, deaths, migration, education H level, employment, and income. Climatologists study both C the short-term and long-term effects of change in climatic conditions. How are the concepts of average rate of change and instantaneous rate of change used in these two professions to analyse data, solve problems, and make predictions? Prerequisite Skills • MHR 3 Eighth pages 1.1 Rates of Change and the Slope of a Curve The speed of a vehicle is usually expressed in terms of kilometres per hour. This is an expression of rate of change. It is the change in position, in kilometres, with respect to the change in time, in hours. This value can represent an average rate of change or an instantaneous rate of change . That is, if your vehicle travels 80 km in 1 h, the average rate of change is 80 km/h. However, this expression does not provide any information about your movement at different points during the hour. The rate you are travelling at a particular instant is called instantaneous rate of change. This is the information that your speedometer provides. In this section, you will explore how the slope of a line can be used to calculate an average rate of change, and how you can use this knowledge to estimate instantaneous rate of change. You will consider the slope of two types of lines: secants and tangents. (cid:129) Secants are lines that connect two points that lie on the same curve. y Q Secant PQ P x (cid:129) Tangents are lines that run “parallel” to, or in the same direction as, the curve, touching it at only one point. The point at which the tangent touches the graph is called the tangent point . The line is said to be tangent to the function at that point. Notice that for more complex functions, a line that is tangent at one x-value may be a secant for an interval on the function. y y Q x P P Tangent to f(x) at the tangent point P x 4 MHR • Calculus and Vectors • Chapter 1 Eighth pages Investigate What is the connection between slope, average rate of change, and instantaneous rate of change? Imagine that you are shopping for a vehicle. One of the cars you are Tools considering sells for $22 000 new. However, like most vehicles, this car loses • grid paper value, or depreciates, as it ages. The table below shows the value of the car • ruler over a 10-year period. Time Value (years) ($) 0 22 000 1 16 200 2 14 350 3 11 760 4 8 980 5 7 820 6 6 950 7 6 270 8 5 060 9 4 380 10 4 050 A: Connect Average Rate of Change to the Slope of a Secant 1. Explain why the car’s value is the dependent variable and time is the independent variable . 2. Graph the data in the table as accurately as you can using grid paper. Draw a smooth curve connecting the points. Describe what the graph tells you about the rate at which the car is depreciating as it ages. 3. a) Draw a secant to connect the two points corresponding to each of the following intervals, and determine the slope of each secant. i) year 0 to year 10 ii) year 0 to year 2 iii) year 3 to year 5 iv) year 8 to year 10 b) Reflect Explain why the slopes of the secants are examples of average rate of change. Compare the slopes for these intervals and explain what this comparison tells you about the average rate of change in value of the car as it ages. 4. Reflect Determine the first differences for the data in the table. What do you notice about the first differences and average rate of change? B: Connect Instantaneous Rate of Change to the Slope of a Tangent 1. Place a ruler along the graph of the function so that it forms a tangent to the point corresponding to year 0. Move the ruler along the graph keeping it tangent to the curve. a) Reflect Stop at random points as you move the ruler along the curve. What do you think the tangent represents at each of these points? b) Reflect Explain how slopes can be used to describe the shape of a curve. 1.1 Rates of Change and the Slope of a Curve • MHR 5 Eighth pages 2. a) On the graph, use the ruler to draw a tangent through the point corresponding to year 1. Use the graph to find the slope of the tangent you have drawn. b) Reflect Explain why your calculation of the slope of the tangent is only an approximation. How could you make this calculation more accurate? C: Connect Average Rate of Change and Instantaneous Rate of Change 1. a) Draw three secants corresponding to the following intervals, and determine the slope of each. i) year 1 to year 9 ii) year 1 to year 5 iii) year 1 to year 3 b) What do you notice about the slopes of the secants compared to the slope of the tangent you drew earlier? Make a conjecture about the slope of the secant between years 1 and 2 in relation to the slope of the tangent. c) Use the data in the table to calculate the slope for the interval between years 1 and 2. Does your calculation support your conjecture? 2. Reflect Use the results of this investigation to summarize the relationship between slope, secants, tangents, average rate of change, and instantaneous rate of change. Determine Average and Instantaneous Rates of Example 1 Change From a Table of Values A decorative birthday balloon is being filled with t(s) V (cm3) helium. The table shows the volume of helium in the 0 0 balloon at 3-s intervals for 30 s. 3 4.2 1. What are the dependent and independent variables 6 33.5 for this problem? In what units is the rate of change 9 113.0 12 267.9 expressed? 15 523.3 2. a) Use the table of data to calculate the slope of the 18 904.3 secant for each of the following intervals. What 21 1436.0 does the slope of the secant represent? 24 2143.6 i) 21 s to 30 s ii) 21 s to 27 s iii) 21 s to 24 s 27 3052.1 30 4186.7 b) Reflect What is the significance of a positive rate of change in the volume of the helium in the balloon? 3. a) Graph the information in the table. Draw a tangent at the point on the graph corresponding to 21 s and calculate the slope of this line. What does this graph illustrate? What does the slope of the tangent represent? b) Reflect Compare the secant slopes that you calculated in question 2 to the slope of the tangent. What do you notice? What information would you need to calculate a secant slope that is even closer to the slope of the tangent? 6 MHR • Calculus and Vectors • Chapter 1 Eighth pages Solution 1. In this problem, volume is dependent on time, so V is the dependent variable and t is the independent variable. For the rate of change, V is V expressed with respect to t, or . Since the volume in this problem is t expressed in cubic centimetres, and time is expressed in seconds, the units for the rate of change are cubic centimetres per second (cm3/s). 2. a) Calculate the slope of the secant using the formula ΔV V (cid:4)V (cid:5) 2 1 Δt t (cid:4)t 2 1 i) The endpoints for the interval 21 (cid:2) t (cid:2) 30 are (21, 1436.0) and (30, 4186.7). ΔV 4186.7(cid:4)1436.0 (cid:5) (cid:2)306 CO N N E C T I O N S Δt 30(cid:4)21 The symbol (cid:24) indicates that an ii) The endpoints for the interval 21 (cid:2) t (cid:2) 27 are (21, 1436.0) and answer is approximate. (27, 3052.1). ΔV 3052.1(cid:4)1436.0 (cid:5) (cid:2)269 Δt 27(cid:4)21 iii) The endpoints for the interval 21 (cid:2) t (cid:2) 24 are (21, 1436.0) and (24, 2143.6). ΔV 2143.6(cid:4)1436.0 (cid:5) (cid:2)236 Δt 24(cid:4)21 The slope of the secant represents the average rate of change, which in this problem is the average rate at which the volume of the helium is changing over the interval. The units for these solutions are cubic centimetres per second (cm3/s). b) The positive rate of change during these intervals suggests that the volume of the helium is increasing, so the balloon is expanding. 3. a) Volume of Helium in a Balloon V 2500 2000 3m) c e (1500 m u P(21, 1436) ol V1000 500 Q(16.5, 500) 0 3 6 9 12 15 18 21 24 t Time (s) 1.1 Rates of Change and the Slope of a Curve • MHR 7