Table Of ContentCalculus 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
