Download Chapter One - Fundamentals

Math 100
Section 1.1
Sets and Intervals
Intervals are sets of real numbers that occur frequently in pre-calculus and
calculus.
Open intervals consist of all the numbers that fall between two numbers, but
not the end numbers themselves. That the endpoints are not included is
indicated by round brackets.
Exercise 1:
(2,6) is an open interval. It consists of all the numbers that fall between 2
and 6, but not the endpoints 2 and 6 themselves. The interval can be
denoted in three ways:
Interval Notation:
(2,6)
Set description:
Graph on number line:
Which of the following numbers belong to the interval (2,6)?
5.7
6.01
3
2.01
1.99

6
2
8
7.3
5.9
5.9999
Is there a largest number in the interval (2,6)?
Is there a smallest number in the interval (2,6)?
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Closed intervals consist of all the numbers that fall between two numbers,
including the numbers themselves. That the endpoints are included is
indicated by square brackets.
Example 2:
[ -4, 4 ] is a closed interval. It consists of all the numbers that fall between
–4 and 4, including the endpoints –4 and 4 themselves.
This interval can be denoted in three ways:
Interval Notation:
[ -4 , 4]
Set description:
Graph on number line:
Which of the following numbers belong to the interval [ -4 , 4 ]?
4.5
0.5
-3
-2
-4.01
-3.99
5
4
3.999
0
-7
7
What is the largest number in the interval [ -4 , 4 ]?
What is the smallest number in the interval [ -4 , 4]?
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Some intervals are neither open nor closed, since they include one endpoint
but not the other.
Exercise 3:
The interval [ 0 , 3 ) is neither open nor closed. Give the set description and
graph for this interval:
Set description:
Graph on number line:
Which of the following numbers belong to the interval?
0
3
2.56
3.2
-1
0.1
What is the largest number in the interval [ 0 , 3 ) ?
What is the smallest number in the interval [ 0 , 3 ) ?
Exercise 4:
The interval ( 2 , 5 ] is neither open nor closed. Give the set description and
graph for this interval:
Set description:
Graph on number line:
Which of the following numbers belong to the interval?
2
5
3
1.5
1
6
4.5
What is the largest number in the interval ( 2 , 5 ] ?
What is the smallest number in the interval ( 2 , 5 ] ?
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Some intervals extend indefinitely far in one direction. This is indicated by
using the symbol  as one of the “endpoints”.
Exercise 5:
The interval [ 3 ,  ) extends infinitely far in the positive direction. Give
the set description and graph for this interval:
Set description:
Graph on number line:
Which of the following numbers belong to the interval?
3
2.999
4

100
1,000,000
What is the largest number in the interval [ 3 ,  )?
What is the smallest number in the interval [ 3 ,  )?
Exercise 6:
The interval ( - , 0 ) extends infinitely far in the negative direction. Give
the set description and graph for this interval:
Set description:
Graph on number line:
Which of the following numbers belong to the interval?
.01
0
-.001

100,000
-1,000,000
What is the largest number in the interval ( - , 0 )?
What is the smallest number in the interval ( - , 0 )?
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Unions and Intersections of Intervals
 UNIONS: The symbol  means “OR”
 INTERSECTIONS: The symbol  means “AND”
Exercise 7:
Which of the following numbers belong to the set [ -2, 0 )  ( 1, 3 ] ?
-3
-2
2
3
0
0.5
4
5
-1.5
Graph the set [ -2, 0 )  ( 1, 3 ]
Exercise 8:
Which of the following numbers belong to the set [ 1, 7 ]  ( 3,  ) ?
-3
1
2
4
7
7.5
100
1,000,000
3
3.001
Graph the set [ 1, 7 ]  ( 3,  ) by first graphing the two intervals
separately, then seeing where they overlap.
Exercise 9:
Which of the following numbers belong to the set [ 1, 7 ]  ( 8,  ) ?
-3
1
2
4
7
7.5
100
1,000,000
3
3.001
Graph the set [ 1, 7 ]  ( 8,  ) by first graphing the two intervals
separately, then seeing where (if anywhere) they overlap.
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Sometimes the answer to a question is an interval of numbers, sometimes an
answer is individual numbers:
Exercise 10: Suppose the number line below represents a river, and the
curved line represents a road. Where the road crosses the number line,
there is a bridge.
road
north
west
east
south
a) On what intervals is the road north of (above) the river (number line)?
b) On what intervals is the road south of (below) the river (number line)?
c) Where does the road cross the number line?
Exercise 11: Suppose a ladybug is walking along the curve shown below,
from west to east.
north
west
east
south
a) On what intervals is the ladybug going generally northward (up)?
b) On what intervals is the ladybug going generally southward (down)?
c) Where does the bug change
i) from going northward to southward?
ii) from going southward to northward?
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Objectives and Suggested Exercises for Section 1.1
Suggested Exercises are on page 13 of the textbook
1. To express an interval given in interval notation using inequalities and
graphs (#33, 35, 37)
2. To express an inequality with interval notation and with a graph. (#39, 41,
43)
3. To graph the union or intersection of two intervals (#45, 47, 49)
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Math 100
Section 1.8
Coordinate Geometry
The Cartesian coordinate system (named for Rene Descartes (1596 - 1650),
the French mathematician and philosopher) has two perpendicular number
lines. The horizontal line is called the x-axis and the vertical line is called
the y-axis. The point of intersection of the axes is known as the origin. Any
point in the plane can be identified with an ordered pair of numbers, (x,y).
Distance Formula
We can find the distance between any two points in the plane by using the
Pythagorean Theorem for right triangles:
c
b
a2 + b2 = c2
a
Exercise 1
a) Plot the points A(3,2) and
B(9,10) and find the distance
between A and B using the
Pythagorean Theorem.
b) Plot the points A(-9,-2) and
B(7,4) and find the distance
between A and B using the
Pythagorean Theorem.
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Exercise 2 Use the Pythagorean Theorem to find the distance between the
points A(x1, y1) and B(x2, y2) in terms of x1, y1 , x2, and y2.
B(x2, y2)
A(x1, y1)
Midpoint Formula
The midpoint of a line segment with endpoints A(x1, y1) and B(x2, y2) is the
point on the line segment that is an equal distance from A and from B.
Exercise 3 Show that the midpoint of the line segment with endpoints
A(x1, y1) and B(x2, y2) is the point
 x  x2 y1  y2 
,
M 1

2 
 2
B(x2, y2)
M(x, y)
A(x1, y1)
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Exercise 4 Plot the points A(2 , 8) and B(-8 , -4)
a) Find the distance between A and B
b) Find the midpoint, M, of the
line segment AB
c) Find the distance between M
and A
c) Find a point on the x-axis that is the same distance from A as it is from
B
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Graphs of Equations
The graph of an equation is the set of all the points (x,y) that make the
equation true.
1
Exercise 5 The graph of the equation y  x 2  1 is given below.
2
i)
Verify that the following points lie on the curve, and also make sure
1
that they satisfy the equation y  x 2  1 :
2
a) (4, 7)
b) (-3, 3.5)
c) (0, -1)
d) ( 2 , 0)
ii)
On which intervals (intervals of x) are:
a) the y-values greater than zero? ___________ ___________
b) the y-values less than zero?_____________
c) for which x-values is y = 0 ?___________ ___________
iii)
Imagine travelling along the curve (going generally from left to right).
a) On which interval are the y-values decreasing? __________
b) On which interval are the y-values increasing? __________
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Circles
By definition, a circle of radius r is the set of all the points whose distance
from the centre (h,k) is r.
Exercise 6 Use the distance formula
to show that any point P(x,y) that lies
P(x, y)
on the circumference of a circle
r
centred at C(h,k) and having radius r
C(h, k)
satisfies the equation:
(x – h)2 + (y – k)2 = r2
Exercise 7 Find an equation of the circle that satisfies the given conditions
and sketch a graph of each circle.
a) Centre at (-5,6) and radius 3
b) Centre at (0, 0) and passing through the
point (3,-4)
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Exercise 8 Find the centre and radius of the circles represented by the
following equations by transforming the equations into the standard form:
(x – h)2 + (y – k)2 = r2. Sketch a graph of each circle.
a) x2 + y2 – 6x + 10y + 18 = 0
b) x2 + y2 + 8x = 0
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Suggested Exercises and Summary of Objectives for Section 1.8
Suggested exercises are on pages 92-94
Objectives:
1) To understand and become skilled at:
a) Plotting points in the Cartesian coordinate system (#3a, 7a)
b) Finding the distance between points (#3b, 7b)
c) Find the mid-point of a line segment (#3c, 7c, 27)
2) To understand what is meant by the graph of an equation (definition on page 84)
3) To find the equation of a circle, given centre and radius (#63, 67, 69)
4) To find the centre and radius of a circle, given its equation (#71, 73)
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Math 100
Section 1.10
Lines
The slope of a straight line is a measure of its steepness. Slope also tells us
whether the line goes downhill or uphill (as we move from left to right). If
the slope is positive, the line goes uphill, if the slope is negative, the line
goes downhill.
Slope of a Line
If a line passes through the points A(x1,y1) and
B(x2,y2) then it has a slope of:
y  y1 rise
m 2

x2  x1
run
Exercise 1 Find the slope of the line that passes through given points and
plot the line:
a) A(6,9) and B(1,-6)
b) A(-7,9) and B(9,1)
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Exercise 2: The slope of a line is the same no matter which two points on
the line we select. In the formula for slope, let B(x,y) be any point on the
line and A(x1,y1) be a fixed point on the line, then use the slope formula to
derive the point-slope form of the equation of a line:
y - y1 = m(x - x1)
Exercise 3 Find the equation of a line that passes through the given points.
Plot the line.
a) A(-8,-3) and (2,-5)
b) A(8,6) and B(0,2)
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The y-intercept of a line is the place where the line crosses the y-axis. The
x-coordinate of this point is always 0. If we use the y-intercept (0,b) as
(x1,y1) in the point-slope formula and solve the resulting equation for y, we
get the slope-intercept form of the equation of a line:
slope
y = mx + b
y-intercept
This form is useful because when an equation is in this form, we can see at a
glance what the slope and y-intercept are and use this information to graph
the line.
Exercise 4 Find the slope and intercept of the given straight lines and
graph the lines:
a) 2x – 5y + 10 = 0
b) 4x = 12 – 3y
c) y = 2
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Parallel and Perpendicular Lines
Two lines are said to be parallel if they have the same slope.
Exercise 5 Find the equation of the line that passes through the point (2,6)
and is parallel to the line 2x – 4 y = 8. Plot both lines.
Two lines are said to be perpendicular if they intersect each other at 90.
When a line is rotated through 90, the rise and run exchange rolls.
Furthermore, the slope switches from positive to negative or vice versa.
1
Thus if two lines with slopes m1 and m2 are perpendicular, then m1  
m2
1
And m2  
.
m1
The only exception are vertical and horizontal lines. A horizontal line has a
slope of zero. A line perpendicular to a horizontal line is vertical, and has an
undefined slope.
Exercise 6
a) Find the equation of the line that passes through the point (-8,5) and is
perpendicular to the line 4x – y = 6. Plot both lines.
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b) Find the equation of the line that passes through the point (–2,4) and is
perpendicular to the line y = -5. Plot both lines.
Exercise 7 Find the equation of each line whose graph is given.
a)
b)
c)
L1
L2
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Section 1.10 Lines
Suggested exercises are on pages 112-114
Objectives:
To understand and become skilled at:
1) Finding the slope of a line given two points on the line (#1, 3)
2) Finding the slope of a line from the graph of a line (#7)
3) Finding the equation of a line given:
a) the graph of a line (#9)
b) one point and the slope (#13, 15)
c) two points on the line (#17, 21)
4) Finding the equation of a line through a given point and:
a) parallel to a given line (#25, 27)
b) perpendicular to a given line (#29)
5) Drawing the graph of a line, given its equation (#39, 47)
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