Calculus and

Analytical Geometry

Instructor
LECTURES for week # 01
Dr. Muhammad Arif Hussain

Sweden
 Topics to be Covered in week 1

Real numbers, real line, intervals, open and closed intervals,

lower and upper bounds, absolute value inequalities,
rectangular/Cartesian coordinate systems, some linear and
non-linear equations and their graphs
BOOK
CALCULUS
EARLY TRANSCENDENTALS

By
HOWARD ANTON
Book
Grading Policy

Assignments (3 - 5) 10%
Quizzes (3 - 5) 10%
Midterm 30%
Final Exam 50%
Definitions

Variable A symbol that can vary in value.

Constant A symbol that does not vary in

value.
Expression A mathematical relationship that
does not contain an equal sign

Equation A mathematical relationship that

contains an equal sign.
Inequality A mathematical relationship that
contains an inequality symbol (, <,
>, , or ).
Absolute Value A given number’s distance from 0
on a number line. Notation = | |.
 Set Notation

Set A collection of objects

Set Notation { }
Fractions Radicals
Set A collection of objects.
Set Notation { }
Natural Counting numbers {1,2,3, …}
numbers
Whole Natural numbers and 0.
Numbers {0,1,2,3, …}
Integers Positive and negative natural numbers
and zero. {… -2, -1, 0, 1, 2, 3, …}
Rational A real number that can be expressed
Number as a ratio of integers (fraction)
Irrational Any real number that is not rational.
Number ( 2 , )
Real Numbers All numbers associated with
the number line.
|3|= 3 Absolute Value

| 2.5 | = 2.5
|0|= 0
|-7 |= 7
| - 4.8 | = 4.8
 To which set(s) of numbers does −6
belong?

a) Natural Numbers
b) Whole numbers
c) Integers
d) Rational numbers
e) Irrational numbers
f) Real numbers
Copyright © 2011 Pearson Education, Inc.
Slide 1- 10
 To which set of numbers does −6
belong?

a) Natural Numbers
b) Whole numbers
c) Integers
d) Rational numbers
e) Irrational numbers
f) Real numbers
Copyright © 2011 Pearson Education, Inc.
Slide 1- 11
 Simplify |7|.

a) 7

b) −7
c) 0

d) 1/7

Copyright © 2011 Pearson Education, Inc.

Slide 1- 12
Simplify |7|.

a) 7

b) −7
c) 0

d) 1/7
 Which statement is false?

a) 7 > 4

b) −2.4 > −1.4

c) 10 < 22

d) −3.6 > −6.4

Copyright © 2011 Pearson Education, Inc.

Slide 1- 14
 Which statement is false?

a) 7 > 4

b) −2.4 > −1.4

c) 10 < 22

d) −3.6 > −6.4

Copyright © 2011 Pearson Education, Inc.

Slide 1- 15
 Set of integers
• If S is a set, the notation a  S means that
a is an element of S.

• b S means that b is not an element of S.

– If Z represents the set

of integers, then –3 Z but π Z.
 Set-Builder Notation
• We can write A in set-builder notation as:
• A = {x | x is an integer and 0 < x < 7}

– This is read:
“A is the set of all x such that x is an integer
and 0 < x < 7.”
 Empty Set
• The empty set, denoted by Ø,
is:

– The set that contains no element.

 Real Numbers
• The real numbers can be represented
by points on a line, as shown.

– The positive direction (toward the right)

is indicated by an arrow.
 Symbol a ≤ b
• The symbol a ≤ b (or b ≥ a):

– Means that either a < b or a = b.

– Is read “a is less than or equal to b.”

 Open Interval
• If a < b, the open interval from a to b
consists of all numbers between a
and b.

– It is denoted (a, b).

 Closed Interval
• The closed interval from a to b
includes the endpoints.

– It is denoted [a, b].

 Open & Closed Intervals
• Using set-builder notation,
we can write:

(a, b) = {x | a < x < b}

[a, b] = {x | a ≤ x ≤ b}
 Open Intervals
• Note that parentheses ( ) in the interval
notation and open circles on the graph in
this figure 5 indicate that:

– Endpoints are excluded from the interval.

 Closed Intervals
• Note that square brackets and solid
circles in this figure 6 indicate that:

– Endpoints are included.

 Types of Intervals
• The table lists the possible types of
intervals.
Absolute Value
 Absolute Value—Definition
• If a is a real number, the absolute
value of a is:

a if a  0
a =
 −a if a  0
 Evaluating Absolute Values
a) |3| = 3

b) |–3| = –(–3) = 3

c) |0| = 0
 Absolute Value
• The absolute value of a number a,
denoted by |a|, is:

– The distance from a to 0 on the real number

line.
 Properties of Absolute Value
• When working with absolute values, we use
these properties.
 Inequalities
These are true inequalities:

7  7.4  7.5
−  −3
22
22
 Graphing Inequalities
a) x < 3

b) x ≥ –2
 Graphing Intervals
• Express each interval in terms of
inequalities, and then graph the interval.

a) [–1, 2)
b) [1.5, 4]
c) (–3, ∞)
SOLUTION (a)

• [–1, 2)

= {x | –1 ≤ x < 2}
SOLUTION (b)

• [1.5, 4]

= {x | 1.5 ≤ x ≤ 4}
SOLUTION (c)

• (–3, ∞)

= {x | –3 < x}
CARTESIAN COORDINATE
SYSTEM
 The Rectangular Coordinate System

Rectangular (or Cartesian) Coordinate System

y
8
y-axis
Origin 6
x-axis
Quadrant II 4Quadrant I
2

x
0 0
-8 -6 -4 -2 0 -2 2 4 6 8

Quadrant III -4
Quadrant IV
-6
-8
 The Rectangular Coordinate System

Ordered Pair
Quadrant
(x, y)
A
D A (5, 3) Quadrant I
x
B (2, –1) Quadrant IV
B
C C (–2, –3) Quadrant III

D (–4, 2) Quadrant II
 The Rectangular Coordinate System

Completing Ordered Pairs

Complete each ordered pair for 3x + 4y = 7.

(a) (5, ? )

We are given x = 5. We substitute into the equation to find y.

3x + 4y = 7
3(5) + 4y = 7 Let x = 5.
15 + 4y = 7
4y = –8
y = –2

The ordered pair is (5, –2).

 4.1 The Rectangular Coordinate System

Completing Ordered Pairs

Complete each ordered pair for 3x + 4y = 7.

(b) ( ? , –5)

Replace y with –5 in the equation to find x.

3x + 4y = 7
3x + 4(–5) = 7 Let y = –5.
3x – 20 = 7
3x = 27
x=9

The ordered pair is (9, –5).

 The Rectangular Coordinate System

A Linear Equation in Two Variables

A linear equation in two variables can be written in the form

Ax + By = C,

where A, B, and C are real numbers (A and B not both 0). This form is
called standard form.
The Rectangular Coordinate System

Intercepts

y-intercept (where the line intersects

the y-axis)

x-intercept (where the

line intersects
the x-axis)
x
 The Rectangular Coordinate System

Finding Intercepts

When graphing the equation of a line,

let y = 0 to find the x-intercept;
let x = 0 to find the y-intercept.
 The Rectangular Coordinate System

EXAMPLE Finding Intercepts

2
Find the x- and y-intercepts of 2x – y = 6, and graph the equation.
We find the x-intercept We find the y-intercept
by letting y = 0. by letting x = 0.

2x – y = 6 2x – y = 6
2x – 0 = 6 Let y = 0. 2(0) – y = 6 Let x = 0.
2x = 6 –y = 6
x=3 x-intercept is (3, 0). y = –6 y-intercept is (0, –6).

The intercepts are the two points (3,0) and (0, –6).
 The Rectangular Coordinate System

EXAMPLE Finding Intercepts

2
Find the x- and y-intercepts of 2x – y = 6, and graph the equation.
The intercepts are the two points (3,0) and (0, –6). We show these ordered
pairs in the table next to the figure below and use these points to draw the
graph. y

x y
x
3 0

0 –6
 The Rectangular Coordinate System

EXAMPLE Graphing a Horizontal Line

3
Graph y = –3.
Since y is always –3, there is no value of x corresponding to y = 0, so the
graph has no x-intercept. The y-intercept is (0, –3). The graph in the figure
below, shown with a table of ordered pairs, is a horizontal line.
y

x y
x
2 –3

0 –3

–2 –3
 The Rectangular Coordinate System

EXAMPLE Graphing a Vertical Line

3
Graph x +con’t
2 = 5.
The x-intercept is (3, 0). The standard form 1x + 0y = 3 shows that every
value of y leads to x = 3, so no value of y makes x = 0. The only way a straight
line can have no y-intercept is if it is vertical, as in the figure below.
y

x y
x
3 2

3 0

3 –2
 Student Activity
Drawing a point is similar to reading a point. You start by moving
across the X axes, then move up the Y axis until you get where you
need to be. Plot the following points on the grid.
Then draw a straight line to connect
from one to the next
1. (8 , 6) to (4, 10)
2. (4 , 2) to (8 , 2)
3. (4 , 4) to (10 , 4)
4. (2 , 4) to (4 , 4)
5. (4 , 6) to (4 , 10)
6. (8 , 6) to (4 , 6)
7. (4 , 2) to (2 , 4)
8. (8 , 2) to (10 , 4)
9. (4 , 4) to (4 , 6)
 The Rectangular Coordinate System
EXAMPLE Graphing a Line That Passes
4 through the Origin

Graph 3x + y = 0.
We find the x-intercept We find the y-intercept
by letting y = 0. by letting x = 0.
3x + y = 0 3x + y = 0
3x + 0 = 0 Let y = 0. 3(0) + y = 0 Let x = 0.
3x = 0 0+y=0
x=0 x-intercept is (0, 0). y=0 y-intercept is (0, 0).

Both intercepts are the same ordered pair, (0, 0). (This means
the graph goes through the origin.)
 The Rectangular Coordinate System
EXAMPLE Graphing a Line That Passes
4 through the Origin

Graph 3x + y = 0.
To find another point to graph the line, choose any nonzero
number for x, say x = 2, and solve for y.
Let x = 2.
3x + y = 0
3(2) + y = 0 Let x = 2.
6+y=0
y = –6
This gives the ordered pair (2, –6).
 The Rectangular Coordinate System
EXAMPLE Graphing a Line That Passes
4 through the Origin

Graph 3x + y = 0.
These points, (0, 0) and (2, –6), lead to the graph shown below.
As a check, verify that (1, –3) also lies on the line.
y x-intercept
and
y-intercept
x y
x
0 0

2 –6

1 –3
 The Rectangular Coordinate System

Use the midpoint formula

If the endpoints of a line segment PQ are (x1, y1) and
(x2, y2), its midpoint M is

 x1 + x2 y1 + y2 
 , .
 2 2 
 The Rectangular Coordinate System
EXAMPLE Finding the Coordinates of a Midpoint
5

Find the coordinates of the midpoint of line segment PQ with

endpoints P(6, −1) and Q(4, −2).

Use the midpoint formula with x1 = 6, x2 = 4, y1 = −1, y2 = −2:

 6 + 4 −1 + (−2)   10 −3   −3 
 ,  =  ,  =  5, 
 2 2   2 2   2 

Midpoint
 Function Notation

y = f (x )
Input
Output Name of
Function
 f
x
x y

y
x

X Y

DOMAIN RANGE
Graph of linear function
Absolute function
Example
Sketch the graph. Using the points:
the graph follows.
Plots of two lines using computer
application WOLFRAM ALPHA
 Open a browser and go to
www.wolframalpha.com

Wolfram
Command plot 2x -3, 2x
window
Computer Output
Slope-Intercept Form
y = mx + b

slope y-intercept
 Intercepts
X-intercept Y-intercept
• Graph crosses the x- axis • Graph crosses the y-axis
• It occurs when y=0 • It occurs when x=0
• (x-intercept, 0) • (0, y-intercept)
 y
4

2
x - intercept
B
1

-6 -4 -2 2 4
X 6

-1

-2
y - intercept

A
-3

-4
 Find the slope and y-intercept
of
y = -5x + 6.

Slope: m = −5 y-intercept: b=6

Slope

Slope describes the

direction of a line.
Find the slope and y-intercept
of
2y = 7x - 4
2 2 2
7
y = x−2
2

7
m= b = −2
2
 You can draw a line using a table of values.

Example. Draw the line with equation y = 2x + 1

1. Choose some values for x such as –3, -2, -1, 0, 1, 2, 3

2. Draw a table like this:

x -3 -2 -1 0 1 2 3
y -5 -3 -1 1 3 5 7

3. Work out 2x + 1 for each value of x

For example, when
These are our coordinate pairs
x = -1, y = –1
 y = 2x + 1
5
•Choose some values for x
4
•Draw a table for your values of x
3

2 •Work out the y values using the

equation
1
•Plot the x and y coordinates
-4 -3 -2 -1 0 1 2 3
-1 •Join up the points to form a
straight line
-2

-3 •Label your line

-4 x -3 -2 -1 0 1 2
-5 y -5 -3 -1 1 3 5
 Finding x-, and y-intercepts

Put y= 0, in the given equation to get x- intercept

Put x= 0, in the given equation to get y- intercept

Equations of the form ax + by = c are called
linear equations in two variables.
y
This is the graph of the (0,4)
equation 2x + 3y = 12.
(6,0)
x
-2 2

The point (0, 4) is the y-intercept.

The point (6, 0) is the x-intercept.

73
Graph of linear function

(0, 3)

(7.5, 0)
 Graph of Quadratic Equation

Graph y = 2x2 – 4. y

(–2, 4) (2, 4)
x y

2 4
1 –2 x
(–1, – 2) (1, –2)
0 –4
–1 –2
(0, –4)
–2 4
System of Equations
 Solving Systems of Linear Equations by Graphing

Solve the system of equations by graphing

 y = 2x − 4

 1 •
 y = − 3 x + 3
Solution : ( 3, 2)
1
2 = 2 ( 3) − 4 2 = − ( 3) + 3
3
2=2 2=2