LECTURE 1Functions

Instructor: Dr. Tarek EmamLocation: C5 301-rightOffice hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30

pmE- mail: tarek.emam@guc.edu.eg

Textbooks: Calculus (An Applied Approach), 7th edition,

by Larson and Edwards Lecture notes (presentations).

INSTRUCTOR AND TEXTBOOKS

COURSE OUTLINES

Math 101

Basic functions

Limits and continuity

Derivative and its

applications

Function of several variables

Sequences and series

Assessment will be based on homework assignments, announced quizzes, midterm exam, and final exam.• 15% Homework assignments.• 15% announced quizzes.• 25% Midterm exam.• 45% Final exam.

Important Notice: 75% of the lectures and tutorials must be attended.

COURSE ASSESSMENT

The Cartesian plane is formed by using two real number lines intersecting at right angles. The horizontal line is usually called x-axis,and the vertical line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants.

The Cartesian plane

Each point in the plane corresponds to an ordered pair (x, y) of realnumbers x and y, called coordinates of the point. The x-coordinate represents the directed distance from the y-axis to the point, and the y-coordinate represents the directed distance from the x-axis to the point.

Distance between two pointsConsider the two points in the Cartesian plane (x1, y1) and (x2, y2).The distance between the two points is given by the formula

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SKETCHING RELATIONS IN THE CARTESIAN PLANE Given a relation between two variables

x and y in the plane xy, we can make a sketch to that relation by these easy steps,

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SKETCHING RELATIONS IN THE CARTESIAN PLANE

Pick enough number of values of one variable (x or y).

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SKETCHING RELATIONS IN THE CARTESIAN PLANE

Pick enough number of values of one variable (x or y).

For each value x (or y), calculate the corresponding value of the other dependent value y (or x).

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SKETCHING RELATIONS IN THE CARTESIAN PLANE

Pick enough number of values of one variable (x or y).

For each value x (or y), calculate the corresponding value of the other dependent value y (or x).

Make a table for these ordered pairs of points.

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SKETCHING RELATIONS IN THE CARTESIAN PLANE

Pick enough number of values of one variable (x or y).

For each value x (or y), calculate the corresponding value of the other dependent value y (or x).

Make a table for these ordered pairs of points.

Plot these points.

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SKETCHING RELATIONS IN THE CARTESIAN PLANE

Pick enough number of values of one variable (x or y).

For each value x (or y), calculate the corresponding value of the other dependent value y (or x).

Make a table for these ordered pairs of points.

Plot these points.Make the sketch by joining between the

points.

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EXAMPLE A Sketch the relation y = 2x + 1Solution Here it is easier to take x as

independent variable and calculate the corresponding values of y

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EXAMPLE 2 Sketch the relation y = 2x + 1Solution Here it is easier to take x as

independent variable and calculate the corresponding values of y

Choose x = -2, 0, 2

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EXAMPLE 2 Sketch the relation y = 2x + 1Solution Here it is easier to take x as

independent variable and calculate the corresponding values of y

Choose x = -2, 0, 2 The corresponding values of y are: -3,

1, 5 respectively.

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WE GET THE TABLE

PLOT THE POINTS (-2,3), (0,1), (2,5)

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WE JOIN THE POINTS TO GET THE SKETCH

3 2 1 1 2 3

4

2

2

4

6

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EXAMPLE 3 Sketch the relation y2 –x=1

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EXAMPLE 3 Sketch the relation y2 –x=1 This is easier to be written as: x = y2 -1

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EXAMPLE 3 Sketch the relation y2 –x=1 This is easier to be written as: x = y2 -

1 Choose y = -3, 0, 4 Calculate the corresponding values x = 8, -1, 15

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WE GET THE TABLE

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PLOT THE POINTS (8,-3), (-1,0), (15,4)

5 10 15

3

2

1

1

2

3

4

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WE JOIN THE POINTS TO GET THE SKETCH

5 10 15

4

2

2

4

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MORE EXAMPLESPlot

y = x2, y = x4

y = x3, y = x5

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THE X-INTERCEPT AND THE Y-INTERCEPT These simply give the intersections of the

curve of the relation with the x-axis and the y-axis

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THE X-INTERCEPT AND THE Y-INTERCEPT These simply give the intersections of the

curve of the relation with the x-axis and the y-axis

The x-intercept is given by setting y = 0 and getting the value of x

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THE X-INTERCEPT AND THE Y-INTERCEPT These simply give the intersections of the

curve of the relation with the x-axis and the y-axis

The x-intercept is given by setting y = 0 and getting the value of x

The y-intercept is given by setting x = 0 and getting the value of y

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EXAMPLE 4 Find the x and y intercepts for the

curves of the relations in examples A, B

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SOLUTION 3: Y = 2X + 1 10 yxThe line intersects with the y-axis at y=1.

2

10 xy

The line intersects with the x-axis at 2

1x

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SOLUTION 4: Y2 –X=1

The curve intersects with the y-axis twice at

10 yx

1y

10 xy The curve intersects with the x-axis at x = -1

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A function is an operation performed on an input (x) to produce an output (y = f(x) ).In other words : A function is a machine that takes a value x in the domain and gives you a value y=f(x) in the range

 The Domain of f is the set of all allowable inputs (x values)The Range of f is the set of all outputs (y values)

 

f

x y =f(x)

Domain

Functions

Range

TYPES OF FUNCTIONS Polynomial Functions (Polynomials)

A function f(x) is called a polynomial if it is of the form:

Where n is a non-negative integer and the numbers a0,a1,…,an are constants called coefficients of the polynomial.

on

nn

n axaxaxaxf 1

11 ...)(

n is called the degree of the polynomial is called the leading coefficient is called the absolute coefficient

naoa

Example 6For each of the following polynomials, determine the degree, the leading coefficient, and the absolute coefficient

SPECIAL POLYNOMIALSTHE ZERO DEGREE POLYNOMIAL

(THE CONSTANT FUNCTION)

SOME BASIC POLYNOMIALS

POLYNOMIALS Notes 1- A linear function f(x) = mx + c is a

polynomial of degree 1 2- A constant function f(x) = c, where c

is constant is a polynomial of degree 0

RATIONAL FUNCTION DEFINITION

DOMAIN OF FUNCTION The domain of a function y = f(x) is the

set of values that the variable x can take.

DOMAIN OF A POLYNOMIAL From the definition of a polynomial, it is

easy to realize that the domain of a polynomial is the set of all Real numbers R

DOMAIN OF A RATIONAL FUNCTION

TO FIND DOMAIN OF A RATIONAL FUNCTION

EXAMPLE 7