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Transcript of Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to...
![Page 1: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/1.jpg)
LECTURE 1Functions
![Page 2: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/2.jpg)
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: [email protected]
Textbooks: Calculus (An Applied Approach), 7th edition,
by Larson and Edwards Lecture notes (presentations).
INSTRUCTOR AND TEXTBOOKS
![Page 3: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/3.jpg)
COURSE OUTLINES
Math 101
Basic functions
Limits and continuity
Derivative and its
applications
Function of several variables
Sequences and series
![Page 4: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/4.jpg)
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
![Page 5: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/5.jpg)
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.
![Page 6: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/6.jpg)
![Page 7: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/7.jpg)
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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GUC - Wniter 2009 8
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,
![Page 9: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/9.jpg)
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SKETCHING RELATIONS IN THE CARTESIAN PLANE
Pick enough number of values of one variable (x or y).
![Page 10: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/10.jpg)
GUC - Wniter 2009 10
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).
![Page 11: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/11.jpg)
GUC - Wniter 2009 11
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.
![Page 12: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/12.jpg)
GUC - Wniter 2009 12
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.
![Page 13: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/13.jpg)
GUC - Wniter 2009 13
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.
![Page 14: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/14.jpg)
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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.
![Page 17: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/17.jpg)
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WE GET THE TABLE
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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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GUC - Wniter 2009 21
EXAMPLE 3 Sketch the relation y2 –x=1 This is easier to be written as: x = y2 -1
![Page 22: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/22.jpg)
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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
![Page 24: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/24.jpg)
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PLOT THE POINTS (8,-3), (-1,0), (15,4)
5 10 15
3
2
1
1
2
3
4
![Page 25: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/25.jpg)
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WE JOIN THE POINTS TO GET THE SKETCH
5 10 15
4
2
2
4
![Page 26: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/26.jpg)
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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
![Page 28: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/28.jpg)
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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
![Page 29: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/29.jpg)
GUC - Wniter 2009 29
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
![Page 30: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/30.jpg)
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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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![Page 33: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/33.jpg)
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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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![Page 35: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/35.jpg)
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
![Page 36: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/36.jpg)
![Page 37: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/37.jpg)
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
![Page 38: Functions. Instructor: Dr. Tarek Emam Location: C5 301-right Office hours: Sunday: from 1:00 pm to 3:00pm Monday : from 2:30 pm to 4:30 pm E- mail: tarek.emam@guc.edu.eg.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d425503460f94a1df55/html5/thumbnails/38.jpg)
Example 6For each of the following polynomials, determine the degree, the leading coefficient, and the absolute coefficient
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SPECIAL POLYNOMIALSTHE ZERO DEGREE POLYNOMIAL
(THE CONSTANT FUNCTION)
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SOME BASIC POLYNOMIALS
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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
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RATIONAL FUNCTION DEFINITION
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DOMAIN OF FUNCTION The domain of a function y = f(x) is the
set of values that the variable x can take.
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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
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DOMAIN OF A RATIONAL FUNCTION
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TO FIND DOMAIN OF A RATIONAL FUNCTION
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EXAMPLE 7