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: Textbooks:  Calculus (An Applied Approach), 7 th edition, by Larson and Edwards  Lecture notes (presentations).

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.

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

GUC - Wniter  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,

GUC - Wniter  Pick enough number of values of one variable (x or y).

GUC - Wniter  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).

GUC - Wniter  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.

GUC - Wniter  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.

GUC - Wniter  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.

GUC - Wniter  Sketch the relation y = 2x + 1 Solution  Here it is easier to take x as independent variable and calculate the corresponding values of y

GUC - Wniter  Sketch the relation y = 2x + 1 Solution  Here it is easier to take x as independent variable and calculate the corresponding values of y  Choose x = -2, 0, 2

GUC - Wniter  Sketch the relation y = 2x + 1 Solution  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.

GUC - Wniter

GUC - Wniter

GUC - Wniter  Sketch the relation y 2 –x=1

GUC - Wniter  Sketch the relation y 2 –x=1  This is easier to be written as: x = y 2 -1

GUC - Wniter  Sketch the relation y 2 –x=1  This is easier to be written as: x = y 2 -1  Choose y = -3, 0, 4  Calculate the corresponding values x = 8, -1, 15

GUC - Wniter

GUC - Wniter

GUC - Wniter

GUC - Wniter Plot  y = x 2, y = x 4  y = x 3, y = x 5

GUC - Wniter These simply give the intersections of the curve of the relation with the x-axis and the y-axis

GUC - Wniter 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

GUC - Wniter 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

GUC - Wniter  Find the x and y intercepts for the curves of the relations in examples A, B

GUC - Wniter The line intersects with the y-axis at y=1. The line intersects with the x-axis at

GUC - Wniter

GUC - Wniter The curve intersects with the y-axis twice at The curve intersects with the x-axis at x = -1

GUC - Wniter

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 xy =f(x) Domain Functions Range

 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 a 0,a 1,…,a n are constants called coefficients of the polynomial. n is called the degree of the polynomial is called the leading coefficient is called the absolute coefficient

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

 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

 The domain of a function y = f(x) is the set of values that the variable x can take.

 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