1. Functions

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

3. Math 101 Basic functions Limits and continuity Derivative and its applications Function of several variables Sequences and series

4. 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.

5. 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.

7. 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

8. GUC - Wniter 2009 8  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,

9. GUC - Wniter 2009 9  Pick enough number of values of one variable (x or y).

10. GUC - Wniter 2009 10  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).

11. GUC - Wniter 2009 11  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.

12. GUC - Wniter 2009 12  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.

13. GUC - Wniter 2009 13  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.

14. GUC - Wniter 2009 14  Sketch the relation y = 2x + 1 Solution  Here it is easier to take x as independent variable and calculate the corresponding values of y

15. GUC - Wniter 2009 15  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

16. GUC - Wniter 2009 16  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.

17. GUC - Wniter 2009 17

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20. GUC - Wniter 2009 20  Sketch the relation y 2 –x=1

21. GUC - Wniter 2009 21  Sketch the relation y 2 –x=1  This is easier to be written as: x = y 2 -1

22. GUC - Wniter 2009 22  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

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25. GUC - Wniter 2009 25

26. GUC - Wniter 2009 26 Plot  y = x 2, y = x 4  y = x 3, y = x 5

27. GUC - Wniter 2009 27 These simply give the intersections of the curve of the relation with the x-axis and the y-axis

28. GUC - Wniter 2009 28 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

29. GUC - Wniter 2009 29 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

30. GUC - Wniter 2009 30  Find the x and y intercepts for the curves of the relations in examples A, B

31. GUC - Wniter 2009 31 The line intersects with the y-axis at y=1. The line intersects with the x-axis at

32. GUC - Wniter 2009 32

33. GUC - Wniter 2009 33 The curve intersects with the y-axis twice at The curve intersects with the x-axis at x = -1

34. GUC - Wniter 2009 34

35. 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

37.  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

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

41.  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

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

44.  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