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Lecture 1

Lecture 1. Functions. Instructor and Textbooks. 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:

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Lecture 1

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  1. Lecture 1 Functions

  2. Instructor and Textbooks 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), 7th edition, by Larson and Edwards • Lecture notes (presentations).

  3. Course outlines

  4. Course Assessment 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. TheCartesianplane 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. 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.

  6. 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

  7. 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, GUC - Wniter 2009

  8. Sketching Relations in the Cartesian Plane • Pick enough number of values of one variable (x or y). GUC - Wniter 2009

  9. 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). 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). • Make a table for these ordered pairs of points. 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. • Plot these points. 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. • Make the sketch by joining between the points. GUC - Wniter 2009

  13. Example A • 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 2009

  14. Example 2 • 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 2009

  15. Example 2 • 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 2009

  16. We get the table GUC - Wniter 2009

  17. Plot the points (-2,3), (0,1), (2,5)

  18. We join the points to get the sketch GUC - Wniter 2009

  19. Example 3 • Sketch the relation y2 –x=1 GUC - Wniter 2009

  20. Example 3 • Sketch the relation y2 –x=1 • This is easier to be written as: x = y2-1 GUC - Wniter 2009

  21. 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 GUC - Wniter 2009

  22. We get the table GUC - Wniter 2009

  23. Plot the points (8,-3), (-1,0), (15,4) GUC - Wniter 2009

  24. We join the points to get the sketch GUC - Wniter 2009

  25. More Examples Plot • y = x2, y = x4 • y = x3, y = x5 GUC - Wniter 2009

  26. 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 GUC - Wniter 2009

  27. 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 GUC - Wniter 2009

  28. 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 GUC - Wniter 2009

  29. Example 4 • Find the x and y intercepts for the curves of the relations in examples A, B GUC - Wniter 2009

  30. Solution 3: y = 2x + 1 The line intersects with the y-axis at y=1. The line intersects with the x-axis at GUC - Wniter 2009

  31. GUC - Wniter 2009

  32. Solution 4: y2 –x=1 The curve intersects with the y-axis twice at The curve intersects with the x-axis at x = -1 GUC - Wniter 2009

  33. GUC - Wniter 2009

  34. f x y =f(x) Functions 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) Domain Range

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

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

  37. Special polynomialsThe zero degree polynomial(the constant function)

  38. Some basic polynomials

  39. 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

  40. Rational Function Definition

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

  42. 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

  43. Domain of a Rational Function

  44. to find Domain of a Rational function

  45. Example 7

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