Dr. Tayeb Brahimi Office: ENG Chap Preview: Properties of Functions Chap 1: Limits and Continuity MATH 127: Calculus MATH 127: Calculus

1 Math /tb MATH 127- Mathematics for Engineers 1  Review the Properties of Functions  Introduce the concepts of Limit & Continuity  Introduce the concepts of Derivatives & integrals  Apply of these concepts to a wide range of practical problems including the equations of motion, related rates, and curve sketching  Have a working knowledge of the fundamental definitions and theorems of calculus and be literate in the language and notation of calculus. Math 127 Course Objectives Prescribed Textbooks: Calculus (10th ed.) Larson and Edwards Brooks/Cole, CENGAGE, 2014, ISBN-13:

1 Math 127 Course Learning Outcomes 3 /tb MATH 127- Mathematics for Engineers 1 Course Learning Outcomes IQRA Core Values Learning Domains ABETCLOs PLO – ECE ABET Program Indicators ITQAN Knowledge (a) an ability to apply knowledge of mathematics, science, and engineering – (KNOWLEDGE). CLO1. Recognize the properties of a function from an algebraic, numerical, and graphical perspective ECE – K1 a3. Express concepts in Mathematical forms or equations ITQAN Cognitive skills a) an ability to apply knowledge of mathematics, science, and engineering – (COGNITIVE). CLO2. Evaluate the limits and test the continuity of a function graphically, numerically, and algebraically. ECE – Ca4. Apply analytical, graphical or numerical methods CLO3. Apply the basic rules of differentiation to various types of functions CLO4. Sketch graphs using first and second derivatives and solve problems involving rate of change. CLO5. Evaluate integrals by using different integration techniques. STEWARDSHIP Interpersonal skill & responsibility (d) an ability to function on multidisciplinary teams CLO6. Demonstrate collaborative learning through assignments and project participation ECE – IS1 d1. Participate in team meetings by providing expertise, contributing to the project workload, and showing willingness to work with others

1 Math 127 Preparation for Calculus P Copyright © Cengage Learning. All rights reserved. MATH 127- Mathematics for Engineers 1 4 /tb

1 Math 127 Math 127 Chapter Preview  P1:Sketch a graph and the Eight Basic Functions  P2: Intercepts of a graph and the points of intersection of two graphs  P3: Equation of a Line and Slopes  P4: Test of Symmetry, Odd and Even Functions  P5: Composite Fcts & Evaluation of Fcts  P6: Domain and Range of Functions 5 /tb MATH 127- Mathematics for Engineers 1

1 Math 127 Sketch the graph by point plotting: 3x + y = 7 & y = x 2 – 2. Solution: Create a table of values, then plot and connect the points. P1. Sketching Graphs MATH 127- Mathematics for Engineers 1 6 /tb Figure P.2 Figure P.1

1 Math 127 Figure P.27 shows the graphs of eight basic functions. Figure P.27 The Eight Basic Functions MATH 127- Mathematics for Engineers 1 7 /tb Figure P.27

1 Math 127 Sketching a Graph by Transformation Some families of graphs have the same basic shape. Example, compare the graph of y = x 2 with the graphs of the four other quadratic functions shown in Figure P.28. Figure P.28 MATH 127- Mathematics for Engineers 1 8 /tb

1 Math 127 P2. Intercepts of a Graph Find the x-and y-intercepts of the graph of y = x 3 – 4x. Solution: To find the x-intercepts, let y be zero and solve for x. Let y be zero. Factor. Solve for x. To find the y-intercepts, let x be zero. Which produces y = 0. (0, 0). y-intercept MATH 127- Mathematics for Engineers 1 9 /tb

1 Math 127 Figure P.10 Find all points of intersection of the graphs of x 2 – y = 3 and x – y = 1. Solution: Begin by sketching the graphs of both equations in the same rectangular coordinate system, as shown in Figure P.10. Finding Points of Intersection MATH 127- Mathematics for Engineers 1 10 /tb

1 Math 127 From the figure, it appears that the graphs have two points of intersection. You can find these two points as follows. Solution Solve first equation for y. Solve second equation for y. Equate y-values. Write in general form. Factor. Solve for x. cont’d (2, 1) and (–1, –2). Points of intersection MATH 127- Mathematics for Engineers 1 11 /tb

1 Math /tb MATH 127- Mathematics for Engineers 1 P3. Equation of a Line and Slopes

1 Math 127 Figure P.13 shows four lines: one has a positive slope, one has a slope of zero, one has a negative slope, and one has an “undefined” slope. Figure P.13 The Slope of a Line MATH 127- Mathematics for Engineers 1 13 /tb

1 Math 127 Finding an Equation of a Line Find an equation of the line that has a slope of 3 and passes through the point (1, –2). Then sketch the line. Solution: MATH 127- Mathematics for Engineers 1 14 /tb

1 Math 127 Parallel and Perpendicular Lines The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular, as shown in Figure P.19. Figure P.19 MATH 127- Mathematics for Engineers 1 15 /tb

1 Math 127 Finding Parallel and Perpendicular Lines Find the general forms of the equations of the lines that pass through the point (2, –1) and are (a)parallel to the line 2 x – 3 y = 5 -3 y = -2 x + 5 y = 2 x /3 + 5/3  m = 2/3 The line parallel to y = 2 x /3 + 5/3 will have the same slope m = 2/3 (b) perpendicular to the line 2 x – 3 y = 5 y = 2 x /3 + 5/3  m = 2/3 The line parallel to y = 2 x /3 + 5/3 will have m = -3/2 MATH 127- Mathematics for Engineers 1 16 /tb

1 Math 127 Figure P.7(a) 1.Symmetric with respect to the y-axis P4. Symmetry, Even, & Odd Functions Symmetric with respect to the x-axis Symmetric with respect to the origin MATH 127- Mathematics for Engineers 1 17 /tb

1 Math /tb MATH 127- Mathematics for Engineers 1 Examples: Symmetry

1 Math 127 Odd and Even Functions MATH 127- Mathematics for Engineers 1 19 /tb Test the graph of y = 2x 3 – x for symmetry with respect to the y-axis. Solution: Write original equation. Replace x by –x. Simplify. It is not an equivalent equation. The function is not symmetric with respect to y-axis

1 Math 127 Whether the graph of eventually rises or falls can be determined by the function’s degree (odd or even) and by the leading coefficient a n, as indicated in Figure P.29. Figure P.29 Polynomial Functions MATH 127- Mathematics for Engineers 1 20 /tb

1 Math 127 P5. Evaluating Fcts & Composite Fcts For the function f defined by f (x) = x 2 + 7, evaluate each expression. Solution: 21 /tb MATH 127- Mathematics for Engineers 1

1 Math 127 Two functions can be combined in various ways to create new functions. For example, given you can form the functions shown. You can combine two functions in yet another way, called composition. The resulting function is called a composite function. Combinations of Functions MATH 127- Mathematics for Engineers 1 22 /tb

1 Math 127 Finding Composite Functions For f (x) = 2x – 3 and g(x) = cos x, find each composite function a. f ◦ g b. g ◦ f 23 /tb MATH 127- Mathematics for Engineers 1 = ?

1 Math 127 P6. Finding the Domain and Range of a Function The domain of the function is the set of all x-values for which x – 1 ≥ 0, which is the interval [1, ). To find the range, observe that is never negative. Figure P.23(a) So, the range is the interval [0, ), as indicated in Figure P.23(a). 24 /tb MATH 127- Mathematics for Engineers 1

Dr. Tayeb Brahimi Office: CoE Fall 2017 Properties of Functions Practice on Chap Preview MATH 127: Calculus MATH 127: Calculus

1 Math /tb MATH 127- Mathematics for Engineers 1 Exercises #1: P1 - Sketch Graphs

1 Math /tb MATH 127- Mathematics for Engineers 1 Exercises #2: P1, P2 & P3: Graphs, Intercepts Equation of a line, & Slopes

1 Math /tb MATH 127- Mathematics for Engineers 1 2. Exercises #3: P1-P2-P3: Intercepts and Equation of a line 1. 3.

1 Math /tb MATH 127- Mathematics for Engineers 1 Exercises #4: P4: Even & Odd Functions

1 Math /tb MATH 127- Mathematics for Engineers 1 Two functions can be combined in various ways to create new functions. For example, given Evaluate each composite value 1. Find: Exercises #5: P5: Combination and Composite Functions

1 Math /tb MATH 127- Mathematics for Engineers 1 Exercises #6: P6: Domain and Range Graphically

1 Math /tb MATH 127- Mathematics for Engineers 1 Exercises #7: Properties of Functions

1 Math /tb MATH 127- Mathematics for Engineers 1 g) Exercises #8: P1-P5-P6: Properties of Functions Evaluate: f(-1) = f(2) = f(0) =

Dr. Tayeb Brahimi Office: CoE Fall 2017 Next Chapter Chap 1: Limits and Continuity MATH 127: Calculus MATH 127: Calculus

1 Math 127 Chapter 1 35 /tb MATH 127- Mathematics for Engineers 1