UNIVERSITY OF THE PHILIPPINES MANILA COLLEGE OF ARTS AND SCIENCES DEPARTMENT OF PHYSICAL SCIENCES AND MATHEMATICS MATHEMATICAL AND COMPUTING SCIENCES UNIT Course Code: Math 73 Course Title: Fundamentals of Analysis 1 Credit Units: 3 Lecture Unit(s): 2 units; 2 hrs/week Laboratory Unit(s): 1 unit; 3hrs/week Course Description: Lines and conics, functions and their graphs, limits and continuity, derivatives, application to simple differential equations, related rates problem, maxima and minima problems, curve sketching, antidifferentiation and the definite integral. Instructional Materials and References: 1. Leithold, L. The Calculus 7 2. Purcell, E. Calculus with Analytic Geometry 3. Anton, H., et al. Calculus: Early Transcendentals, 10th Edition Student Outcomes: 1. 2. 3. 4. Demonstrate application of knowledge of mathematics, science, and engineering in practical situations Exhibit moral, ethical and social responsibilities as a professional and as a Filipino citizen Can work both independently and with a group Communicate effectively in oral and written form Course Outcomes: At the end of the course, students shall be able to: 1. Understand the concept of limits (from informal, intuitive notion to precise mathematical definitions) and its applications; calculate and evaluate limits 2. Know the concept of the derivative of a function – its definition and applications, and the precise mathematical methods in differentiation 3. Discern the concept of antidifferentiation and integration, and understand the fundamental theorems of Calculus COURSE OUTCOMES Understand the concept of limits from informal, intuitive notion to precise mathematical definitions; calculate and evaluate limits TOPICS 1. Review of Functions a. Definition of a Function b. Domain and Range c. Type of Functions TIME FRAME 1.5 hours INTENDED LEARNING OUTCOMES TEACHING-LEARNING ACTIVITIES Explain class policies and university rules, course topics and requirements and the grading system Syllabi Discussion Discuss the main objective/s of the course, its importance as a prerequisite to higher Math subjects and its relevance to the Computer Science Education Exemplification in the Laboratory ASSESSMENT TASKS 1. Give supplementary problems as exercises and assignment Recall lecture on functions 2. Short quiz at the start of next meeting regarding Chapter 1 3. Boardwork and oral recitation in the laboratory Recall the notion of a function, domain and range, the different types of functions and their graphs 2. Lines and Circles 1 hour a. Slope of a line b. Forms of equations of a line c. Parallel and perpendicular lines d. Forms of equations of a circle Determine the important points and solve all relevant quantities of a line/circle given its equation in any form Lecture/Class Discussion 1. Give supplementary problems as exercises and assignment Exemplification in the Laboratory 2. Short quiz at the start of next meeting regarding Chapter 2 Determine the equation of a line and a circle given sufficient information and sketch its graph 3. Written exercises and oral recitation in the laboratory Determine if two lines are parallel or perpendicular 3. Conics 2 hours a. Parabola b. Ellipse c. Hyperbola 4. Limit of Functions a. Intuitive Notion b. Formal Definition (ɛ-δ) c. Limit Theorems d. One-sided limits e. Infinite limits, Limits at infinity and Infinite Limits at Infinity f. Asymptotes g. Curve Tracing Determine the important points and solve all relevant quantities of a conic given its equation in any form Lecture/Class Discussion Exemplification in the Laboratory 2. Boardwork and written exercises in the laboratory Determine the equation of a conic given sufficient information and sketch its graph. 3 hours Define the limit of a function Lecture/Class Discussion Evaluate limits using limit theorems Exemplification in the Laboratory Interpret limits geometrically Prove limits using the Formal Definition Determine the asymptotes of a function analytically 1. Give supplementary problems as exercises and assignment 1. Give supplementary problems as exercises and assignment 2. Short quiz at the start of next meeting regarding Chapter 4 3. Boardwork and written exercises in the laboratory Sketch the graph of a function using limits 5. Continuity 2 hours a. Definitions b. Types of Discontinuities c. Theorems on continuity Define the continuity of a function at a point, on an interval and on its domain Lecture/Class Discussion 1. Give supplementary problems as exercises and assignment Exemplification in the Laboratory Discuss the continuity of a given function, i.e., the intervals for which it is continuous and its points of discontinuity. 2. Short quiz at the start of next meeting regarding Chapter 5 3. Boardwork and reporting in the laboratory 4. Comprehensive Exam Know the concept of the derivative of a function – its definition and applications, and the precise mathematical methods in differentiation 6. Derivatives 4 hours a. Slope of a Tangent Line b. Definition of Derivative c. Differentiability of a Function d. Differentiability and Continuity e. Rules on Differentiation f. The Chain Rule g. Implicit Differentiation h. Higher Order Derivatives Define the derivative of a function and the derivative of a function at a point Lecture/Class Discussion 1. Give supplementary problems as exercises and assignment Exemplification in the Laboratory Find the derivative of a given algebraic function using the rules on differentiation and the chain rule 2. Short quiz at the start of next meeting regarding Chapter 6 3. Boardwork and oral recitation in the laboratory Discuss the differentiablity of a given function, i.e., the intervals for which it is differentiable and the points where the derivative DNE Interpret derivatives geometrically Contrast differentiability from continuity Extend the idea differentiation to curves which are not functions through implicit differentiation Calculate higher order derivatives 7. Applications of Derivatives a. Tangent and Normal Lines b. Rectilinear Motion b. Related rates d. Relative Extrema of a Function e. Absolute Extrema of a function f. Extreme Value Theorem g. Rolle’s and Mean Value Theorem h. Optimization Problems i. Curve Tracing 5 hours Determine the equations of tangent and normal lines to a curve using derivative Lecture/Class Discussion 1. Give supplementary problems as exercises and assignment Exemplification in the Laboratory Analyze a particle moving in a rectilinear motion 2. Short quiz at the start of next meeting regarding Chapter 7 Determine the relative and absolute extrema of a function on a given interval 3. Boardwork and reporting in the laboratory Solve related rates and optimization problems using derivatives 4. Comprehensive Exam Employ the idea of derivatives to sketch the graph of a function Discern the concept of antidifferentiation and integration, and understand the fundamental theorems of Calculus 8. Antidifferentiation 2 hours a. Antidifferentiation Formulas b. Integration by substitution c. Simple Differential Equations 9. The Definite Integral Determine the anti-derivative of a given function Lecture/Class Discussion Exemplification in the Laboratory 3 hours a. Summation Notation and Riemann Sum b. Area under a Curve c. Definition of the Definite Integral d. Properties of the Definite Integral e. The Mean Value Theorem for Integrals f. The Fundamental Theorems of Calculus g. Applications of the Definite Integral Evaluate antiderivatives using Integration by Substitution 2. Short quiz at the start of next meeting regarding Chapter 8 Solve simple differential equations using antidifferentiation 3. Boardwork and oral recitation in the laboratory Define the Definite Integral Lecture/Class Discussion Apply the properties of the Definite Integral, Mean Value Theorem and the Fundamental Theorems of Calculus to solve problems involving Definite Integrals Exemplification in the Laboratory Laboratory Grade = ¾ ( Average of 3 Laboratory Exams ) + ¼ ( Laboratory Exercises ) Lecture Grade = 2/3 [( 9/10 Average of 3 Lecture Exams ) + 1/10 (Non departmental items)] +1/3 (Final Exam) Final Grade = ½ ( Lecture Grade ) + ½ ( Laboratory Grade ) Grading Scale 87–89 1.5 84–86 1.75 80–83 2.0 2. Short quiz at the start of next meeting regarding Chapter 9 4. Comprehensive Exam Computation of Grade 90–92 1.25 1. Give supplementary problems as exercises and assignment 3. Boardwork and reporting in the laboratory Calculate the area of a plane region Grading System: ≥ 93 1.0 1. Give supplementary problems as exercises and assignment 75–79 2.25 70–74 2.5 65–69 2.75 60–64 3.0 55–59 4.0 ≤ 55 5.0
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