UNIVERSITY OF THE PHILIPPINES MANILA COLLEGE OF ARTS AND SCIENCES DEPARTMENT OF PHYSICAL SCIENCES AND MATHEMATICS MATHEMATICAL AND COMPUTING SCIENCES UNIT Course Code: Mathematics 74 Course Title: Fundamentals of Analysis II Credit Units: 3 Lecture Unit(s): 2 units; 2 hrs/week Laboratory Unit(s): 1 unit; 3hrs/week Course Description: Derivatives and integrals of transcendental functions, techniques of integration, applications of integration to area of a plane region, volumes and solids of revolution, arclength, area of surfaces of revolution, center of mass, and polar coordinates Instructional Materials and References: 1. Leithold, L. The Calculus 7 Harper Collins College Publishers 1996 2. Anton, H. et al. Calculus Early Transcendentals Hoboken NJ: Wiley 2009 Ninth 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 Mapping of Course Outcomes vis-à-vis Program Outcomes 1. 2. 3. 4. Understand the extension of the concepts of limits, derivatives, and integrals to transcendental functions Handle and evaluate integrals using additional integration techniques Apply the principle of integral evaluation in solving physical problems Discuss a new kind of coordinate system and relate the rectangular coordinate system COURSE OUTCOMES Understand the extension of the concepts of limits, derivatives, and integrals to transcendental functions TOPICS Limits of Trigonometric Functions Derivatives and Integrals of Trigonometric Functions Derivatives of Exponential and Logarithmic Functions Integrals Yielding Exponential and TIME FRAME 7.5 hrs lecture 12 hrs laboratory INTENDED LEARNING OUTCOMES TEACHING-LEARNING ACTIVITIES ASSESSMENT TASKS Calculate the limit of a transcendental function at a point numerically using appropriate techniques and manipulation Lecture/Class Discussion Lecture Exam Laboratory Exercises Quiz Understand how hyperbolic functions and inverse hyperbolic functions are defined from exponential and logarithmic functions, respectively Computer-aided graphing and discussions Laboratory Exam Laboratory Exercises Problem set discussion Boardworks 2|P a g e F u n d a m e n t a l s o f A n a l y s i s Logarithmic Functions Obtain fundamental hyperbolic identities Derivatives of Inverse Trigonometric Functions Find the derivatives of exponential and logarithmic functions, trigonometric and inverse trigonometric functions, and hyperbolic and inverse hyperbolic functions in implicit and/or explicit forms Integrals Yielding Inverse Trigonometric Functions Derivatives and Integrals of Hyperbolic Functions Boardwork discussion by the students II OBE SYLLABUS Problem Sets Homework Find the general antiderivatives of exponential and logarithmic functions, trigonometric and inverse trigonometric functions, and hyperbolic and inverse hyperbolic functions Derivatives and Integrals of Inverse Hyperbolic Functions Evaluate a definite integral with a transcendental function as integrand using an appropriate antiderivative Integration by Parts Integration of Powers of Trigonometric Functions Integration by Trigonometric Substitution 5.5 hrs lecture 9 hrs laboratory Find the general antiderivative of a product of functions using a simple intergration by parts, repeated integration by parts, or tabular integration by parts Recognize and implement appropriate techniques to find the general antiderivative of product of trigonometric functions Lecture/Class Discussion Lecture Exam Laboratory Exercises Quiz Computer-aided graphing and discussions Laboratory Exam Laboratory Exercises Problem set discussion Integration by Partial Fractions Miscellaneous Techniques Boardworks Devise and apply a trigonometric substitution in finding the general antiderivative of certain forms of functions Boardwork discussion by the students Problem Sets Homework Evaluate the integral of a rational function by decomposing the rational integrand into partial fractions Apply other substitution techniques to transform the integral into a form easier to integrate 3|P a g e F u n d a m e n t a l s o f A n a l y s i s Apply the principle of integral evaluation in solving physical problems Area of a Plane Region 5 hrs lecture Interpret the area enclosed between curves as a definite integral and compute its value 9 hrs laboratory Find the length of arc of a curve defined by the graph of a function using a definite integral Volumes of Solids of Revolution Length of Arc Surface Area of a Solid of Revolution Interpret the volume of a solid defined by the graph of a function revolved about an axis as a sum of volumes of disks, washers, or cylindrical shells; convert such sum to a definite integral, and compute its value Center of Mass of a Plane Region II OBE SYLLABUS Lecture/Class Discussion Lecture Exam Laboratory Exercises Quiz Computer-aided graphing and discussions Laboratory Exam Laboratory Exercises Problem set discussion Boardworks Boardwork discussion by the students Express the surface area of a solid defined by the graph of a function revolved about an axis as a sum of areas of rings; convert such sum to a definite integral, and compute its value Problem Sets Homework Use definite integration to find the center of mass of a thin rod of uniform and non-uniform density Apply definite integration to find the center of mass of region of uniform area density and to find the centroid of a plane region Discuss a new kind of coordinate system and relate the rectangular coordinate system Polar Coordinate System 3 hrs lecture Equations and Graphs in Polar Form Area in Polar Coordinates 4.5 hrs laboratory Relate the Polar and the Rectangular Cartesian Coordinates, converting from one form to another Lecture/Class Discussion Lecture Exam Laboratory Exercises Quiz Identify and sketch the graphs of certain equations in polar form Computer-aided graphing and discussions Laboratory Exam Apply definite integration in finding the area of a plane region bounded by polar curves Problem set discussion Laboratory Exercises Boardworks Boardwork discussion by the students Problem Sets Homework 4|P a g e F u n d a m e n t a l s o f A n a l y s i s Grading System: Final Lecture Grade = [2/3*(90% Average of 3 Lecture Exams)+ 10% (Homework & Quiz)] + [1/3* Final Exam] Final Laboratory Grade = [(3/4) Average of 3 Laboratory Exams + 1/4 (Laboratory Exercises & Boardworks)] FINAL GRADE = (50% Final Lecture Grade) + (50% Final Laboratory Grade) GRADING SCALE ≥ 93 1.0 90–92 1.25 87–89 1.5 84–86 1.75 80–83 2.0 75–79 2.25 70–74 2.5 65–69 2.75 60–64 3.0 55–59 4.0 ≤ 55 5.0 II OBE SYLLABUS
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