TEACHING GUIDE FOR SENIOR HIGH SCHOOL Basic Calculus

1 TEACHING GUIDE FOR SENIOR HIGH SCHOOL Basic Calculus CORE SUBJECT This TEACHING GUIDE was collaboratively developed and reviewed by educators from public and private schools, colleges, and universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Commission on Higher Education, K to 12 Transition Program Management Unit - SENIOR High SCHOOL Support Team at We value your feedback and on Higher Education in collaboration with the Philippine Normal UniversityThis TEACHING GUIDE by the Commission on Higher Education is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike International License.

2 This means you are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material. The licensor, CHED, cannot revoke these freedoms as long as you follow the license terms. However, under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. NonCommercial You may not use the material for commercial purposes. ShareAlike If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.

3 Printed in the Philippines by EC-TEC Commercial, No. 32 St. Louis Compound 7, Baesa, Quezon City, by the Commission on Higher Education, 2016 Chairperson: Patricia B. Licuanan, Commission on Higher Education K to 12 Transition Program Management Unit Office Address: 4th Floor, Commission on Higher Education, Garcia Ave., Diliman, Quezon City Telefax: (02) 441-1143 / E-mail Address: DEVELOPMENT TEAM Team Leader: Jose Maria P. Balmaceda, Writers: Carlene Perpetua P. Arceo, Richard S. Lemence, Oreste M. Ortega, Jr., Louie John D. Vallejo, Technical Editors: Jose Ernie C. Lope, Marian P. Roque, Copy Reader: Roderick B.

4 Lirios Cover Artists: Paolo Kurtis N. Tan, Renan U. Ortiz CONSULTANTS THIS PROJECT WAS DEVELOPED WITH THE PHILIPPINE NORMAL UNIVERSITY. University President: Ester B. Ogena, VP for Academics: Ma. Antoinette C. Montealegre, VP for University Relations & Advancement: Rosemarievic V. Diaz, Ma. Cynthia Rose B. Bautista, , CHED Bienvenido F. Nebres, , , Ateneo de Manila University Carmela C. Oracion, , Ateneo de Manila University Minella C. Alarcon, , CHED Gareth Price, Sheffield Hallam University Stuart Bevins, , Sheffield Hallam University SENIOR HIGH SCHOOL SUPPORT TEAM CHED K TO 12 TRANSITION PROGRAM MANAGEMENT UNIT Program Director: Karol Mark R.

5 Yee Lead for SENIOR High SCHOOL Support: Gerson M. Abesamis Lead for Policy Advocacy and Communications: Averill M. Pizarro Course Development Officers: Danie Son D. Gonzalvo, John Carlo P. Fernando Teacher Training Officers: Ma. Theresa C. Carlos, Mylene E. Dones Monitoring and Evaluation Officer: Robert Adrian N. Daulat Administrative Officers: Ma. Leana Paula B. Bato, Kevin Ross D. Nera, Allison A. Danao, Ayhen Loisse B. DalenaiContentsIntroduction .. ivDepEd Basic Calculus Curriculum GUIDE .. vii1 Limits and Continuity .. 1 Lesson 1: The Limit of a Function: Theorems and Examples .. 2 Topic : The Limit of a Function.

6 3 Topic : The Limit of a Function at c versus the Value of a Function at c .. 17 Topic : Illustration of Limit Theorems ..22 Topic : Limits of Polynomial, Rational, and Radical Functions .. 28 Lesson 2: Limits of Some Transcendental Functions and Some Indeterminate Forms .. 38 Topic : Limits of Exponential, Logarithmic, and Trigonometric Functions ..39 Topic : Some Special Limits .. 46 Lesson 3: Continuity of Functions .. 52 Topic : Continuity at a Point .. 53 Topic : Continuity on an Interval .. 58 Lesson 4: More on Continuity .. 64 Topic : Different Types of Discontinuities .. 65 Topic : The Intermediate Value and the Extreme Value Theorems.

7 75 Topic : Problems Involving Continuity .. 852 Derivatives .. 89 Lesson 5: The Derivative as the Slope of the Tangent Line .. 90 Topic : The Tangent Line to the Graph of a Function at a Point .. 91 Topic : The Equation of the Tangent Line ..100 Topic : The Definition of the Derivative .. 107 Lesson 6: Rules of Differentiation ..119 Topic : Differentiability Implies Continuity .. 120iiTopic : The Differentiation Rules and Examples Involving Algebraic, Exponential, and Trigonometric Functions .. 126 Lesson 7: Optimization ..141 Topic : Optimization using Calculus .. 142 Lesson 8: Higher-Order Derivatives and the Chain Rule.

8 156 Topic : Higher-Order Derivatives of Functions .. 157 Topic : The Chain Rule .. 162 Lesson 9: Implicit Differentiation .. 168 Topic : What is Implicit Differentiation? ..169 Lesson 10: Related Rates .. 180 Topic : Solutions to Problems Involving Related Rates .. 1813 Integration ..191 Lesson 11: Integration ..192 Topic : Illustration of an Antiderivative of a Function .. 193 Topic : Antiderivatives of Algebraic Functions .. 196 Topic : Antiderivatives of Functions Yielding Exponential Functions and Logarithmic Functions .. 199 Topic : Antiderivatives of Trigonometric Functions ..202 Lesson 12: Techniques of Antidifferentiation.

9 204 Topic : Antidifferentiation by Substitution and by Table of Integrals ..205 Lesson 13: Application of Antidifferentiation to Differential Equations .. 217 Topic : Separable Differential Equations .. 218 Lesson 14: Application of Differential Equations in Life Sciences .. 224 Topic : Situational Problems Involving Growth and Decay Problems ..225 Lesson 15: Riemann Sums and the Definite Integral ..237 Topic : Approximation of Area using Riemann Sums .. 238 Topic : The Formal Definition of the Definite Integral .. 253 Lesson 16: The Fundamental Theorem of Calculus .. 268 Topic : Illustration of the Fundamental Theorem of Calculus .

10 269 Topic : Computation of Definite Integrals using the Fundamental Theorem of Calculus .. 273iiiLesson 17: Integration Technique: The Substitution Rule for Definite Integrals ..280 Topic : Illustration of the Substitution Rule for Definite Integrals .. 281 Lesson 18: Application of Definite Integrals in the Computation of Plane Areas ..292 Topic : Areas of Plane Regions Using Definite Integrals .. 293 Topic : Application of Definite Integrals: Word Problems .. 304 Biographical Notes .. 309ivIntroductionAs the Commission supports DepEd s implementation of SENIOR High SCHOOL (SHS), it upholds the vision and mission of the K to 12 program, stated in Section 2 of Republic Act 10533, or the Enhanced Basic Education Act of 2013, that every graduate of Basic education be an empowered individual, through a program rooted competence to engage in work and be productive, the ability to coexist in fruitful harmony with local and global communities, the capability to engage in creative and critical thinking, and the capacity and willingness to transform others and oneself.