Course Title | ## Calculus for the Sciences | ||

Course Code | ## MH1802 | ||

Offered | Study Year 1, Semester 1 | ||

Course Coordinator | Yang Bo (Asst Prof) | YANG.BO@ntu.edu.sg | |

Pre-requisites | None | ||

AU | 4 | ||

Contact hours | Lectures: 39, Tutorials: 13 | ||

Approved for delivery from | AY 2022/23 semester 1 | ||

Last revised | 27 Nov 2022, 19:26 |

This course aims to equip you with

• mathematical knowledge and analytical skills so that they are able to apply techniques of calculus (along with your existing mathematical skills) to solve scientific problems whenever applicable;

• mathematical reading skills so that you can read and understand related mathematical content in the basic and popular scientific and engineering literature; and

• mathematical communication skills so that you can effectively and rigorously present your mathematical ideas to mathematicians, scientists and engineers.

Upon successfully completing this course, you should be able to:

- Explain the common terms used in the discussion of different types of numbers and functions
- Cite examples of the applications of polynomial, trigonometric, logarithmic and exponential functions in Science and conversely identify the appropriate functions that best describe given scientific phenomena or experiments
- Apply basic algebraic (including Binomial theorem), trigonometric, inverse trigonometric, logarithmic and exponential identities to prove identities in general
- Apply basic concepts in complex numbers (including Euler’s formula and de Moivre’s theorem) to solve related problems
- Explain the meaning of different types of limits of a function and evaluate them
- Explain the concept of continuity of a function and apply it to estimate root of a function (Intermediate Value Theorem).
- Apply the concept of limits of a function to solve related problems and derive formulas for the derivative of algebraic, trigonometric, inverse trigonometric, logarithmic and exponential functions
- Apply the appropriate techniques to solve derivatives and higher order derivatives in general
- Provide a graphical interpretation of derivatives, classify critical points and apply the appropriate techniques for curve sketching
- Apply the appropriate techniques in solving single variable optimization and mean value problems
- Compute and apply Taylor series of functions such as algebraic, trigonometric, inverse trigonometric, logarithmic and exponential functions
- Cite examples of the applications of derivatives in Science (such as in kinematics and chemical kinetics) and conversely formulate descriptions of relevant scientific phenomena or experiments using derivatives
- Apply derivatives to numerical approximations (such as Newton’s method, linear approximations and differentials)
- Apply differentiation to L’hospital rule
- Compute numerical derivatives
- Apply the concept of Riemann sum, fundamental theorem of calculus and use them to determine integrals
- Apply the appropriate techniques (such as substitution, integration by parts, using partial fractions, using complex numbers) to solve integrals
- Apply integration to compute area under graph, arc lengths, volume of revolution, mean values
- Cite examples of the applications of integrals in Science (such as in kinematics, determining center of mass) and conversely formulate descriptions of relevant scientific phenomena or experiments using integrals
- Apply numerical methods to make approximation of integrals
- Perform simple classification of differential equations
- Apply the appropriate techniques (such as separation of variables, use of integrating factors) to solve basic first order ordinary differential equations
- Identify and apply techniques to solve second order linear ordinary differential equations with constant coefficients (homogeneous and nonhomogeneous)
- Cite examples of the applications of differential equations in Science (such as in mechanics, chemical kinetics) and conversely formulate descriptions of relevant scientific phenomena or experiments using differential equations
- Recognise that power series can be used in solving of general differential equations

Types of Numbers; Functions and Graphs

Algebraic, trigonometric, logarithmic and exponential functions and identities

Basic Complex Numbers

Limits & Continuity

Derivatives & Techniques of Differentiation

Applications of Differentiation; Numerical Approximation of differentiation

Indefinite Integrals and Definite Integral, Fundamental Theorem of Calculus, Techniques of Integration 1

Techniques of Integration 2; Applications of Integration

Applications of Integration in Science; Numerical Approximation of integration

Introduction to Differential Equations; First Order Ordinary Differential Equations

Second Order Linear Differential Equations with constant coefficients

Power Series Method

Revision

Component | Course ILOs tested | SPMS-MAS Graduate Attributes tested | Weighting | Team / Individual | Assessment Rubrics |
---|---|---|---|---|---|

Continuous Assessment | |||||

Tutorials | |||||

Homework | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | 1. a, b, c, d | 20 | individual | See Appendix for rubric |

Mid-semester Quiz | |||||

Common Test 1 | 1, 2, 3, 4, 5, 6, 7, 8 | 1. a, b, c3. a | 20 | individual | See Appendix for rubric |

Common Test 2 | 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | 1. a, b, c3. a | 20 | individual | See Appendix for rubric |

Examination (2 hours) | |||||

Final Examination | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | 1. a, b, c3. a | 40 | individual | See Appendix for rubric |

Total | 100% |

These are the relevant SPMS-MAS Graduate Attributes.

## 1. Competence

a. Independently process and interpret mathematical theories and methodologies, and apply them to solve problems

b. Formulate mathematical statements precisely using rigorous mathematical language

c. Discover patterns by abstraction from examples

d. Use computer technology to solve problems, and to communicate mathematical ideas

## 3. Communication

a. Present mathematics ideas logically and coherently at the appropriate level for the intended audience

Formative feedback is given through discussion within tutorial lessons and lectures. It is also given after each common test on the common mistakes and level of difficulty of the problems.

Lectures (39 hours) | Train you to be independent learners who are able to derive ideas/concepts from first principles and take ownership of your own learning. Help you understand the motivation behind mathematical theorems, definitions and formulae. Develop the train of thoughts in problem solving and presentation skills in presenting mathematical ideas and solutions. |

Tutorials (13 hours) | Train you to be independent learners who are able to derive ideas/concepts from first principles and take ownership of your own learning. Help you understand the motivation behind mathematical theorems, definitions and formulae. Develop the train of thoughts in problem solving and presentation skills in presenting mathematical ideas and solutions. |

Text Books:

Thomas, GB Jr., Weir MD, Hass J and Giordano FR , Thomas’ Calculus, Pearson-Addison-Wesley, 13th Edition in SI Units , 2016. ISBN-13 978-1-292-08979-9.

James Stewart: Calculus (International Student Edition, Metric Version), 7th Edition, Thomson, Brooks/Cole, Cengage Learning. 2016. ISBN-13: 978-0538497817.

Other References:

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, Cambridge University Press, 3rd edition (March 13, 2006). ISBN 0521861535.Serge Lang, A First Course in Calculus, Addison-Wesley Pub Co, 3rd edition (January 1973). ISBN-13: 978-0201042238.

Donald Trim, Calculus for Engineering, Prentice Hall Canada, 2nd edition (March 7, 2001). ISBN-13: 978-0130856036.

Tom M Apostol, Calculus (Vol 1 and 2) Wiley 2nd edition (2016 and 2007). ISBN-13: 978-0471000051 (Vol 1), ISBN-13: 978-8126515202 (Vol 2).

Absence Due to Medical or Other Reasons

If you are sick and unable to attend your class (particularly the mid-terms), you have to:

1. Send an email to the instructor regarding the absence.

2. Submit the original Medical Certificate* to administrator.* The medical certificate mentioned above should be issued in Singapore by a medical practitioner registered with the Singapore Medical Association.

Good academic work depends on honesty and ethical behaviour. The quality of your work as a student relies on adhering to the principles of academic integrity and to the NTU Honour Code, a set of values shared by the whole university community. Truth, Trust and Justice are at the core of NTU’s shared values.

As a student, it is important that you recognize your responsibilities in understanding and applying the principles of academic integrity in all the work you do at NTU. Not knowing what is involved in maintaining academic integrity does not excuse academic dishonesty. You need to actively equip yourself with strategies to avoid all forms of academic dishonesty, including plagiarism, academic fraud, collusion and cheating. If you are uncertain of the definitions of any of these terms, you should go to the Academic Integrity website for more information. Consult your instructor(s) if you need any clarification about the requirements of academic integrity in the course.

Instructor | Office Location | Phone | |
---|---|---|---|

Yang Bo (Asst Prof) | SPMS-PAP-04-09 | YANG.BO@ntu.edu.sg |

Week | Topic | Course ILO | Readings/ Activities |
---|---|---|---|

1 | Types of Numbers; Functions and Graphs | 1, 2 | Reading assigned pre-lecture notes and relevant book chapters |

2 | Algebraic, trigonometric, logarithmic and exponential functions and identities | 2, 3, 4 | Reading assigned pre-lecture notes and relevant book chapters |

3 | Basic Complex Numbers | 5 | Reading assigned pre-lecture notes and relevant book chapters |

4 | Limits & Continuity | 6, 7, 8 | Reading assigned pre-lecture notes and relevant book chapters |

5 | Derivatives & Techniques of Differentiation | 8, 9 | Reading assigned pre-lecture notes and relevant book chapters; |

6 | Applications of Differentiation; Numerical Approximation of differentiation | 9, 10, 11, 12 | Reading assigned pre-lecture notes and relevant book chapters |

7 | Indefinite Integrals and Definite Integral, Fundamental Theorem of Calculus, Techniques of Integration 1 | 13, 14 | Reading assigned pre-lecture notes and relevant book chapters |

8 | Techniques of Integration 2; Applications of Integration | 15 | Reading assigned pre-lecture notes and relevant book chapters |

9 | Applications of Integration in Science; Numerical Approximation of integration | 16, 17, 18 | Reading assigned pre-lecture notes and relevant book chapters; Second Midterm test. |

10 | Introduction to Differential Equations; First Order Ordinary Differential Equations | 19, 20 | Reading assigned pre-lecture notes and relevant book chapters |

11 | Second Order Linear Differential Equations with constant coefficients | 21, 22 | Reading assigned pre-lecture notes and relevant book chapters |

12 | Power Series Method | 23, 24 | Reading assigned pre-lecture notes and relevant book chapters |

13 | Revision | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | Reading assigned pre-lecture notes and relevant book chapters |

The students complete homeworks (weekly and biweekly) either on their own or by forming self-organised study groups.

Most of the homework questions will be MCQs from MyMath Lab or short answers. The answers will be graded by MyMath Lab.