Engineering Mathematics - Calculus
ePrep Course for University
Also Available at SF@NS LXP

textbook used in engineering maths - calculus eprep course
Engineering Mathematics – Calculus eprep course is one of the ten specially designed e_prep courses by NTU to help NSF, NSmen, and others to better prepare themselves for their further studies, whether in the universities in Singapore or overseas. 
This Engineering Mathematics – Calculus eprep course is developed in collaboration with the book publishers, Cengage.  In addition to providing the latest 9th Edition of the popular textbook “Calculus” by James Stewart at no additional cost, this Engineering Mathematics – Calculus eprep course also comes with excellent learning materials provided by the publishers on calculus and other branches of mathematics.  
There are also lots of materials on other subjects such as physics, mechanics, engineering economy, economics, biotechnology, life science, business finance, corporate finance, Python programming, discrete mathematics, etc., so that the students not only get to build up a strong foundation on mathematics, they also get to strengthen their knowledge on many other subjects as well.  Samples of materials provided can be found below.  Most of these materials can be downloaded for later studies. 
A retired NTU professor acts as the personal tutor to all students taking this Engineering Mathematics – Calculus eprep course.  He can be reached via email or WhatsApp messaging.  Students are free to consult him, not only during the duration of the course, but until they enter universities, and even after they have started their university studies.
Please note that this course, as well as the other ePrep courses provided by NTU, is now also available at SF@NS LXP, the SkillsFuture@NationalService Learning eXperience Platform.

Engineering Mathematics – Calculus ePrep Learning Contents

Due to the constraints of time, only the first ten chapters are compulsory for certification purposes, but materials and support for the remaining seven chapters are also available.

I. Compulsory Chapters

Chapter 1: Functions and Limits

1.1: Four Ways to Represent a Function 

1.2: Mathematical Models: A Catalog of Essential Functions

1.3: New Functions from Old Functions 

1.4: The Tangent and Velocity Problems 

1.5: The Limit of a Function 

1.6: Calculating Limits Using the Limit Laws 

1.7: The Precise Definition of a Limit 

1.8: Continuity

Chapter 2: Derivatives

2.1: Derivatives and Rates of Change 

2.2: The Derivative as a Function 

2.3: Differentiation Formulas 

2.4: Derivatives of Trigonometric Functions 

2.5: The Chain Rule

2.6: Implicit Differentiation 

2.7: Rates of Change in the Natural and Social Sciences 

2.8: Related Rates

2.9: Linear Approximations and Differentials

Chapter 3: Applications of Differentiation

3.1: Maximum and Minimum Values 

3.2: The Mean Value Theorem 

3.3: How Derivatives Affect the Shape of a Graph 

3.4: Limits at Infinity; Horizontal Asymptotes 

3.5: Summary of Curve Sketching 

3.6: Graphing with Calculus and Calculators 

3.7: Optimization Problems 

3.8: Newton’s Method 

3.9: Antiderivatives 

Chapter 4: Integrals

4.1: Areas and Distances 

4.2: The Definite Integral 

4.3: The Fundamental Theorem of Calculus 

4.4: Indefinite Integrals and the Net Change Theorem 

4.5: The Substitution Rule

Chapter 5: Applications of Integration

5.1: Areas Between Curves 

5.2: Volumes 

5.3: Volumes by Cylindrical Shells

5.4: Work 

5.5: Average Value of a Function 

Chapter 6: Inverse Functions

6.1: Inverse Functions 

6.2: Exponential Functions and Their Derivatives

6.2*: The Natural Logarithmic Function 

6.3: Logarithmic Functions 

6.3*: The Natural Exponential Function 

6.4: Derivatives of Logarithmic Functions 

6.4*: General Logarithmic and Exponential Functions 

6.5: Exponential Growth and Decay 

6.6: Inverse Trigonometric Functions 

6.7: Hyperbolic Functions 

6.8: Indeterminate Forms and l’Hospital’s Rule 

Chapter 7: Techniques of Integration

7.1: Integration by Parts 

7.2: Trigonometric Integrals 

7.3: Trigonometric Substitution 

7.4: Integration of Rational Functions by Partial Fractions 

7.5: Strategy for Integration 

7.6: Integration Using Tables and Computer Algebra Systems 

7.7: Approximate Integration 

7.8: Improper Integrals 

Chapter 8: Further Applications of Integration

8.1: Arc Length

8.2: Area of a Surface of Revolution 

8.3: Applications to Physics and Engineering 

8.4: Applications to Economics and Biology 

8.5: Probability

Chapter 9: Differential Equations

9.1: Modeling with Differential Equations 

9.2: Direction Fields and Euler’s Method 

9.3: Separable Equations 

9.4: Models for Population Growth 

9.5: Linear Equations 

9.6: Predator-Prey Systems 

Chapter 10: Parametric Equations and Polar Coordinates

10.1: Curves Defined by Parametric Equations 

10.2: Calculus with Parametric Curves 

10.3: Polar Coordinates

10.4: Areas and Lengths in Polar Coordinates 

10.5: Conic Sections 

10.6: Conic Sections in Polar Coordinates 

II. Optional Chapters

Chapter 11: Infinite Sequences and Series

11.1: Sequences 

11.2: Series 

11.3: The Integral Test and Estimates of Sums 

11.4: The Comparison Tests 

11.5: Alternating Series 

11.6: Absolute Convergence and the Ratio and Root Tests 

11.7: Strategy for Testing Series 

11.8: Power Series 

11.9: Representations of Functions as Power Series 

11.10: Taylor and Maclaurin Series 

11.11: Applications of Taylor Polynomials 

Chapter 12: Vectors and the Geometry of Space

12.1: Three-Dimensional Coordinate Systems 

12.2: Vectors 

12.3: The Dot Product 

12.4: The Cross Product 

12.5: Equations of Lines and Planes 

12.6: Cylinders and Quadric Surfaces 

Chapter 13: Vector Functions

13.1: Vector Functions and Space Curves 

13.2: Derivatives and Integrals of Vector Functions 

13.3: Arc Length and Curvature 

13.4: Motion in Space: Velocity and Acceleration

Chapter 14: Partial Derivatives

14.1: Functions of Several Variables 

14.2: Limits and Continuity 

14.3: Partial Derivatives 

14.4: Tangent Planes and Linear Approximations 

14.5: The Chain Rule 

14.6: Directional Derivatives and the Gradient Vector 

14.7: Maximum and Minimum Values 

14.8: Lagrange Multipliers 

Chapter 15: Multiple Integrals

15.1: Double Integrals over Rectangles 

15.2: Iterated Integrals 

15.3: Double Integrals over General Regions 

15.4: Double Integrals in Polar Coordinates 

15.5: Applications of Double Integrals 

15.6: Surface Area 

15.7: Triple Integrals 

15.8: Triple Integrals in Cylindrical Coordinates 

15.9: Triple Integrals in Spherical Coordinates 

15.10: Change of Variables in Multiple Integrals 

Chapter 16: Vector Calculus

16.1: Vector Fields 

16.2: Line Integrals 

16.3: The Fundamental Theorem for Line Integrals 

16.4: Green’s Theorem

16.5: Curl and Divergence 

16.6: Parametric Surfaces and Their Areas 

16.7: Surface Integrals 

16.8: Stokes’ Theorem 

16.9: The Divergence Theorem 

16.10: Summary

Chapter 17: Second-Order Differential Equations

17.1: Second-Order Linear Equations 

17.2: Nonhomogeneous Linear Equations

17.3: Applications of Second-Order Differential Equations 

17.4: Series Solutions 

What You Get in this Engineering Mathematics – Calculus ePrep Course

I. Free Textbook
“Calculus” is a very popular Calculus Textbook, authored by James Stewart, Daniel Clegg and Saleem Watson, 9th Ed.  Millions of students worldwide have used the textbooks by the late James Stewart.
II. Free Consultation
A retired NTU professor is acting as the tutor. You can consult him via email. He provides very personalized guidance according to the student’s needs.
III. Materials Online
1. Video lessons and PowerPoint files.
2. Answers/solutions to all questions/problems in the textbook.
3. Online exercises.
4. Problems, answers and solutions in the same file.
5. Bonus learning materials in other branches of mathematics, including algebra, geometry, trigonometry, and discrete mathematics.
IV. Digital Certificate
A digital certificate will be issued if you have successfully completed this ePrep course and passing all the tests at the end of each of the ten compulsory chapters. While this certificate may not be used as the main criterion for university admission, there are university admission officers willing to take this certification for consideration under the ASA or Discretionary Admission consideration.

Engineering Mathematics – Calculus ePrep Course: Sample Materials

1. Video Lesson (Area between Curves)

This video lesson discusses the determination of the area between two curves, by first finding the points of intersection, and then determine the area of the region bounded by the two curves by integrating the areas of the tiny triangles within the region. More Sample Videos.

2. Problem and Solution (Integration)


Evaluate the integral

math equation in engineering maths - calculus eprep course


Solution to a math problem in engineering maths - calculus eprep course

3. Review of Algebra (Binomial Theorem)

Binomial theorem in engineering mathematics - calculus eprep course

Algebra is a very important and more fundamental branch of mathematics and it provides a useful tool for solving calculus problems. A review of algebra is therefore performed before the study of calculus.

Engineering Mathematics – Calculus ePrep Course

Sample of Bonus Course Material – Discrete Mathematics

Discrete Mathematics (Euclidian Algorithm)


Euclidian Algorithm


Discrete Mathematics
Shown above is a sample of the bonus materials on discrete mathematics which had become very important in the digital world.
It is a gentle persuasion to NSFs not to short-change yourselves wasting your precious time and energy on those low-grade courses prepared by any “Tom-Dick-And-Harry” who self-claim to be an industry expert, especially if you are proceeding to further academic studies!  
It is only wise to go for a high-quality specially-designed academic course such as this Engineering Mathematics – Calculus ePrep course for getting you a head start in mathematics in the university. 
Our two other courses on mathematics are
1. Mathematics for the Managerial, Life and Social Sciences. It is a more general mathematics course covering a wider range of topics. More information about the course can be found at Mathematics.
2. Statistics for Undergraduate Studies. It provides an excellent introduction to probability and statistics at the university level. More details can be found at Statistics.
Where to Sign Up?
1. At SF@NS LXP if you are an NSF eligible for SkillsFuture@NationalService Scheme. You need to activate the SF@NS LXP account first at
2. At NS Portal if you have NS e-PREP Credits.
3. At NTU PaCE directly for all others. 
Who Should Take This ePrep Course
A must for all students doing engineering, computer science, and physical science degrees.
Also useful for students doing biology and other life science, statistics, economics, and business degrees. (Alternative to: Mathematics for Managerial, Life and Soc Sciences)
Even for those not going to any university due to various reasons, this is an opportunity to build up their mathematics foundation and to prove that they are capable of completing a university-level course. In this way, they too will be able to  proceed to obtain a full university education as well.
Everyone is welcome and there is no pre-requisite. This course will provide the fundamentals (till very advanced materials) that you will need in studying calculus and the other branches of mathermatics!
Course Duration
The official duration of the course is three months but may be extended upon request. Unofficially, however, the support by the tutor extends beyond the official course duration. Also, most of the course materials can be downloaded for later study (most of them are proprietary materials by NTU and the book publishers, please use them for your own private studies only and refrain from transmitting them to others).
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