Engineering Mathematics - Calculus
ePrep Course for
University Preparation

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 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.

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, 9th Ed.  Millions of students worldwide have used the textbooks by James Stewart.
II. Free Consultation
A retired NTU professor is acting as the tutor. You can consult him via email or WhatsApp. 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, linear algebra, linear programming, discrete mathematics, probability, and statistics
6. Bonus learning materials in other subjects such as business finance, corporate finance, engineering economy, economics, physics, mechanics, Python programming, life science, biotechnology, and psychology.
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.
Additional Note: NTU is providing an alternative pathway to the part-time B.Eng degree programme where interested students may take this course (and a physics course) and sit for a special exam in early July for consideration to join the special semester. The performance of the courses (which include a math course which is a slightly more advanced version of the this course) during the semester may be used for consideration into NTU’s PT B.Eng degree programme.

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

Sample Supplementary Course Materials 

– From Other Mathematics ePrep Courses

1. Video Lesson on Maths for Managerial, Life and Soc Sc (Exponential Functions)

This video lesson illustrates the solution steps in solving a half-life problem as an exponential decay model.

2. Video Lesson on Probability (Probability of Discrete Events)

This video lesson illustrates using the concept of sample space to solve a discrete event probability problem.

3. Video Lesson on Statistics (Confidence Interval)

This video lesson illustrates how we can use the sample mean, standard deviation, and sample size to compute the confidence interval for the population mean.

Engineering Mathematics – Calculus ePrep Course

Samples of Bonus Course Materials

1. Video Lesson on Business Finance (Ordinary Annuity and Annuity Due)

This short video lesson illustrates what annuity is and how annuity due differs.

2. Video Lesson on Corporate Finance (Sustainable Growth Model)

This video lesson illustrates that the Sustainable Growth Model is for a business that wants to maintain a target payout ratio and capital structure without issuing new equity, and it provides an estimate of the annual percentage increase in sales that can be supported.

3. Cross-Word Puzzle on Biotechnology (Principle of Genetic Transfer)

biotech Cross-word puzzle in engineering mathematics - calculus eprep courses

4. Worked Example on Engineering Economy (Project Evaluation)


A retrofitted space-heating system is being considered for a small office building. The system can be purchased and installed for $110,000, and it will save an estimated 300,000 kilowatt-hours (kWh) of electric power each year over a six-year period. A kilowatt-hour of electricity costs $0.10, and the company uses a MARR of 15% per year in its economic evaluations of refurbished systems. The market value of the system will be $8,000 at the end of six years, and additional annual operating and maintenance expenses are negligible. Use the PW method to determine whether this system should be installed.


To find the PW of the proposed heating system, we need to find the present equivalent of all associated cash flows.

The estimated annual savings in electrical power is worth 300,000 kWh × $0.10/kWh = $30,000 per year.

At a MARR of 15%, we get

PW(15%) = −$110,000 + $30,000 (P/A, 15%, 6) + $8,000 (P/F, 15%, 6)

= −$110,000 + $30,000(3.7845) + $8,000(0.4323)

= $6,993.40.

Since PW(15%)≥ 0, we conclude that the retrofitted space-heating system should be installed.


MARR = Minimum Acceptable Rate of Return

PW = Present Worth

(P/A, 15%, 6) and (P/F, 15%, 6) are factors that can be obtained from tables, software, financial calculator, or by applying formulas.

5. Objective Question Exercise on Physics (Electric Power)


Two conductors made of the same material are connected across the same potential difference. Conductor A has twice the diameter and twice the length of conductor B. What is the ratio of the power delivered to A to the power delivered to B?

1.    8

2.    4

3.    2

4.    1

5.    12


Compare resistances:

RA/RB = (ρLA(dA/2)2)/ (ρLB(dB/2)2)

= (LAdB2)/ (LBdA2) = (2LBdB2)/ (LB (2dB)2)

= 2/4 = ½.

Compare powers:

PA/PB = (ΔV2/RA)/ (ΔV2/RB)

= RB/RA = 2.

6. Web Exercise on Psychology (Human Development)


Since having their first child, Bob and Angela notice that they no longer feel they have anything in common with their childless friends and are spending a lot more time with other parents. Psychologists would say that:

a  their ingroup has changed from nonparents to parents.

b  this is an example of group polarization.

c  they have a stereotype about what parents are like.

d  their attribution processes are now different than before.

Answer: (a)

7. Video Lesson on Mechanics (Work and Energy)

This short video explains the relationship between power, rate of work done, total work done, and time taken to perform the work.

8. Screen Record of Animation on Life Science (Translocation)

This is a screen record of animation that illustrates translocation in chromosomes.

9. Python Programming (Histogram)

import matplotlib
import numpy as np
import matplotlib.pyplot as plt


# example data
mu = 100  # mean of distribution
sigma = 20  # standard deviation of distribution
x = mu + sigma * np.random.randn(1000) #normal dist with mu and sigma

subdivisions = 25

fig, ax = plt.subplots()

n, bins, patches = ax.hist(x, subdivisions, density=True) #histogram

y = ((1 / (np.sqrt(2 * np.pi) * sigma)) *
     np.exp(-0.5 * (1 / sigma * (bins – mu))**2)) # create theoretical line

ax.plot(bins, y, ‘.’)
ax.set_ylabel(‘Probability density’)
ax.set_title(r’Histogram: $\mu=100$, $\sigma=20$’)
fig.tight_layout() #provide spacing to prevent clipping of y-axis abel


Histogram Produced by Python in ePrep Course
10. Economics (The Marginal-Cost Curve and the Firm’s Supply Decision)
Marginal Cost Curve and Firm's Supply Decision

1. Cost curves have special features that are important for our analysis.

  • The marginal cost curve is upward slopping.
  • The average total cost curve is u-shaped.
  • The marginal cost curve crosses the average total cost curve at the minimum of average total cost.

2. Marginal and average revenue can be shown by a horizontal line at the market price

3. To find the profit-maximizing level of output, we can follow the same rules that we discussed above.

  • If marginal revenue is greater than the marginal cost, the firm can increase its profit by increasing output.
  • If marginal cost is greater than marginal revenue, the firm can increase its profit by decreasing output.
  • At the profit-maximizing level of output, marginal revenue is equal to marginal cost.


4. If the price in the market were to change to P2, the firm would set its new level of output by equating marginal revenue and marginal cost.

5. Because the firm’s marginal cost curve determines how much the firm is willing to supply at any price, it is the competitive firm’s supply curve.

11. Discrete Mathematics (Euclidian Algorithm)


Euclidian Algorithm


Discrete Mathematics

Shown above are samples of the bonus materials on other subjects and they illustrate how comprehensive and broad-base this Engineering Mathematics – Calculus ePrep course is for preparing students for their university studies or for their career advancement. While not all the bonus course materials may be of interest to the students who take up this ePrep course on Engineering Mathematics – Calculus, they can choose which of these bonus course materials are of interest and download them and ignore the rest. 



It is a gentle persuasion to NSFs not to short-change themselves wasting their precious e-PREP Credits on those low-grade courses prepared by any “Tom-Dick-And-Harry” who self-claim to be an industry expert, especially if they are proceeding to further academic studies!  They do not need thousands of such courses.  As you can see, with this single Engineering Mathematics – Calculus course, you get a good suite of course materials on many other subjects as well.  You also get a hard copy Calculus textbook by James Stewart.
It is only wise to go for a high-quality specially-designed academic course such as this Engineering Mathematics – Calculus e_prep course for getting you a head start in university.
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 prove that they are capable of completing a university-level course. Hopefully, one day, they too will be able to obtain a full university education as well.




Everyone is welcome and there is no pre-requisite. This course will provide the fundamentals that you will need in studying calculus!

Audio: Who should take this Engineering Mathematics – Calculus ePrep course?

How to Sign Up for Engineering Mathematics – Calculus ePrep Course
1 If You Are an NSF or NSmen
If you are an NSF or NSMen and within one year of ORD, please sign up with NS Portal to enjoy the NS e-PREP subsidy.
Please go to “Access ePREP”, search for “NTU” as the course provider, then select “Engineering Mathematics – Calculus” among the e-PREP courses offered by NTU.  
Please note that since the course fee is S$385 inclusive of the textbook, and the maximum NS e-PREP Credits is $350, after registration for the course you will have to pay S$35. For advice on payment matters, please contact the MINDEF ePrep administrator at Tel No: (+65) 63731221,  or eMail:  
2 For Others
Please sign up with NTU PaCE directly. The course fee is S$385, inclusive of the textbook. If you are located outside Singapore you may have to pay for the postage for the textbook.
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 the book publishers, please use them for your own private studies and refrain from transmitting them to others).
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