Engineering Mathematics - Calculus
ePrep Course for University
Also Available at SF@NS LXP
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)
Question:
Evaluate the integral
Solution:
3. Review of Algebra (Binomial Theorem)
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)
Question:
Solution:
Remarks:
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 https://go.gov.sg/activatesfnslxp.
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
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