# Engineering Mathematics - Calculus

ePrep Course for

University Preparation

##### 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.

**Audio**: Intro to Engineering Mathematics – Calculus

**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 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.

**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**

**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)

**4. Worked Example on Engineering Economy** (Project Evaluation)

**Question**:

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.

**Answer**.

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.

**Note**:

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)

**Question**:

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

**Answer**:

Compare resistances:

*R** _{A}*/

*R*= (ρ

_{B}*L*/π

_{A}*(*

*d*/2)

_{A}^{2})/ (ρ

*L*/π

_{B}*(d*/2)

_{B}^{2})

= (*L _{A}*

*d*

_{B}*)/ (*

^{2}*L*

_{B}*d*

_{A}*) = (2*

^{2}*L*

_{B}*d*

_{B}*)/ (*

^{2}*L*(2

_{B}*d*)

_{B}*)*

^{2}= 2/4 = ½.

Compare powers:

*P _{A}*/

*P*= (Δ

_{B}*V*/

^{2}*R*)/ (Δ

_{A}*V*/

^{2}*R*)

_{B}= *R _{B}*/

*R*= 2.

_{A}**6. Web Exercise on Psychology** (Human Development)

**Question**:

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)

**Code:**

import matplotlib

import numpy as np

import matplotlib.pyplot as plt

np.random.seed(1213141516)

# 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_xlabel(‘Scores’)

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

plt.show()**Output:**

**10. Economics **(The Marginal-Cost Curve and the 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 *P*2, 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)

**Question**:

**Solution**:

**Remarks**:

##### 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 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!

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

**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 and refrain from transmitting them to others).

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