# Engineering Mathematics - Calculus ePrep Course forUniversity Preparation

##### 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

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

### 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.

### – 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

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)

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.

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 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)

Question:

Solution:

##### 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?