# Engineering Mathematics - Calculus ePrep Course for UniversityAlso Available at SF@NS LXP

### 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.

Question:

Solution: