MTH101 Handouts by VUBuddy

45-Planning Production Levels: Linear Programming VU

TABLE OF CONTENTS :

Lesson 1 :Coordinates, Graphs, Lines Lesson 2 :Absolute Value Lesson 3 :Coordinate Planes and Graphs Lesson 4 :Lines Lesson 5 :Distance; Circles, Quadratic Equations Lesson 6 :Functions and Limits Lesson 7 :Operations on Functions Lesson 8 :Graphing Functions Lesson 9 :Limits (Intuitive Introduction) Lesson 10:Limits (Computational Techniques) Lesson 11: Limits (Rigorous Approach) Lesson 12 :Continuity Lesson 13 :Limits and Continuity of Trigonometric Functions Lesson 14 :Tangent Lines, Rates of Change Lesson 15 :The Derivative Lesson 16 :Techniques of Differentiation Lesson 17 :Derivatives of Trigonometric Function Lesson 18 :The chain Rule Lesson 19 :Implicit Differentiation Lesson 20 :Derivative of Logarithmic and Exponential Functions Lesson 21 :Applications of Differentiation Lesson 22 :Relative Extrema Lesson 23 :Maximum and Minimum Values of Functions Lesson 24 :Newton’s Method, Rolle’s Theorem and Mean Value Theorem Lesson 25 :Integrations Lesson 26 :Integration by Substitution Lesson 27 :Sigma Notation Lesson 28 :Area as Limit Lesson 29 :Definite Integral Lesson 30 :First Fundamental Theorem of Calculus Lesson 31 :Evaluating Definite Integral by Subsitution Lesson 32 :Second Fundamental Theorem of Calculus Lesson 33 :Application of Definite Integral Lesson 34 :Volume by slicing; Disks and Washers Lesson 35 :Volume by Cylindrical Shells Lesson 36 :Length of Plane Curves Lesson 37 :Area of Surface of Revolution Lesson 38:Work and Definite Integral Lesson 39 :Improper Integral Lesson 40 :L’Hopital’s Rule Lesson 41 :Sequence Lesson 42 :Infinite Series Lesson 43 :Additional Convergence tests Lesson 44 :Alternating Series; Conditional Convergence Lesson 45 :Taylor and Maclaurin Series

Lecture 1

Coordinates, Graphs and Lines

What is Calculus??

Well, it is the study of the continuous rates of the change of quantities. It is the study of how various quantities change with respect to other quantities. For example, one would like to know how distance changes with respect to (from now onwards we will use the abbreviation w.r.t) time, or how time changes w.r.t speed, or how water flow changes w.r.t time etc. You want to know how this happens continuously. We will see what continuously means as well.

In this lecture, we will talk about the following topics: · -Real Numbers · -Set Theory · -Intervals · -Inequalities · -Order Properties of Real Numbers

Let's start talking about Real Numbers. We will not talk about the COMPLEX or IMANGINARY numbers, although your text has something about them which you can read on your own. We will go through the history of REAL numbers and how they popped into the realm of human intellect. We will look at the various types of REALS - as we will now call them. So Let's START.

The simplest numbers are the natural numbers

Natural Numbers

1, 2, 3, 4, 5,…

They are called the natural numbers because they are the first to have crossed paths with human intellect. Think about it: these are the numbers we count things with. So our ancestors used these numbers first to count, and they came to us naturally! Hence the name

NATURAL!!!

The natural numbers form a subset of a larger class of numbers called the integers. I have used the word SUBSET. From now onwards we will just think of SET as a COLLECTION OF THINGS.

This could be a collection of oranges, apples, cars, or politicians. For example, if I have the SET of politicians then a SUBSET will be just a part of the COLLECTION. In mathematical notation we say A is subset of B if

x · A · x · B

.Then we write A

B

Set

The collection of well defined objects is called a set. For example

{George Bush, Toney Blair, Ronald Reagoan}

Subset

A portion of a set B is a subset of A iff every member of B is a member of A. e.g. one subset of above set is

{George Bush, Tony Blair}

The curly brackets are always used for denoting SETS. We will get into the basic notations and ideas of sets later. Going back to the Integers. These are

…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…

So these are just the natural numbers, plus a 0, and the NEGATIVES of the natural numbers.

The reason we didn’t have 0 in the natural numbers is that this number itself has an interesting story, from being labeled as the concept of the DEVIL in ancient Greece, to being easily accepted in the Indian philosophy, to being promoted in the use of commerce and science by the Arabs and the Europeans. But here, we accept it with an open heart into the SET of INTEGERS.

What about these NEGATIVE Naturals??? Well, they are an artificial construction. They also have a history of their own. For a long time, they would creep up in the solutions of simple equations like

x+2 = 0. The solution is x = - 2

So now we have the Integers plus the naturals giving us things we will call REAL numbers. But that's not all. There is more. The integers in turn are a subset of a still larger class of numbers called the rational numbers. With the exception that division by zero is ruled out, the rational numbers are formed by taking ratios of integers.

Examples are

2/3, 7/5, 6/1, -5/2

Observe that every integer is also a rational number because an integer p can be written as a ratio. So every integer is also a rational. Why not divide by 0? Well here is why:

If x is different from zero, this equation is contradictory; and if x is equal to zero, this equation is satisfied by any number y, so the ratio does not have a unique value a situation that is mathematically unsatisfactory.

/0 · 0. · x · y · x · y · x

For these reasons such symbols are not assigned a value; they are said to be undefined.

So we have some logical inconsistencies that we would like to avoid. I hope you see that!! Hence, no division by 0 allowed! Now we come to a very interesting story in the history of the development of Real numbers. The discovery of IRRATIONAL numbers.

Pythagoras was an ancient Greek philosopher and mathematician. He studied the properties of numbers for its own sake, not necessarily for any applied problems. This was a major change in mathematical thinking as