Calculus Fundamentals Explained

CALCULUS FUNDAMENTALS EXPLAINED

Samuel Horelick

Copyright © 2021 by Samuel Horelick. All rights reserved.

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise without prior written permission of the author, except as permitted under section 107 or 108 of the 1976 United States Copyright Act.

Smashwords Edition

About the author: Dr. Samuel Horelick is mathematics professor and higher education consultant. He has graduated from three Universities with four degrees in Mathematics, Philosophy, Mathematical Education, and Theology.

CONTENTS

Introduction

CHAPTER 1 – FUNCTIONS

What is Function?

Functions and Graphs

Vertical Line Test: is it a Function?

Horizontal Line Test: is it One-to-One Function?

Arithmetic Combinations and Composition of Functions

Inverse Functions

Families of Functions

Symmetry, Even and Odd Functions

Zeroes of a Function

CHAPTER 2 – LIMITS

The Concept and Properties of Limits

Continuity

Infinite Limits and Vertical Asymptotes

Limits at Infinity and Horizontal Asymptotes

Tangent Lines, Areas, and Limits

CHAPTER 3 – DERIVATIVES

Derivative as a Limit

Derivatives and Rules of Differentiation

Writing Equations of Lines Tangent to Function f(x)

Average and Instantaneous Velocity

Chain Rule

Implicit Differentiation

Differentials

Increasing and Decreasing Functions, Mean-Value Theorem

First Derivative Test

Second Derivative Test

Concavity and Inflection Points

Absolute Maximum and Minimum, Extreme-Value Theorem

Business Applications

CHAPTER 4 – INTEGRALS

Indefinite Integral: Integration as Anti-differentiation

Integration by Substitution

Initial Conditions and Particular Solutions to Differential Equations

Definite Integral: Area Under the Graph

Area Between Two Graphs

Volume of a Solid of Revolution

Introduction

This textbook is a product of two decades of teaching Calculus. Over this time I have taught students of every imaginable cultural, social, and educational background, from the affluent communities in South Florida to the inner-city New York, not to mention various colleges and innumerable hours of private tutoring. And regardless of student's background and the level of preparation, most students seem to have one deeply ingrained misconception about Calculus: Calculus is something so difficult and ‘far out’ that most people cannot really do it at all. This misconception is so blatantly false, but so widely held, that it is probably the school system itself that propagates this ridiculous, silly notion.

One serious difficulty in learning Calculus today is a lack of simple and clear textbooks. There are dozens of huge, expensive textbooks that are so needlessly complicated as to be practically incomprehensible. It seems that every author of Calculus textbook believes that every student is a born mathematician that wants to become a math professor. They forget that a mark of a good teacher is the ability to make complicated things seem easy and simple. A good textbook must explain and teach its subject matter, helping the student to learn and say: That’s all? That’s easy!

The effectiveness of this textbook was tested in one of the most challenging (and rewarding) teaching environments – the inner-city high schools of New York. And then it was improved, updated, and implemented to teach Science, Math, and Engineering students in college.

In this textbook you will find all mathematics needed for the first Calculus course; clear explanations of all the principal concepts of Calculus; coverage of all course fundamentals; effective problem-solving skills and strategies; and fully worked out problems with complete, step-by-step explanations and answers.

The purpose of this book is to enable any student to master the fundamentals of Calculus in one semester – four months. The only requirement is an honest desire to learn. The material in this textbook is prepared and sequenced in a particular way to allow for easy, logical progression from the simple to the more elaborate concepts and exercises.

By carefully reading all material, working through all examples and solving all exercises, the student will easily master the fundamentals of Calculus in one semester.

This textbook is intended as the main text for the first Calculus course in College. It may also be used as an explanatory or supplemental text with a different textbook.

Calculus is the mathematics of change, and change is represented by functions. The basic operations in Calculus are differentiating and integrating functions. Whatever we do in Calculus, we deal with functions – we find limits of functions, we differentiate functions, we integrate functions. To understand Calculus students must understand functions first: what sort of relation is called function, how to graph functions, how to find slope of a line, how to write equations of lines. These are very basic topics, but the understanding of these basic concepts provides a necessary foundation and a starting point in the study of Calculus. This book explains and teaches Calculus in a logical, handy and succinct format without overwhelming you with unnecessary details.

The ideas and principles of what eventually became known as Calculus can be traced back to the Ancient Greek mathematicians. These ideas were developed by mathematicians and philosophers through the Middle Ages, but the actual rigorous clarification of the fundamental principles of Calculus was made independently by Isaac Newton in England and Gottfried Leibniz in Germany in seventeenth century. We can simplify the fundamentals into three classes of problems:

1. How to find an equation of the line tangent to the graph of a given function at a given point.

2. How to find the area of a given region.

3. How to find velocity and acceleration at an instant, given a formula for the distance traveled by an object in a specified period of time. Conversely, given a formula for acceleration at a given instant, how to find the distance traveled by this object in a specified period of time. Each of these problems involves infinite process:

For the Tangent Line problem, the infinite process is the process of drawing secant lines through two points P and Q on the graph of function f(x) as point Q moves infinitely closer and closer to point P so that the distance between P and Q approaches zero.

For the Area problem, the infinite process is the measurement of the area under the graph of function using more and more rectangles with progressively smaller widths.

For the Instantaneous Velocity and Acceleration problem, the infinite process is the limiting value of the average speed computed over smaller and smaller time intervals, progressively closer to the specified instant of time t.

Calculus is a set of ideas that provides a way of analyzing the world around us. As with all mathematics courses, Calculus involves equations and formulas. But even if you learn all the formulas and solve all the equations but do not master the underlying ideas, you would have learned really nothing. But if you carefully study and understand the interrelated ideas of Calculus, not only will you have the tools to go beyond what other people have done, you will also be amazed that Calculus is not really difficult at all.

CHAPTER 1 – FUNCTIONS

What is function?

Function is simply a relation between two sets of objects (or numbers) where each member of the set called Domain is assigned to exactly one member of another set called Range.

Some functions may assign several members from the Domain to exactly one member of the Range:

Relation That Is Not function:

However, one object in the Domain may not be assigned to more than one object in the Range. Think of a relation between the set of policemen in a precinct and the set of police cars. If there are enough cars, each policeman can be assigned to one car.

If not, two policemen may be assigned to the same car, or three, etc. What is not possible is to assign one policeman to more than one car at the same time.

A relation that assigns one element of the Domain to more than one element of the Range is not function.

Functions and Graphs:

Example:

Let set D = {2, 4, 6, 8}, let function F assign each element of set D (domain) to an element of set R (range) using the rule: an element of set D plus 10 is equal to an element of set R. Then,

F(2) = 2 + 10 = 12

F(4) = 4 + 10 = 14

F(6) = 6 + 10 = 16

F(8) = 8 + 10 = 18 and therefore, set R = {12, 14, 16, 18}.

If we denote the numbers in the set D by x and denote the numbers in the set R by y, then this function F is written F(x) = x + 10 or y = x + 10. The meaning of y is the same as the meaning of F(x) – it is the Range of the function.

Function may be represented by ordered pairs (x, y) or (x, f(x)), where x is an element of the Domain and y