Introductory Calculus

For information:

Fred W. Duckworth, Jr.

c/o Jewels Educational Services

1560 East Vernon Avenue

Los Angeles, CA 90011-3839

E-mail: admin@trinitytutors.com

Website: www.trinitytutors.com

Copyright © 2007 by Fred Duckworth. All rights reserved. This publication is copyrighted and may not be linked to directly, reproduced, stored in a retrieval system, transmitted in any form by any means, electronic, mechanical, photocopying, recording, or otherwise without written permission from the publisher.

You may print this entire publication or portions thereof directly from this website, but may neither download to your computer nor store any text or images comprising this work. Moreover, you may not place on any other website for others to access, nor distribute to anyone else in any manner any of the pages comprising this work. This material is to be utilized only for your personal use in a homeschool or tutoring environment, and you may print out any or all of the pages herein only on an individual, one-time basis.

Furthermore, all copyright notifications must be included and you may not alter them in any way. Classroom use and/or use in a public or private school setting is expressly prohibited. Anyone wishing to use this material must come to this website to access it. Any use beyond these terms requires the written permission of the author/publisher. This publication is being provided at no cost and may not be sold under any circumstances.

Table of Contents

Chapter 1

For information:

Fred W. Duckworth, Jr.

c/o Jewels Educational Services

1560 East Vernon Avenue

Los Angeles, CA 90011-3839

E-mail: admin@trinitytutors.com

Website: www.trinitytutors.com

Copyright © 2006 by Fred Duckworth. All rights reserved. This publication is copyrighted and may not be linked to directly, reproduced, stored in a retrieval system, transmitted in any form by any means, electronic, mechanical, photocopying, recording, or otherwise without written permission from the publisher.

Calculus

INTRODUCTORY

In arithmetic, operations are carried out on numbers. In algebra, operations are carried out on symbols that stand for numbers. However, in calculus, operations are carried out on functions that represent the relationship between two veritable quantities.

1a Integral Calculus

Calculus is a branch of mathematics divided into two closely related parts, integral calculus and differential calculus, both of which deal with the measurement of constantly changing shapes, paths, areas, etc. For this reason, calculus has been described as “the mathematics of change.”

Integral calculus is concerned with calculating the area and/or volume of shapes or figures that have curves which are changing at constantly varying degrees, such as elipses, parabolas, domes (such as the Houston Astrodome, pictured above), etc.

The basic strategy of integral calculus is to calculate the area of a complicated

region – one whose rate of curve is constantly changing – by chopping it up into so many tiny rectangular regions that each little region is virtually flat. The next step is to add up the individual areas of all the “mini regions” in order to determine the total area. Although this does not succeed completely (because there will always be some amount of curve to each tiny region) the more mini regions into which the total area is divided, the closer the approximation will be.

conceptualization

Earth

Venus

Mercury

Mars

Differential calculus is concerned with calculating a moving object's varying rate of speed at any given point along its path, based on the moment in time it was there.

When you travel at the same speed over a given distance, you can determine what that speed was by dividing the distance traveled by the time it took to complete the trip.

1b Differential Calculus

NOTE: The calculus in this textbook is presented with the same level of depth and rigor as are entry-level college and university calculus courses, providing a complete college curriculum in one variable calculus. Treating nothing lightly, it makes the point of taking sufficient time to covering all the bases in an understandable, comprehensible and applicable manner, for example, spending substantial time on both differential equations and infinite sequences and series.

1a Functions

functions

NOTE: Make sure that you are familiar with set description and interval notation. Brackets mean the the enclosed values are included in the domain or range, whichever applies.

Parentheses, on the other hand, indicate that an endpoint is NOT included.

In other words, the domain is all of the values that x can be, and the range is all of the values tht y can be, given x.

LESSON 1

Lesson Review

In a nutshell, calculus takes a fluid, evolving problem that can’t be done with regular math because factors are constantly changing, and breaks it up into limited sections so infinitesimally small that each section is, for all practical purposes, unchanging. Calculus then applies the regular rules of math to each individual section to finish the problem off. By focusing on and solving each little part, then adding them all up, calculus solves the overall problem.

The catch is, you can never break the problem up into small enough chunks to be totally accurate. Calculus handles this problem by zooming in infinitely. If something is constantly changing, from each infinitesimal moment to the next, it is changing infinitely – so everything you do in calculus involves infinity in one way or another.

Given tht calculus requires operations to be carried out on functions that represent the relationship between two variable (changing) quantities – such as speed and time – to understand calculus, you must therefore understand functions, since the concept of functions is fundamental to calculus.

What is Calculus?

Key Concepts

So then, calculus is the mathematics that helps us deal with calculating information about changing quantities. It was developed several hundred years ago, by Newton and Liebniz, to deal with such problems as finding the total distance traveled by an object with changing velocity, or finding the velocity of an object with changing position.

Suppose that a ball has been thrown into the air. Its height, in feet, after tea seconds is given by the equation: h = –16t² + 40t + 5.

To find the height at any time, t, we input the value of t into the equation.

When we do this, we obtained an output that is the height value. So time, t, is the input, and height, h, is the output.

Look at the graph of height versus time. Notice that there is exactly one corresponding height value for each time value.

Since each input value is assigned to exactly one output values, the graph confirms that height is a function of time, so, we say that the height of the ball is a function of time.

In general, any relationship that has only one output value for each input value is a function.

You saw that the height of a ball thrown in the air is a function of the time since it has been thrown. Can we also say that time is a function of high? In other words, is there exactly 1 time value assigned to each fight value?

Look at the graph. Noticed that Lee H. equals 21, he can either be 0.5 or 2. There are two possible outfits for this in place. So time is not a function of high.

In general, any relationship that has only one output value for each input value is a function. For a detailed explanation of functions click on this link.

A graph represents a function if any vertical line intersects the graph only once. This rule is called the vertical line test. (The input value is represented by the x coordinate and the output value, or function of x, is represented by the y coordinate.) The vertical line rule demonstrates that if a ball is thrown in the air, its height is a function of time, but time is not a function of height.

FUNCTION NOTATION

The statement h is a function of t can be shortened to h equals f of t, or shorter yet h = f(t), where h is the y coordinate, t is the x coordinate, and f is the "rule" or computation that is performed on x in order to generate y. In other words, whatever varible happens to be inside of the parentheses is actually representing the x coordinate.

Sometime you will be given the output value and required to find the corresponding input value. In such cases, the quadratic formula is one means of obtaining the solution (for example, the two points at which t satisfies the equation f(t) = 25.

Also, we call the solution of the equation f(t) = 0 a root or a zero of the function (just for your information).

The set of input values for which the function is definded (the "used" portion along the x-axis) is called the domain of the function, and the corresponding set of output values (along the "used" portion of the y-axis) is called the range of the function.

Use the graph to answer the following:

Is it true that f(0) = 5

Is it true that f(21) = 2

Is it true that f(5) = 0

Is it true that f(2) = 21

Does f (1.5) = 29 tell us that 1.5 seconds after the ball was thrown, its height was 29 meters -- or does it tell us that 29 seconds after the ball was thrown, its height was 1.5 meters?

Lets see how you can evaluate f (1.5) using the graph or the equation for f (t).

To use the graph to find an approximate value for f (1.5) first find the link on the graph with a t-coordinate of 1.5.

You can see that the vertical cordon of his point is slightly less than 30 feet. So the value of f (1.5) is approximately 28 or 29.

You can also evaluate f (1.5) by substituting 1.5 for t indict equations. This method gives us the exact value for f (1.5) which is 29 feet.

Another method for evaluating f (1.5) uses your calculator. Make sure that you can use your calculator to evaluate the value of a function for a given conflict. If you don't know how to do so, referred to that Making a Table section in the "Using Your Graphing Calculator" lesson.

You have seen how to evaluate the height of the ball at any time using the graph for the equation. Often the situation is reversed; we know the output value and we want to find the corresponding infant values.

Lets see how to find the time in the fall is 25 feet above the ground or he

In this case, we need to find in the values that he is an outfit of 25. In other words, we need to find t when f(t) = 25.

We can use the grass to estimate the solution. To do so, you will first find the link on the graph that have a vertical coordinate of 25.

You can see that there are two such flights. Has a t-coordinate of approximately 0.7, and the other has a t-coordinate of approximately 1.8.

To find the exact solutions, we can solve the equation algebraicly.

So, the height of the ball is 25 feet at two different times. Once we t = 0.691 seconds, and once when t = 1.809 seconds.

And of course, another method is using your calculator.

Recall tha we defined h to be the height of a ball t seconds after it has been thrown into the air. Although h is a function of t, the value of h is not defined for every possible value of t.

For example, it makes no sense to talk about the height of the ball when t = -2 or t = 4. So, although h is a function of t, this function is only defined for certain values of t.

Also, notice that h, that outfit value of the function, can only attain values between zero and 30.

Is set up in the values for which the function is the fine is called the domain of the function. The corresponding set the outfit values is called the range of the function.

So, the domain of the function in our example is the set of all real numbers between zero and 2.619. This set can also be described using interval notation, as shown.

Racket next 20 means that zero is included in the domain. Similarly, the bracket next to 2.6 and 19 means that 2.619 is included in the domain.

The range of the function is this set of all real numbers between zero and 30. Again, the brackets mean that zero and 30 are included in the range.

Now, consider the function h(x), whose graph is shown. We want to find the domain and range of this function.

By looking at the graph, you can see that the domain of this function is all real numbers. Is set of real numbers can be represented as the open interval from negative infinity to infinity.

The parentheses next to the infinity symbols mean that infinity and negative infinity are not included in the domain. This is because these values are not actually numbers.

Now let's find the range of h(x). The range of this function is all real numbers greater than zero.

The basic strategy of differential calculus however is to take a trip in which you traveled at varying speeds and divide the trip into lots of “mini trips.” You then pretend that your speed didn't change at all during each of those mini trips and compute the speed for each trip individually. But, since the assumption that the speed did not change over each tiny trip is generally wrong, you only approximate the correct answer. Nonetheless, the smaller you make each tiny trip, the more accurate your calculation of the actual speed at any given point along the big trip will be. What is of real importance in this case is not the actual amount of movement, but the rate of movement – or rate of increase.

The problem of finding the rate of increase is the job of differential calculus.

time in seconds

height in feet

h = 16t² + 40t + 5

time in seconds

height in feet

h = 16t² + 40t + 5

LESSON 1: Definition of a function, Vertical line test,Function notation, Finding input and output 0values

What is Calculus?

Absolute value functions are all piecewise defined functions. (They are always shaped like a "v"?)

Review

Integral calculus is concerned with calculating the area and/or volume of shapes or figures that have curves which are changing at constantly varying degrees.

Differential calculus is concerned with calculating a moving object's varying rate of speed at any given point along its path, based on the moment in time it was there.

A function is a relationship in which each input value has only one output value.

The vertical line test states that a graph represents a function if any vertical line intersects the graph only once

FUNCTION NOTATION

The solution of the equation f(t) = 0 is called a root or a zero of the function.

The domain is the set of input values for which the function is definded (the "used" portion along the x-axis).

The range is the corresponding set of output values (along the "used" portion of the y-axis).

Jewels Educational Services for Up-and-coming Scholars