Pdf teach yourself calculus p abbott pdf free download

Precalculus: A Self-Teaching Guide includes an algebra review and complete coverage of exponential functions, log functions, and trigonometry. Whether you are studying precalculus for the first time, want to refresh your memory, or need a little help for a course, this clear, interactive primer will provide you with the skills you need. Precalculus offers a proven self-teaching approach that lets you work at your own pace-and the frequent self-tests and exercises reinforce what you've learned.

Turn to this one-of-a-kind teaching tool and, before you know it, you'll be solving problems like a mathematician! This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process.

As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions. A programmed text designed to introduce the student to the fundamentals of differential and integral calculus.

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The whole point of calculus is to take the conventional rules and principles of mathematics and apply them to dynamic situations where one or more variable is changing. In particular, calculus provides you with the tools to deal with rates of change. These are both techniques that can be applied to algebraic expressions. As you will discover, one is the converse of the other. Algebraically, what this means is that, if you start with some function f x and differentiate it, then integrating the result will get you back, roughly speaking, to your original function f x.

And the same idea is true in reverse — if you first choose to integrate f x , then differentiating the result will get you back to your original function f x. But why might you want to perform a differentiation or an integration?

Well, differentiating a function f x gives a new function that tells you about the rate of change of f x. Integrating the function f x allows you to find the area contained between the graph of f x and the x-axis. This is a useful tool for finding the volumes of awkward shapes. Here is a typical problem of calculus. Consider a small plant which grows gradually and continuously.

If you examine it after an interval of a few days, the growth will be obvious and you can measure it. But if you examine it after an interval of a few minutes, although growth has taken place, the amount is too small to see. If you observe it after an even smaller interval of time, say a few seconds, although you can detect no change, you know that the plant has grown by a tiny amount.

You can see the process of gradual and continuous growth in other situations. What is of real importance in most cases is not the actual amount of growth or increase, but the rate of growth or rate of increase. The problem of finding the rate of growth is the basis of the differential calculus.

Leibnitz published an account in , though his notebooks show that he used it in ; Newton published his book on the subject in , but he told his friends about the calculus in It is generally accepted now that they discovered it independently. The growth is a function of time, but you may not be able to give an equation for the function. As the idea of a function is fundamental to calculus, this section is a brief revision of the main ideas.

When letters are used to represent quantities or numbers in an algebraic formula, some represent quantities which may take many values, while others remain constant. Nugget — Spotting variables In general, when you look at a formula, it is clear that the variables are the letters conventionally written in italics and the constants are the numbers. However, just to keep you on your toes, there are some constants that appear to masquerade as variables.

One of these is pi, written p. Remember that pi is a constant; its value never changes. Another example of a constant disguised as a letter is the acceleration due to gravity, g.

Example 1. In each of the above examples, there are two kinds of variables. That is, the volume V depends on r. Generally, in mathematical formulae, there are two types of variable, dependent and independent. Functions 3 33 the variable in which changes of value produce corresponding changes in the other variable is called the independent variable, as r and t above.

When two variables behave in this way, that is, the value of one variable depends on the value of the other, the dependent variable is said to be a function of the independent variable. In the examples above: 33 the volume of a sphere is a function of its radius; 33 the distance fallen by a body is a function of time. The square of a number is a function of the number.

The volume of a mass of gas is a function of the temperature while the pressure remains constant. The sines and cosines of angles are functions of the angles. The time of beat of a pendulum is a function of its length. Definition: If two variables x and y are related in such a way that, when any value is given to x, there is one, and only one, corresponding value of y, then y is a function of x. When expressing functions of angles, in addition to x, the Greek letters q theta and f phi are often used for angles.

In this notation, the letter f is used to mean the function, while the letter x is the independent variable. Thus f q is a general method of indicating a function of q. Similar forms of this notation are F x and f x.

You would use this notation when you want to substitute a value of x. Functions 5 Example 1. A similar notation is used for other variables. In this case, the letters must not be separated. It is possible when x is increased by 1 that y could be decreased by 1. If the increase is actually a decrease, its sign will be negative. Functions 7 Example 1.

Let x be increased by h and let the corresponding increase in y be k. Then, by the definition of a function, in Section 1. So, by giving a set of values to x you obtain a corresponding set of values for y.

You can plot the pairs of values of x and f x to give the graph of f x. Nugget — What is f x on the graph? When it comes to understanding the big ideas of mathematics, the simplest questions are often the most useful. One query that often crops up with students is how f x can be interpreted on a graph. It is important to remember here that f x corresponds to the value of y.

So, if you were to choose a particular point on the graph of this function and consider the coordinates of that point, the x-coordinate will represent its horizontal distance from the y-axis, and its value for f x i.

The curve, a parabola, is shown in Figure 1. These lines are called ordinates. In Figure 1. The corresponding increase in f x or y is MP. This set of values for x and y is often called a window.

In particular, you will find it is useful to use the zoom function, so that you can magnify and look more closely at a part of the curve. Similarly, you can use a computer for drawing graphs, with a graphing spreadsheet such as Excel, or any other graph-plotting software.

Using Excel, you will need first to create a table of about 30 values for the function, and then, from the table, use the scatterplot graphing facility to draw a conventional graph of the function. Once again you can look at any part of the graph that you wish, and you should be able to look closely at a particular part of the graph by adjusting the scales. You can think of each of them as undoing the effect of the other. Functions 11 For example, the area of a triangle is a function of both its base and height; the volume of a fixed mass of gas is a function of its pressure and temperature; the volume of a rectangular-shaped room is a function of the three variables: length, breadth and height.

This book, however, deals only with functions of one variable. Find the values of f 3 , f 3. Also find the value of f 3. Here are some examples which show how the dependent variable changes when the independent variable changes.

Example 2. It shows how the function changes when x changes. Remembering that the values of x are shown as continuously increasing from left to right, you can see the following features from the graph. This is called a turning point on the curve. For values of x which are negative, but which are numerically very large, y is also very large and positive. As can be seen from its equation, the y-value is found by dividing 1 by the corresponding x-value.

Note that a special feature of reciprocals is that by applying this function twice in succession to a number, it returns to its starting value. Variations in functions; limits 15 Example 2. These numbers, both very large and very small, are numbers which you can write down. They are finite numbers. If, however, you increase x so that it eventually becomes bigger than any finite number, however large, which you might choose, then x is said to be increasing without limit. From the positive branch, you can see the graphical interpretation of the conclusions reached above.

As x approaches infinity, the distance between the curve and the x-axis becomes very small and the curve approaches the x-axis as the value of x approaches infinity.

Geometrically, you could say that the x-axis is tangential to the curve at infinity. A straight line which is tangential to a curve at infinity, is called an asymptote to the curve. Both axes are asymptotes to the curve in negative directions. Notice also that throughout the whole range of numerical values of x, positive and negative, y is always decreasing. The sudden change as x passes through zero is a matter for consideration later. Variations in functions; limits 17 2.

Here is a way to find such a meaning. The idea of a limit is not only fundamental to calculus, but also to nearly all advanced mathematics. If x is given the value 2, both numerator and denominator become 0, and the function is not defined there.

But when x is close to 2, the function takes the form This form is called indeterminate. The form 00 is of great importance and needs further investigation. Start by giving x values which are slightly greater or slightly less than the value which produces the indeterminate form, that is 2.

Using a calculator, you find the following results. You can make the value of the fraction as close to 4 as you choose by taking x to be close enough to 2. The method used is similar to the one above, but in a more general form. Notice that you cannot put x equal to a. The value of this ratio approaches a limit as the numerator and the denominator approach zero.

You can find the limit of this ratio as follows. In Figure 2. Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. Understand calculus Item Preview.

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