Calculus has a rather fearsome reputation with the general public. There is a reason for this reputation. The foundation upon which calculus is built can be quite confusing for beginners. This is because most people are not very comfortable with precise logical thinking, and there is a picky precision to most of the fundamental definitions upon which calculus based.
Easy calculus offers a basis for calculus that is more accessible for beginners in three ways. First of all, mainstream calculus is formulated in terms of limits. These limits are defined by a somewhat complicated mathematical expression. In simple English, this definition amounts to saying a limit is a really good approximation. In easy calculus we cut out the middleman and formulate calculus directly in terms of really good approximation.
More than anything else, calculus is the mathematical study of change, and rates of change. Consider a falling object. In t seconds the object will fall 16 t squared feet or 4.9 t squared meters or 490 t squared centimeters. For simplicity we will say the object falls t squared units in t seconds. We will ask how far the object falls between time t and another nearby point in time. We will also ask how fast the object was falling during that time interval. Since Blogger is not set up well for using superscripts, we will write y = t squared as simply y=tt. We will let t be our first point in time, but we will also need to indicate a second point in time. There are two approaches we could take to do this.
Approach A. We will let T be our second point in time and let Y=TT. The distance the object falls from time t to time T will be Y - y = TT - tt. To find the rate at which the object is falling we must divide TT - tt by T - t. To do this we first factor a difference between two squares so that we get TT - tt = (T + t)(T - t).
We get (TT - tt)/(T - t) = (T + t)(T - t)/(T -t) = (T + t).
If T is very close to t then this rate of T + t will be very close to t + t = 2t. We will call 2t the derivative of the function y = tt at t. It represents a very good approximation for the rate of change from time t to a nearby time T.
Approach B. This time we will call our second point in time t+h. The distance the object falls from t to t+h will be (t+h)(t+h) - tt = (tt + 2th +hh) - tt = 2th + hh.
So far we have done more work than we did with Approach A, but now our next step is now quite simple. In order to divide 2th+hh by h we simply write 2th + hh = (2t+h)h and dividing by h we get a rate of 2t+h. Finally we note that t+h will be close to t when h is close to zero, and in that case 2t+h will be close to 2t. Once again we get 2t as the derivative of the function y = tt, a very good approximation of the rate change from time t to a nearby time t+h.
Mainstream calculus uses Approach B, while we will take Approach A in easy calculus. This is the second way in which easy calculus is different from mainstream calculus. Approach A was easy for y = tt since you probably knew how to factor the difference between to squares. But how do you factor the difference between two cubes or two fourth powers or two fifth powers? Actually the answer to this question can be found in a typical second year high school algebra text--but it is not something students are likely to remember. So we will not assume this is something you know. Instead we will give an example which will illustrate a technique that can always be used in these situations.
Consider z = ttt and Z = TTT. What is the change from ttt to TTT? It is helpful to think of the change as being made up of three parts: a change of ttt to Ttt followed by a change of Ttt to TTt and then a change of TTt to TTT.
(1)The change from ttt to Ttt is tt(T-t).
(2)The change from Ttt to TTt is Tt(T-t).
(3)The change from TTt to TTT is TT(T-t).
The total change is tt(T-t) + Tt(T-t) + TT(T-t).
We can add together these three terms by adding together the parts that are different, and multiplying our sum by the part (T-t) that is common to every term.
We get (tt + Tt + TT)(T-t).
When we divide this by the time difference of (T-t) we get a rate of (tt +Tt + TT).
If T is close to t, our rate of (tt + Tt +TT) will be close to (tt + tt +tt) = 3tt.
3tt is the derivative of the function z = ttt at t, our approximation for the rate of change from time t to a nearby time T.
PART TWO
The third way in which easy calculus differs from mainstream calculus involves the definition of what we mean by a really good approximation. We can formulate a version of easy calculus that is equivalent to mainstream calculus, but there is also a beginner's version of easy calculus where instead of using a really good approximation we use a really really good approximation.
The diameter of a circle is twice the radius or d = 2r. For a second value of the radius R we would get D = 2R. The change in diameter D - d = 2R - 2r = 2(R - r). We can make the change in diameter as small as we wish if we are willing to make the change in radius sufficiently small.
If we want to make the size or absolute value of 2(R - r) less than 0.01, we need only make the size or absolute value of R - r less than 0.01/2. In general if we want to make the size of 2(R-r) less than any positive p, we need only make the size of R-r less than p/2. In calculus we refer to this property as continuity.
The circumference of a circle is pi times the diameter or c = pi d. Then a second value D for the diameter gives us C = pi D. We get C - c = pi D - pi d = pi(D - d). If we want the change in circumference to be less than, say 0.01, we need the change in diameter to be less than 0.01/pi. But there is a problem. Pi is an irrational number. It isn't easy to divide by an irrational number. So what do we do?
We do not actually have to know the exact value of pi to show tht we have continuity. We know that pi is less than 4. If we make the size of D - d less than 0.01/4 then the size of 4(D - d) must be less than 0.01--and if the size of 4(D - d) is less than 0.01, then certainly the size of pi(D - d) must also be less than 0.01. The same argument would hold with any other positive number in place of 0.01, and so our function c = pi d is continuous.
Now look at the function z = ttt or Z = TTT that we considered above. For a given change in t we get a change is z that is Z - z = TTT - ttt = (tt + Tt + TT)(T - t). This is a multiple of (T - t). If we want (Z - z) to be less than some positive p, we only need (T - t) to be less than what we get when we divide p by (tt + Tt + TT).
Unfortunately, we cannot know the value of (tt + Tt + TT) until we know the value of both t and T--and that is something we do not yet know. So this won't work. But we might try the trick we used with the circumference of the circle. We don't actually have to know the value of (tt + Tt + TT). It is enough if we know that the size or absolute value of (tt + Tt + TT) can be no more than some number K.
If t and T are allowed to assume any values, there can be no limit to the size of (tt + Tt + TT), but if we are willing to place limits on the sizes of t and T, we can then place a limit on the size of (tt + Tt + TT). If we restrict t and T to absolute values no more than a thousand, then tt and Tt and TT will each have an absolute value no more than a million, and their sum will have an absolute value no more than 3 million. More generally if we restrict the absolute values of t and T to no more than B, then tt and Tt and TT will each have absolute value no more than BB, and their sum will have absolute value no more than 3BB.
For any positive p, if the change in t is no more than p/(3BB) then the change in our function z will be no more than p--and our function z will be continuous.
We can see that given any function y if the size of (Y - y) is no more than some constant K times the size of (T - t) then we can make the size of (Y - y) smaller than any positive p if we are willing to make the size of (T - t) less than p/K.
The mainstream definition of continuity is based upon being able to make the size of (Y - y) as small as we wish as long as T is sufficiently close to t. In the beginner's version of easy calculus we ask that this can be done in the particularly simple fashion that we have just seen. We ask that the size of the difference (Y - y) be no more than some constant K times the size of (T - t). This will make working with continuous functions easier for beginners.
PART THREE
In Part Two I shifted the focus from the rates of change that we considered in Part One to the question of continuity. I did this because continuity is also something that is important in calculus, and because the points I wished to make in Part Two were easier to make when we talk about continuous function rather than functions which have a derivative. But not I would like to shift our focus back to functions with a derivative.
Recall y = ttt and Y = TTT. We saw (Y - y) = (tt + Tt + TT)(T - t) and we claimed that the rate of change (tt + Tt + TT) was close to our derivative tt + tt + tt = 3tt. We will noe look at this claim more closely. We can change tt + tt + tt into tt + Tt + TT one term at a time. We can change each term one letter at a time. (1) The first term requires no change. (2) The second term requires one t to be changed into T. When tt is changed to Tt the size of the change is t(T - t). (3) The third term requires two changes. The first change is from tt to Tt and the size of that change is t(T-t) as we have just seen. The second change is from Tt to TT and the size of this change is T(T - t).
Altogether we have 0 + 1 + 2 = 3 changes of t(T - t), t(T - t) and T(T- t) which add up to a total change of (2t + T)(T - t) when we change the derivative 3tt = tt + tt + tt into the actual rate of change tt + Tt + TT. As long as we require that the absolute values of t and T be no more than B we get (2t + T) has an absolute value no more than 3B. This means that when we use the derivative as our estimate of the rate of change the size of our error of (2t + T)(T - t) can be no more than 3B times the size of (T-t). For any positive p, we can make the size of this error no more than p as long as we make the size of (T - t) no more than p/(3B).
In mainstream calculus we have shown that the function y = ttt is a differentiable function with derivative 3tt, at least for values of t from -B to +B. This is because the error in our approximation can be made arbitrarily small sufficently small changes in t. The derivative 3tt is a really good approximation for (tt + Tt + TT) which is the actual rate of change.
But we can say more than this. Not only does the error in our approximation become arbitraryily small for small changes in t--but it does so in a particularly nice fashion. If we set L = 3B, we see the size of the error is never more than L times the size of (T-t) for a constant L. This is a really really good approximation. Errors of this type are particularly easy to work with. In practical terms, using such errors results in only a very slight loss of generality--so these are the errors we will use in our beginner's version of easy calculus.
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