# Introduction into two main categories: differentiation which is the

Introduction

In this essay, I shall begin by introducing
the basic ideas of calculus, and providing a brief history of its invention and
the initial controversy surrounding it. I will then move onto to talking about
the solution to this controversy; limits and how they make the results obtained
by calculus accurate, focusing specifically on their application in the process
of integration. Finally, I shall give some examples of real life uses of
calculus in order to solidify, in your mind, its importance to us.

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Calculus

Originally introduced by Newton
and Leibniz, calculus is defined as the “branch of mathematics dealing with
infinitesimal changes to a variable or quantity”.1
It allows us to make sense of the motion and dynamic change in the world around
us2,
and is split into two main categories: differentiation which is the process of
finding a derivative, and integration which is the process of finding an
integral.

Integration:

Differentiation:

Derivatives are concerned with the rate of change of quantities,
whilst integrals are concerned with the area enclosed by a curve and the
horizontal axis. The following general rules are followed:

Notation & Controversy

Both Newton and Leibniz worked
independently, but developed very similar ideas around calculus, representing
their work using different notation. This difference in notation is represented
by figure 1.3

Much debate has revolved around the topic of who came up with the
ideas around calculus first, with Leibniz even being accused of plagiarising Newton’s
work. The Royal Society, of which both Newton and Leibniz were members,
eventually gave credit to Newton for the discovery of calculus, and to Leibniz
for the first publication around calculus. 4

One concept both Newton and Leibniz
made use of in their work was infinitesimals. This in itself caused much
controversy and debate.

Figure 1:
Leibniz’s and Newton’s notation for Calculus

Infinitesimals

Infinitesimals are things which
can be treated like real numbers, but are so small that there is no real way of
measuring them. This concept is thought to have originally been introduced by Gottfried Wilhelm Leibniz when he was
developing his work around calculus, however he attributed the term to German
mathematician Nicholas Mercator. 5
& 6

In differentiation, an
infinitesimal change in x was denoted
by the dx, and one in y was denoted by dy. They ratio of dy to dx (dy/dx)
is the common notation now given to a derivative. In the case of integration,
many infinitesimals are summed together to give an integral which represents an
area under the curve of the relevant function.

This idea of infinitesimals
brought about a lot of controversy. In fact, it has been said that “the
founders knew that their use of infinitesimals was logically incomplete and
This incomplete logic was brought to light by the criticism of Irish
philosopher George Berkeley.

Berkeley’s
Criticism

In 1734, Berkeley published ‘The
Analyst’ wherein he attacked the foundations and logic of calculus. In
particular, he attacked the idea of infinitesimals holding the opinion that
they were self-contradictory. He noted that it was treated as finite in some
cases and zero in others in accordance to the problem at hand.8
The key point he was trying to get across in his work was that calculus was
devoid of a rigorous theoretical foundation. He wrote that: “In every other
Science Men prove their Conclusions by their Principles, and not their
Principles by the Conclusions. But if in yours you should allow your selves
this unnatural way of proceeding, the Consequence would be that you must take
up with Induction, and bid adieu to Demonstration. And if you submit to this,
your Authority will no longer lead the way in Points of Reason and Science”.9

When you analyse the criticism of
Berkeley, you can see that he is making a logical criticism of the foundations
of calculus. If in a calculation, the infinitesimal is at first taken to be
something, and then later to be nothing, this immediately raises concern
because it is known that for any quantity to be involved in a calculation, it
must remain the same throughout the calculation. Based on this, it is clear
that the infinitesimal value used in this case has contradictory properties,
and so Berkeley’s criticism was justified at that level. However, it was noted
that fluxions quantities ‘flowed’, and hence time was meant to play a central
role. Thus, English scientist James Jurin wrote, in response to Berkeley’s
criticism, that “if I say at one time, the increments now exist; and say an
hour after, the increments do not now exist; the latter assertion will neither
be contrary, nor contradictory to the former, because the first now signifies
one time, and the second now signifies another time, so that both assertions
may be true”. In that sense then, Berkeley’s criticism was incorrect.10

What is important to note is that
Berkeley did not dispute the results of calculus, but rather the theory behind
why it worked. This problem was later resolved by the introduction of limits.

Limits

In 1821, French mathematician Augustin-Louis
Cauchy introduced the idea of limits in his publication ‘Cours d’Analyse’.11 This idea was later formalised
in the work of German mathematician Karl Weierstrass, who eliminated the use of
infinitesimals entirely.

What
are limits?

In mathematics, limits are values
that a function or sequences ‘tends’ to. They are essential to calculus and are
used to define both derivatives and integrals. To give you an understanding of
limits, imagine you had the equation

. For x=1,
you would get

; an
answer we cannot quantify. So, let us approach this problem using a different
method.

work out a value for y when x=1, let us find values for y using x values which are approaching 1.
The results are shown in table 1.

x

y

0.5

1.5000

0.7

1.7000

0.9

1.9000

0.99

1.9900

0.999

1.9990

0.9999

1.9999

Table 1:
x-values and corresponding y-values

These results shows that as x gets closer to 1, y gets closer to 2. Thus, we can say that ‘the limit
of y as x approaches 1 is 2’. This can be represented by the
following notation:

.

Since I do not wish to drag this
piece on for too long, I shall focus solely on the use of limits in integration
and not go into detail regarding their use in differentiation.12

Limits
in Integration

In order to understand what is to
come, it is important to begin by clarifying what an integral is. An integral
is defined as the “sum of a large number of small quantities”.13
That is to say that it represent the sum of the areas of the small shapes that
a larger shape has been divided into.

In calculus, this is specifically
regarding the area enclosed by a graph and the horizontal axis. This idea of
integration therefore can alternatively be defined as “a means of finding areas
using summation and limits”. In order to understand this statement, it is best
to go through an example.

Consider the curve shown in
figure 2. Let’s say that we are required to find the area contained by the
curve, the horizontal x-axis, the
vertical y-axis and the line x=a. The first thing we would do would
be to separate the area under the curve into strips, each of width ?x and of area ?A (as shown in figure

P

Q

3).

Figure 2                                                                                                                            Figure
3

Take the point P, shown on the
curve in figure 3, which has coordinates (x,y). The inner rectangle therefore would
have an area of y?x. The point Q,
also shown in figure 3, has coordinates (x+?x,y+?y) because a small change in the x coordinate will cause a small change
in the y coordinate, and so the total
area of the rectangle would be given by (y+?y)
?x =y?x+?y?x.

In order to obtain an accurate
estimation for the area now, we would have to let ?x to tend to zero. This is so
that the strips get thinner, because the more strips you have, the greater the
accuracy of your value for the area. As ?x
tends to zero, ?y also tends to zero
and so the product of ?y?x would be
very small, so small in fact that it can be considered to be negligible.

Therefore, the area can be
written as

, which represents the sum of the
areas of all of the rectangles formed between x=0 and x=a, as ?x tends to zero. This
can be rewritten as

which is ‘the
integral of y with respect to x, between the limits 0 and a’.14

So, as you can see from the
explanation above, the integral sign is simply another way of representing the
sum of the areas of rectangles of height y
and horizontal sides of infinitesimally small width, denoted by the term dx. In fact, when Leibniz originally
introduced the sign, it was what is now an obsolete letter which was
essentially an elongated ‘s’, and this was meant to represent the idea that an
integral is a continuous sum of quantities. The use of limits ensures that only
the area required is found, and that the answer obtained is accurate.15

Calculus in the
Real World

Calculus plays a role in multiple
disciplines spanning from the sciences to engineering. In the final section of
this essay, I intend to give some examples of real world applications of
calculus, thereby demonstrating its great importance to us.

Maximising Profits

Suppose
that a block of flats has 250 flats up for rent.  If the owners were to rent x flats,
then their monthly profit can be found using the following:

Based on
this model, how many flats should they rent in order to maximise their profit?

So, the aim is to maximise the profit whilst sticking to the restriction
that x must be in the range

First, we need to find the derivative. This can
then be used to calculate the critical point that falls in the required range.

Since we
have a continuous function and an interval with finite bounds, the maximum
value can be found by entering in the critical point and the end points of the
range back into the original equation.

So, we have
found that if the owners were to rent out 200 of the flats, it would be more
profitable to them than renting out all 250.16

Newton’s Second Law of Motion (Science)

Newton’s
second law of motion states that the rate of change of momentum of an object is
proportional to the force acting on it. It can be represented by the
equation:

Here, we
see that the force is considered the rate of change of momentum, meaning it can
be expressed as

. It is also know that acceleration is
the time derivative of the velocity of an object (

. Hence, we can see that this principle
makes use of differential calculus.

Force Acting on the Wall of a Tank (Engineering)

Water is contained in a large tank with a depth of 6 meters (as show by
figure 4). An engineer is given the job of finding the equivalent force the
water creates on a unit width of the wall of the tank. How would he approach
this problem?

Firstly, it is important to visualise the problem at hand and this is
done by drawing a sketch which is something similar to that shown by figure 5.

6m

Unit width = 1

dh

Figure 4                                                                   Figure
5

If h is taken to be the height of water (the maximum of which is 6),
then the pressure generated will act over a fraction of h, and so therefore the height of the section being studied has
been denoted as dh.

Force is given by the equation

, and pressure by the equation

. Using our sketch, it can be seen that
the area of the studied section is given by 1·dh.
Therefore

and this can be converted into an integral
with limits 0 and 6,

, which eventually gives

.

Conclusion

From the controversies around who
first came up with the idea, to those surrounding the use of infinitesimals,
calculus has enjoyed a rocky ride to the top. The introduction of limits helped
solidify the foundations upon which it had been built, and thereby its validity
in the minds of many. It has now become an integral part of the mathematics in
countless disciplines.

1 Collins, 2006. Collins English Dictionary. 1st ed. Glasgow:
HarperCollins.

2 Mastin, L., 2010. 17TH CENTURY MATHEMATICS – NEWTON. Online
Available at: http://www.storyofmathematics.com/17th_newton.html
Accessed 27 November 2017.

3 Mastin, L., 2010. 17TH CENTURY MATHEMATICS – LEIBNIZ. Online
Available at: http://www.storyofmathematics.com/17th_leibniz.html
Accessed 27 November 2017.

4 Mastin, L., 2010. 17TH CENTURY MATHEMATICS – LEIBNIZ. Online
Available at: http://www.storyofmathematics.com/17th_leibniz.html
Accessed 27 November 2017.

5 Katz, M. & Sherry, D., 2013. Leibniz’s Infinitesimals: Their
Fictionality, Their Modern Implementations, and Their Foes from Berkeley to
Russell and Beyond. Erkenntnis, 78(3), pp. 571-625.

6 Wikipedia, 2017. Infinitesimal. Online
Available at: https://en.wikipedia.org/wiki/Infinitesimal
Accessed 26 November 2017.

7 Stroyan, K. D., 1997. Mathematical
Background: Foundations of Infinitesimal Calculus. 2nd ed. s.l.:Academic
Press.

8 Wisdom, J., 1953. Berkeley’s
criticism of the infinitesimal. British Journal for the Philosophy of
Science, 4(13), pp. 22-25.

9 Wikipedia, 2017. George Berkeley. Online

Available at: https://en.wikipedia.org/wiki/George_Berkeley
Accessed 25 November 2017.

10 Vickers, P., 2007. Was the Early
Calculus an Inconsistent Theory?. Online
Available at: http://philsci-archive.pitt.edu/3477/1/Early_calculus_August07.pdf
Accessed 27 November 2017.

11 Wikipedia, 2017. Augustin-Louis
Cauchy. Online
Available at: https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy
Accessed 27 November 2017.

12 Math is Fun, 2017. Limits (An
Introduction). Online
Available at: https://www.mathsisfun.com/calculus/limits.html
Accessed 28 November 2017.

13 Collins, 2006. Collins English
Dictionary. 1st ed. Glasgow: HarperCollins.

14 mathcentre, 2009. Integration as
Summation. Online
Accessed 4 December 2017.

15 What does/did the Integral Sign represent?
Online

Available at: http://math.ucr.edu/~res/math009B-2012/integral-sign.pdf

Accessed 7 December 2017

16 Dawkins, P., 2003. CALCULUS I –
NOTES. Online