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.

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

could lead to incorrect results”.7

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.

Instead of trying to

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

(Business)

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

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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

Available at: http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-intassum-2009-1.pdf

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

Available at: http://tutorial.math.lamar.edu/Classes/CalcI/BusinessApps.aspx

Accessed 29 November 2017.