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My favourite topic in Maths is calculus (or as we call it, differentiation and integration). I don’t find this topic easy however, the effect that it has had in many areas of the world we see today continues to fascinate me. In this article, I want to portray the surprising ways in which I have found differentiation and integration to be useful in ‘real life’ situations, that I never would have associated with mathematics. I believe that many of these uses are overlooked or not taught enough in schools leading to many myths about the topic’s relevance outside of maths.Differentiation and integration, otherwise known as calculus, is an area of maths which enables us to understand the infinitely small changes between values that are related by a function. I was originally intrigued by this topic due to the origin of the word ‘calculus’. Calculus originated from the Latin word for ‘small stone’, which left me puzzled as to why this was the case and how it related to differentiation and integration. I later discovered that pebbles were once used by mathematicians for counting which explained their relationship with differentiation (cutting something up into small pieces to discover how it changes) and integration (joining small pieces together to make them whole). After completing various questions on this topic in class, I couldn’t understand how it was useful or applicable to the ‘real world’ until my teacher introduced the class to examples of where differentiation and integration could be used in day to day life.The example I remember was the one of smoke rooms that were once used in restaurants to allow people to smoke inside, but away from customers. However, the issue was that smoke would escape every time the door was opened, which reached customers in the restaurant. Differentiation could be used here to form an equation in order to work out how much smoke escaped every time the door was opened. This could allow the restaurant to better locate the smoke room in relation to customers to reduce the volume of smoke reaching them. Although it was fascinating to see how calculus could be used ‘out of the classroom’, I wanted to know how it could be applied to me or relate to my other A-level subjects (Business and PE). After doing some research, I discovered that differentiation was particularly useful in business in numerous ways. I found a couple of examples from ; the first being to minimise production costs. For example, if a chocolate manufacturer has the capability of producing 60,000 chocolate bars in a day and the total cost of producing x number of chocolate bars in a day is given by the equation:C(x) = 250,000 + 0.08 x + 200000000xwe can work out how many chocolate bars per day they should produce in order to minimise production costs through the use of differentiation, as shown below:C’(x) = 0.08 - 200000000x2 = 0So, x2 = 2,500,000,000Therefore, x = 50,000So, the chocolate manufacturers should produce 50,000 chocolate bars a day to minimise production costs. Another use for differentiation in business is to help businesses maximise their profits. An example of this would be to calculate how many boats a boat manufacturing business, that has the capacity to produce 1500 units a year, should try to sell in order to maximise profits. If they sell x number of boats in a year, then their profits (given in pounds) will be given by the equation:P(x) = 30,000,000 – 360,000 x + 750x2 - 13 x3After differentiation,P’(x) = -360,000 + 1,500 x - x2= (x -1200)( x -300) = 0By substituting x = 1,200 and x = 300 back into the equation, we find that producing 1,200 units in a year will maximise the company’s profits (they will generate ?102,000,000 in profit). These are just two examples of how differentiation can be used by businesses and there are many more, such as calculating how demand changes with respect to price. These uses for differentiation are pivotal to the success and competitiveness of businesses, by allowing them to minimise unit costs and maximise profits to reinvest back into the company. Having such a strong passion for business and economics, I found the idea that differentiation could enhance business performance exciting, as I hope to be able to use it in the future when setting up a business of my own. I still wondered whether differentiation and integration could be used elsewhere, for example, in sport. I never considered maths playing any role in sport, but calculus has proved that it does. The sport that appeared to rely most heavily on this was Formula 1 racing, where differentiation and integration are used to calculate the fastest route that should be taken on a track, the optimal shape for a car to reach maximum velocity, and many other performance enhancing methods. I felt that using differentiation to work out the optimal shape for the race car was especially noteworthy as this could save substantial sums of money and lengths of time spent on creating and testing prototypes. Additionally, using this method would produce the highest performing racing vehicle.A huge success-determining factor in Formula 1 racing is the ability to save time. Cars can’t carry high volumes of fuel as this can cost 0.03 seconds per kg of fuel per lap. However, taking a stop to refuel is also time consuming, so must be completed at the optimum stage in the race to reduce time wasted. The best lap to refuel on can be calculated through differentiation. For example (using information from the royal academy of engineering - ):4708805166606Lend00Lend4762264395324L200L2123640417721L200L21233229396284L100L1Total race time = t0 l+ L2Wl- Wl22 + Lend-L2×tf+tp + t0l+ LendWl- Wl22 Where W = fuel lap weight effect, tp = extra time taken per lap with a pit stop and no refuel, Lend = total number of laps, L2 = stop lap, L1= 0 laps and tf = time taken to add 1 lap of fuel.We differentiate with respect to L2 and setting LHS to zero to find the stop lap, L2, that gives the minimum value for race time:0 = 2WL2- tf- LendWAfter substituting in the known values we calculate that:L2= 502 + 0.52×0.09 = 25 + 2.78 = 27.78So, rounding up to the nearest lap, we know that stopping to refuel on lap 28 will ensure the fastest race time.Having always thought of Formula 1 racing as a sport entirely controlled by the driver, it was incredible to see how maths plays such a vital role in the success of sporting role models such as Louis Hamilton. This really made me question how often differentiation is used in other sports and I found that research has been done to work out the optimum angle for a basketball to be released at in order to maximise the chances of reaching the basket. It can also be used to calculate statistics in baseball or football. With a huge passion for playing football, I found it insane that differentiation could be used to enhance sporting performance. Differentiation could help top level performers to achieve marginal gains in order to improve success against opponents; another incredible use for differentiation in the real world!In conclusion, a relatively unpopular mathematics topic with huge myths and misconceptions about its relevance to ‘life outside of the classroom’, is my favourite topic because it can be used to enhance performance in many different aspects of life, many of which are still yet to be discovered. I found that differentiation can be applied to areas of life that I had never associated with mathematics therefore, I genuinely believe that it can be of use to anyone. I hope that some of the ‘everyday’ examples of where differentiation is used can be more recognised in school to break the stigma attached to such a powerful area of maths. ................
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