Slide 1 : WHAT IS CALCULUS ALL ABOUT?
Let us take a few commonplace examples first : Let us take a few commonplace examples first A BOOKSTORE CAN OBTAIN A BOOK FROM A PUBLISHER AT RS. 60 PER COPY AND SELL IT AT RS. 100 PER COPY. AT THIS SELLING PRICE THE STORE HAS BEEN ABLE TO SELL ONLY 150 COPIES PER MONTH. TO STIMULATE THE SALES THE STOREKEEPER DECIDES TO OFFER A DISCOUNT AND CALCULATES THAT IF HE OFFERED A DISCOUNT OF 10% HE COULD SELL 25 COPIES MORE PER MONTH. WHAT SHOULD BE THE SELLING PRICE SO THAT THE PROFIT IS MAXIMUM?
Slide 3 : A CABLE IS TO BE LAID FROM A POWER PLANT ON ONE SIDE OF A RIVER 1200 METRES WIDE TO THE OTHER SIDE 1500 METRES DOWNSTREAM. THE COST OF LAYING THE CABLE UNDER WATER IS RS.2500 PER METRE WHILE THE COST ON LAND IS RS.2000 PER METRE. WHAT IS THE MOST ECONOMICAL ROUTE TO RUN THE CABLE?
Here is a third problem: From a rectangular piece of card, a box has to be made by cutting off the pink squares and folding the sheet up. What size squares have to be cut off so as to make the volume of the box maximum? : Here is a third problem: From a rectangular piece of card, a box has to be made by cutting off the pink squares and folding the sheet up. What size squares have to be cut off so as to make the volume of the box maximum?
The answer to all these questions can be found in CALCULUS : The answer to all these questions can be found in CALCULUS
Before introducing the subject you may be interested to know who the brain behind Calculus is? : Before introducing the subject you may be interested to know who the brain behind Calculus is?
During 1684, a mathematics paper packed with Latin words and mathematical symbols appeared in a journal at the end of which there was the word CALCULI which means set of rules. The author was Leibniz. This was the first published paper on Calculus and it was entitled, A new method for calculation of maxima and minima as well as tangents. : During 1684, a mathematics paper packed with Latin words and mathematical symbols appeared in a journal at the end of which there was the word CALCULI which means set of rules. The author was Leibniz. This was the first published paper on Calculus and it was entitled, A new method for calculation of maxima and minima as well as tangents.
Slide 8 : GOTTFRIED WILHELM LEIBNIZ
1642-1727 A.D.
But the real ‘super star’ of the mathematical world was not Leibniz. It was Sir Isaac Newton who developed the subject at least 20 years earlier (in 1664) under the name ‘FLUXIONS’ which also meant ‘maxima and minima’’ : But the real ‘super star’ of the mathematical world was not Leibniz. It was Sir Isaac Newton who developed the subject at least 20 years earlier (in 1664) under the name ‘FLUXIONS’ which also meant ‘maxima and minima’’
Slide 10 : SIR ISAAC NEWTON
1642 – 1727 A.D.
Who invented Calculus – the English mathematician Newton or the German mathematician Leibniz?This controversy lasted several centuries. Many committees appointed to probe this issue failed to find satisfactory results.Whoever be the inventor of such an important and useful subject, the question for us is : WHAT did they find? : Who invented Calculus – the English mathematician Newton or the German mathematician Leibniz?This controversy lasted several centuries. Many committees appointed to probe this issue failed to find satisfactory results.Whoever be the inventor of such an important and useful subject, the question for us is : WHAT did they find?
In any traditional undergraduate program Calculus has come to serve as an entry (or is it barrier?) to higher mathematics. The subject has become an indispensable tool for engineers, chemists, physicists, economists etc.Only in the 17th century, several ideas were put together and resulted in the Calculus of today. : In any traditional undergraduate program Calculus has come to serve as an entry (or is it barrier?) to higher mathematics. The subject has become an indispensable tool for engineers, chemists, physicists, economists etc.Only in the 17th century, several ideas were put together and resulted in the Calculus of today.
Slide 13 : Calculus, sometimes called the ‘mathematics of change’ is the branch of math which describes precisely how one change in a variable quantity is related to a corresponding change in another variable quantity. In almost every human activity we encounter 2 types of variables – those which we can control directly and those that we cannot. Fortunately those which is beyond our control often depend in some way with those which can be controlled by us. For example, the acceleration of a car responds to the way in which we control the flow of petrol to the engine, the level of antibiotic in a patient’s bloodstream responds to the dosage and timing of the drug, the rate of inflation of an economy to the way in which the government controls the money supply and so on.
Thus Calculus has an enormous, often unnoticed, impact on our daily lives. Whether it is Fingerprint analysis, weather prediction. Noise control in a music recording, air flow past an aircraft, DNA structure or Space missions Calculus has an important role to play.
To those who have a background of some elementary geometry, here is what Calculus is about…. : To those who have a background of some elementary geometry, here is what Calculus is about…. The concept of slope of line which had all along been a part of Geometry was brought to Calculus. You know from elementary Analytic Geometry that if you are given 2 points on a line it is possible to find the slope of the line. Can you find the slope of a line when just one point on the line is given?Mr. Geometry says ‘No’ while Mr. Calculus says ‘yes’!
Let us see what this ‘slope’ is all about. : Let us see what this ‘slope’ is all about. In these diagrams how does y change when x changes?
In the first case, y increases slowly when x increases while in the second case y increases fast as x increases. We say that the second line has greater ‘slope’ than the first line. : In the first case, y increases slowly when x increases while in the second case y increases fast as x increases. We say that the second line has greater ‘slope’ than the first line.
Slope of a line is measured by a RATIO, namely rise / run In the language of Trigonometry, slope of a line is the tangent of its inclination.Slope of a curve is defined as the slope of the tangent to the curve and to calculateslopes we need Calculus. The Calculus used to calculate slopes is called the Differential Calculus. : Slope of a line is measured by a RATIO, namely rise / run In the language of Trigonometry, slope of a line is the tangent of its inclination.Slope of a curve is defined as the slope of the tangent to the curve and to calculateslopes we need Calculus. The Calculus used to calculate slopes is called the Differential Calculus.
Slide 18 : Here is a business problem which can be solved using Calculus. Suppose this graph shows the profits made by a Firm during a
5-year period. Production level is shown on the x axis while profits are shown on the y axis. What is the production level at which the profit is highest? This question is same as ‘What is the highest point on the curve?’
The point at which the TANGENT is horizontal is the highest point
and Calculus helps in finding this point.
Another common business problem is what is called ‘marginal cost’. Suppose the total monthly cost of manufacturing x cars is C(x). The cars are so priced that every car made during the month is sold in the same month. If the Company decides to increase production, then what will be the additional cost involved for each additional car made? The answer can be found in Calculus. : Another common business problem is what is called ‘marginal cost’. Suppose the total monthly cost of manufacturing x cars is C(x). The cars are so priced that every car made during the month is sold in the same month. If the Company decides to increase production, then what will be the additional cost involved for each additional car made? The answer can be found in Calculus.
Here is another problem which can be solved by using Calculus: Suppose M(x) represents the marks you will score when you study for x hours per day. Let us suppose that this function is given by M(x) = (16x + 2)/(3x + 1). The question is, if you increase your study hours from the present 4 hours a day to 5 hours a day, how much more can you score? : Here is another problem which can be solved by using Calculus: Suppose M(x) represents the marks you will score when you study for x hours per day. Let us suppose that this function is given by M(x) = (16x + 2)/(3x + 1). The question is, if you increase your study hours from the present 4 hours a day to 5 hours a day, how much more can you score?
Graphs of functions play an important role in Calculus.We will be discussing functions and graphs in another sessionIn the following slides you can see some interesting curves which are all graphs of some functions. Some of the curves occur in nature. : Graphs of functions play an important role in Calculus.We will be discussing functions and graphs in another sessionIn the following slides you can see some interesting curves which are all graphs of some functions. Some of the curves occur in nature.
Here is a view of archimedian spiral named after the famous mathematician Archimedes : Here is a view of archimedian spiral named after the famous mathematician Archimedes
Here is another view of archimedian spiral : Here is another view of archimedian spiral
This is an equiangular spiral : This is an equiangular spiral
Spirals in nature : Spirals in nature
This is also an equiangular spiral : This is also an equiangular spiral
Calculus deals with 2 basic problems1. Calculation of slopes of tangents. 2. Calculation of areas of regions.These two totally unrelated problems are linked together by the DERIVATIVE concept. : Calculus deals with 2 basic problems1. Calculation of slopes of tangents. 2. Calculation of areas of regions.These two totally unrelated problems are linked together by the DERIVATIVE concept. Calculation of slope is a part of DIFFERENTIAL CALCULUS while the problem of calculation areas is a part of INTEGRAL CALCULUS
What is Integral Calculus all about?Let us take an example: : What is Integral Calculus all about?Let us take an example: Suppose the fish population in a river is 250, 000 at
present. Due to water pollution by humans this
population is expected to decrease slowly.
Environmentalists predict that x months from now the
fish population will be reducing at the rate of 0.2x – 10
thousand per month. What will be the population of fish
in the river 10 months from now?
Here is another problem:What is the area of the region shown in this diagram? : Here is another problem:What is the area of the region shown in this diagram?
The answer to all the above questions can be found in INTEGRAL CALCULUS : The answer to all the above questions can be found in INTEGRAL CALCULUS By now you may have some idea of what Calculus is all about……