Вариант № 10

The differential calculus

A The differential calculus is concerned with rates of change. In practical life we constantly come across pairs of quantities which are related, so that after both have been measured, when we know one, we know the other. Thus if we know the distance along a road from a fixed point we can find the height above sea level from a map with contours. If we know the time of day we can determine the air temperature on any particular day from a record of a thermometer made on that day. In such cases we often want to know the rate of change of one relative to the other.

B If x and y are the two quantities, then the rate of change of y to x is dy/dx. For example if x is the distance of a point on a railway from London, measured in feet, and y the height above the sea level, dy/dx is the gradient of the railway. If the height y increases by 1 foot while the distance x increases by 172 feet, the average value of dy/dx is 1/172. We say that the gradient is 1 in 172.

C The mathematical part of the calculus is the art of calculating dy/dx if y has some mathematical relation to x, for example is equal to its square or logarithm. The rules have to be learned like those for the area and volume of geometrical figures, and have the same sort of value. No area is absolutely square, and no volume is absolutely cylindrical. But there are things in real life like enough to squares and cylinders to make the rules about them worth learning. So with the calculus it is not exactly true that the speed of a falling body is proportional to the time it has been falling. But this is close enough to the truth for many purposes.

D The differential calculus goes a lot further. Think of a bus going up a hill which gradually gets steeper. If x is the horizontal distance, and y the height, this means that the slope dy/dx is increasing. The rate of change of dy/dx with regard to y is written d² y / dx². In this case it gives a measure of the curvature of the road surface. In the same way if x is time and y distance, d² y / dx² is the rate of change of speed with time, or acceleration. This is a quantity which good drivers can estimate pretty well, though they do not know they are using the basic ideas of the differential calculus.

E If one quantity depends on several others, the differential calculus shows us how to measure this dependence. Thus the pressure of a gas varies with the temperature and the volume. Both temperature and volume vary during the stroke of a cylinder of a steam or petrol engine, and the calculus is needed for an accurate theory of their action.

contours – геодез. горизонталь	slope - уклон
gradient – градиент, уклон	estimate - оценивать
steeper – более крутой	stroke – ход

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1. What is the purpose of differential calculus?

A … to measure the dependence on other disciplines

B … to find the value of one unknown

C … to calculate related quantities

D … to calculate the product of two quantities

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2. The passage A is about

A …the applying differential calculus to our everyday life problems

B… the way of finding the height above the sea level

C… the way to determine the air temperatures

D … the formulas used for differential calculus usage