Introduction to differential equations
Differential equations are any equations that include derivatives and arise in many situations. This free course, Introduction to differential equations, considers three types of first-order differential equations. Section 1 introduces equations that can be solved by direct integration and section 2 the method of separation of variables. Section 3 looks at applications of differential equations for solving real world problems. Section 4 introduces the integrating factor method for solving linear
Modelling and estimation
This free course is concerned with modelling and estimation and looks in particular at the binomial distribution. Section 1 starts by defining probability, introduces relevant notation and briefly discusses basic properties of probabilities. The section concludes by considering some of the general features of and ideas about modelling discrete random variables. Section 2 looks at one particular probability model for discrete data, the binomial distribution. Section 3 investigates how data can be
Modelling events in time
This free course develops ideas about probability and random processes. Sections 1 and 2 introduce the fundamental ideas of random processes through a series of examples. Section 3 describes a model that is appropriate for events occurring ‘at random’ in such a way that their rate of occurrence remains constant. Section 4 derives the main results from Section 3. Section 5 introduces the multivariate Poisson process in which each event may be just one of several different types of event. Sect
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3.20 Multiplication rules The rules for multiplying positive and negative numbers can be illustrated by the table below. Multiplying a positive number by a positive number gives a positive answer. Multiplying a negative number by a positive number gives a negative answer. Multi 3.17 Subtraction of negative numbers Next consider subtraction of a negative number. In terms of Thomas’s piggy bank, subtracting a negative number is the same as taking away one of his IOUs. If his mother says ‘you have been a good boy today so I’ll take away that IOU for £3’ this is equivalent to him being given £3. So, − (−3) = 3. Does this correspond with the number line interpretation of subtracting a negative number? Consider the evaluation of 8 − −3. Continue to think o Try some yourself Evaluate each of the following and give an example from everyday life to illustrate the sum (e.g. Thomas's piggy bank). (a) −4 − 6 (b 3.13 More division with fractions How about other fractions? What is 6 ÷ Try some yourself Evaluate each of the following. 3.8 Order of decimal calculations The same rules about the order of calculations apply to decimals as apply to whole numbers. Brackets; Powers (e.g. squaring or cubing a number); Division and Multiplication (performed in the order written, left to Try some yourself Insert brackets in the following calculations to emphasise the order in which a scientific calculator would perform them, then do the calculations by hand and on your calculator, with and without the bracke 3.4 Order of calculations You may have noticed that sometimes the order in which calculations are carried out seems to matter and sometimes it does not. When using a calculator, it is very important to know the order in which it will do calculations. It is not always the order in which you enter them. Although written English is read from left to right, this is not the case for all written languages (Chinese is read top to bottom, right to left). With mathematics, the order of the written operations does not alw Try some yourself Carry out the following calculations, without using a calculator. (a) A million pound lottery prize minus a three hundred pound administrative charge. 2.5 Measuring capacity The basic metric unit for capacity is the litre, usually denoted by the symbol l (though sometimes an uppercase L is used to avoid confusion with the number 1). In the SI system, units such as cubic metres (m3), cubic centimetres (cm3) and cubic millimetres (mm3) are used. These two systems are linked because: 1 ml = 1 cm3 The animation below i 2.3 Measuring mass The basic SI unit for mass is the kilogram, symbol kg The tonne (t) which is equivalent to 1000 kg and is a metric unit is often used alongside the SI units. The animation below illustrates how to convert between the most commonly used units of mass, the metric tonne (t); the kilogram (kg); the gram (g); the milligram (mg) and the microgram (μg). 2.2 Converting units A great advantage of the metric system of units is that conversion between units within the system is particularly easy. For example, ‘£1 is worth 100p’ is converting one pound into pence. To convert pounds to pence, you multiply by 100. So £2 is 200p, and £2.63 is 263p. (Remember that to multiply by 100, you move the digits two places to the left in the place value table.) To convert from pence to pounds, you need to reverse this process, i.e. to divide by 100 (moving the 2.1.2 Try some yourself Suggest appropriate units for each of the following: (a) the age of the kitten when it is weaned; (b) the distance between one train station and the Try some yourself Contour lines on a map show all the points at a given height above sea level. The lines are drawn for each height at 50-metre intervals, and points below sea level are shown by negative heights. The diagra 1.9 Negative numbers Numbers can be positive or negative, i.e. greater than or less than zero. Negative numbers have several uses; for example, to measure temperatures below zero, such as −3°C (‘minus 3 degrees Celsius’). They are also used to represent debts and overdrawn accounts: a bank balance of −£84.33 means ‘overdrawn by £84.33’. Negative numbers are shown on the number line to the left of 0. The animation below shows −8, −7, − 1.7 Fractions A fraction is written as one number over another (such as
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? This means how many Author(s):
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Author(s):
Calculations are performed in the following order:
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) and means the top number divided by the bottom number. The top number, 3, is called the numerator and the bottom number, 10, is call
