In Part 1 of this programme on Complex Numbers, we discovered how to manipulate them in adding, subtracting, multiplying and dividing. We also finished Part 1 by seeing that a complex number a + jb can also be expressed in Polar Form, which is always of the form r(cos θ + j sin θ).
When you were first introduced to trigonometry, it is almost certain that you defined the trig, ratios — sine, cosine and tangent — as ratios between the sides of a right-angled triangle. You were then able, with the help of trig, tables, to apply these new ideas from the start to solve simple right-angled triangle problems ..... and away you went.
You are quite familiar with the method of solving a pair of simultaneous equations by elimination. We could first find the value of x by eliminating y. To do this, of course, we should multiply (i) by 4 and (ii) by 3 to make the coefficient of y the same in each equation.
Here is a revision list of the standard differential coefficients which you have no doubt used many times before. Copy out the list into your notebook and memorize those with which you are less familiar — possibly Nos. 4, 6, 10, 11, 12. Here they are:
You already know that the symbol sin−1x (sometimes referred to as ‘arcsine x’) indicates ‘the angle whose sine is the value x’. e.g. sin−1 0.5 = the angle whose sine is the value 0.5 = 30°
In the first part of the programme on partial differentiation, we established a result which, we said, would be the foundation of most of the applications of partial differentiation to follow.
The purpose of this programme is eventually to devise a reliable method for establishing the relationship between two variables, corresponding values of which have been obtained as a result of tests or experimentation. These results in practice are highly likely to include some errors, however small, due to the imperfect materials used, the limitations of the measuring devices and the shortcomings of the operator conducting the test and recording the results.
In the first programme (No. 11) on series, we saw how important it is to know something of the convergence properties of any infinite series we may wish to use and to appreciate the conditions in which the series is valid.
You are already familiar with the basic principles of integration and have had plenty of practice at some time in the past. However, that was some time ago, so let us first of all brush up our ideas of the fundamentals.
I. Consider the integral From our work in Part 1 of this programme on integration, you will recognize that the denominator can be factorized and that the function can therefore be expressed in its partial fractions.
In an earlier programme on integration, we dealt with the method of integration by parts, and you have had plenty of practice in that since that time. You remember that it can be stated thus: So just to refresh your memory, do this one to start with.
We now look at some of the applications to which integration can be put. Some you already know from earlier work: others will be new to you. So let us start with one you first met long ago.
In the previous programme, we saw how integration could be used
(a)
to calculate areas under plane curves,
(b)
to find mean values of functions,
(c)
to find r.m.s. values of functions.
We are now going to deal with a few more applications of integration: with some of these you will already be familiar and the work will serve as revision; others may be new to you. Anyway, let us make a start, so move on to frame 2.
The amount of work that an object of mass m, moving with velocity v, will do against a resistance before coming to rest, depends on the values of these two quantities: its mass and its velocity.
In previous programmes, we have seen how to deal with various types of integral, but there are still some integrals that look simple enough, but which cannot be determined by any of the standard methods we have studied.
A differential equation is a relationship between an independent variable, x, a dependent variable, y, and one or more differential coefficients of y with respect to x.
Many practical problems in engineering give rise to second order differential equations of the form where a, b, c are constant coefficients and f(x) is a given function of x. By the end of this programme you will have no difficulty with equations of this type.
Statistics is concerned with the collection, ordering and analysis of data. Data consist of sets of recorded observations or values. Any quantity that can have a number of values is a variable. A variable may be of one of two kinds:
(a)
Discrete — a variable that can be counted, or for which there is a fixed set of values,
(b)
Continuous — a variable that can be measured on a continuous scale, the result depending on the precision of the measuring instrument, or the accuracy of the observer.
In very general terms, probability is a measure of the likelihood that a particular event will occur in any one trial, or experiment, carried out in prescribed conditions. Each separate possible result from a trial is called an outcome.
