The trigonometric functions can all be associated with ratios of the lengths of the various sides of a right-angled triangle. Consider the following right- angled triangle containing the angle θ.
©©©© ©©©© ©© θ h a b
The values of the various trigonometrical functions (sine, cosine and tangent) are defined as follows:
sin θ = a/h = opposite/hypotenuse cos θ = b/h = adjacent/hypotenuse tan θ = a/b = opposite/adjacent
5.7. TRIGONOMETRIC RATIOS 57 Notice that as θ → 0, sin θ → θ and cos θ → 1, because a → hθ and
b → h. See section 5.5 concerning limiting values.
Using the definitions above, together with Pythagoras’ theorem, other useful relationships can be found such as:
sin2θ + cos2θ = 1
and tan θ = sin θ
cos θ
In order to convince yourselves that you understand the trigonometric ratios prove the two relationships above for yourselves.
Radiation on a Surface
The intensity of radiation is usually measured in terms of the amount of radiation falling on an area perpendicular to the direction of radiation. Thus radiation of 100 W m−2 implies that an area of one square meter at right-
angles to the radiation would receive 100 W of radiative energy.
However, on a surface which is not at right-angles to the direction, for example the solar panel on your house roof, should you decide to go green:
?I ©©©© ©©©© ©©©© ©©©© ©©©© θ ? ? ? ? ? ? ? ? ? ?
intensity on inclined surface = Radiation Intensity ×area at right-angle area of shadow = I × cos θ
Chapter 6
Differential Calculus
6.1
Introduction
The word calculus is Latin for pebble, and in its generic form means any method of calculation. Therefore you have been making use of calculus for years! Early forms of the abacus used pebbles in grooves marked out in sand, hence the association.
However, in more recent times the calculus has come to mean the branch of mathematics which is concerned with the behaviour of dynamic systems, that is with systems in which objects move or change - like all living things: bacteria, cows and humans! It was developed by Fermat, Newton and others in order to study the motion of planets, pendulums. . . and falling apples?. Current applications include the modelling of plant/animal development and aspects of population growth, epidemiology etc. Most computer models are based on the methods of calculus, though they use numerical approximations in order to solve the (much) more complex equations necessary to describe these systems. The ability to construct differential equations that define such systems will allow you to make use of the many computer packages that can produce solutions.
Calculus comprises two processes; differentiation in which we know the equations defining the state of a system and use them to work out the rate at which the system will change as its independent variables change, and integration in which we know the equations defining the rate of change and we use them to predict the state of the system at specific values of its variables. Many people, especially on the island at the other side of the Atlantic, refer to integration as anti-differentiation - a hideous but usefully accurate term!
We begin by describing differential calculus, because differentiation can be defined using one formula - though working with it can be extremely tedious, whilst integration is more of an art form relying to a large extent on guesswork and experience gained from differentiation.
60 CHAPTER 6. DIFFERENTIAL CALCULUS
6.1.1 What is differentiation?
The following maths-speak introduces some complicated notation - don’t panic! Just accept it for what it is - you will soon understand.
If we know a relationship y = f (x), it is often possible to derive a formula that defines the slope of its graph at any point x. This formula, which we denote by dy/dx or f0(x), is variously called the derivative or the differential or the differential coefficient of the function y = f (x) and the
process that we go through in order to find it is known (to mathematicians) as differentiation.
“Differentiation” is a word which causes me, and I suspect many of you, great difficulty, because its colloquial meaning gives little clue to its mathematical use. I am more at home with the word “differential”, because I know that the differential on a car allows the right and left hand wheels to turn at different speeds when cornering. I am also familiar with the term “differential” referring to bicycles where the differential is the ratio of wheel velocity to pedal or crank velocity.
The differential on a bicycle is the result of dividing the number of teeth on the chainwheel (the one connected to the pedals), by the number of teeth on the rear wheel sprocket and tells you the relative speed (rpm) of the wheels corresponding to the speed (rpm) of the pedals. Thus if the chainwheel has 48 teeth and the rear wheel has 16 teeth, then the differential coefficient is 3, because for each turn of the pedals the wheels go round three times.
Also if we plot the angular speed of the wheels against the speed of the pedals we would see a straight line graph whose slope is 3. The differential coefficient is the slope of such a graph.
Thus the derivative of the function defining wheel velocity in terms of pedal velocity is the same as the differential, which is also the slope of the graph of wheel angular velocity against pedal velocity.
Differential coefficients are quantities or expressions that determine the relative change in a variable as its independent variable changes. Usually the independent variable will be time. Thus the growth rate is the change in mass, divided by the corresponding change in time. We would refer to this as
dm/dt. Acceleration or the rate of change of velocity may also be expressed
as a differential coefficient (dv/dt) i.e. the relative change in velocity as time changes. The independent variable will not necessarily always be time. For example we could ask what is the relative change in the area of a circle as the radius changes, in this case we would require an expression for dA/dr.
In general, if we can define a function which specifies the size, position, concentration, etc. of an object at a given time, then differentiation will enable us to derive equivalent functions for the growth rate, velocity, reaction rate, etc. of the object.