text:
Planet Earth/1b. Earth System Science: Gaia or Medea?

Earth as a Puddle
"“Imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’ This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything’s going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.”"

- Douglas Adams

In December 1968, Astronaut William A. Anders on Apollo 8 took a picture of the Earth rising above the Moon’s horizon. It captured how small Earth is when viewed from space, and this image suddenly had a strange effect on humanity. Our planet, our home, is a rather small, insignificant place when viewed from the great distance of outer space. Our lives are collectively from the perspective of Earth looking outward, but with this picture, taken above the surface of the Moon looking back at us, we realized that our planet is really just a small place in the universe.

Earth system science was born during this time in history, in the 1960s, when the exploration of the moon and other planets, allowed us to turn the cameras back on Earth and study it from afar. Earth system science is the scientific study of Earth’s component parts and how these components —the solid rocks, liquid oceans, growing life-forms, and gaseous atmosphere—function, interact, and evolve, and how these interactions change over long timescales. The goal of Earth system science is to develop the ability to predict how and when those changes will occur from naturally occurring events, as well as in response to human activity. Using the metaphor of Douglas Adams’ salient puddle above, we don’t want to get surprised if our little puddle starts to dry up!

A system is a set of things working together as parts of a mechanism or an interconnecting network, and Earth system science is interested in how these mechanisms work in unison with each other. Scientists interested in these global questions simplify their study into global box models. Global box models are analogies that can be used to help visualize how matter and energy move and change across an entire planet from one place or state to another.

For example, the global hydrological cycle can be illustrated by a simple box model, where there are three boxes, representing the ocean, the atmosphere, and lakes and rivers. Water evaporates from the ocean into the atmosphere where it forms clouds. Clouds in the atmosphere rain or snow on the surface of the ocean and land filling rivers and lakes (and other sources of fresh water), which eventually drain into the ocean. Arrows between each of the boxes indicate the direction that water moves between these categories. Flux is the rate at which matter moves from one box into another, which can change depending on the amount of energy. Flux is a rate, which means that it is calculated as a unit over time, in the case of water, this could be determined as volume of water that it rains or snows per year.

There are three types of systems that can be modeled. Isolated systems, in which energy and matter cannot enter into the model from the outside. Closed systems, in which energy, but not matter, can enter into the model, and open systems, in which both energy and matter can enter into the model from elsewhere.

Global Earth systems are regarded as closed systems since the amount of matter entering Earth from outer space is a tiny fraction of the total matter that makes up the Earth. In contrast, the amount of energy from outer space, in the form of sunlight, is large. Earth is largely open to energy entering the system and closed to matter. (Of course, there are rare exceptions to this when meteorites from outer space strike the Earth.)

In our global hydrological cycle, if our box model was isolated, allowing no energy and no matter to enter the system, there would be no incoming energy for the process of evaporation, and the flux rate between the ocean and atmosphere would decrease to zero. In isolated systems with no exchange of energy and matter, over time they will slow down and eventually stop functioning, even if they have an internal energy source. We will explore why this happens when we discuss energy. If the box model is open, such as if ice-covered comets frequently hit the Earth from outer space, there would be a net increase in the total amount of water in the model, or if water was able to escape into outer space from the atmosphere, there would be a net decrease in the total amount of water in the model over time. So it is important to determine if the model is truly closed or open to both matter and energy.

In box models we also want to explore all possible places where water can be stored, for example, water on land might go underground to form groundwater and enter into spaces beneath Earth’s surface, hence we might add an additional box to represent groundwater and its interaction with surface water. We might want to distinguish water locked up in ice and snow by adding another box to represent frozen water resources. You can begin to see how a simple model can, over time, become more complex as we consider all the types of interactions and sources that may exist on the planet.

A reservoir is a term used to describe a box which represents a very large abundance of matter or energy relative to other boxes. For example, the world’s ocean is a reservoir of water because most of the water is found in the world’s oceans. A reservoir is relative and can change if the amount of energy or matter in the source decreases in relation to other sources. For example, if solar energy from the sun increased and the oceans boiled and dried away, the atmosphere would become the major reservoir of water for the planet since the portion of water locked in the atmosphere would be more than found in the ocean. In a box model, a reservoir is called a sink when more matter is entering the reservoir than is leaving it, while a reservoir is called a source if more matter leaves the box than is entering it. Reservoirs are increasing in size when they are a sink and decreasing in size when they are a source.

Sequestration is a term used when a source becomes isolated and the flux between boxes is a very slow rate of exchange. Groundwater, which represents a source of water isolated from the ocean and atmosphere, can be considered an example of sequestration. Matter and energy which is sequestrated have very long residence times, the length of time energy and matter reside in these boxes.

Residence times can be very short, such as a few hours when water from the ocean evaporates, then falls back down into the ocean as rain; or very long, such as a few thousand years when water is locked up in ice sheets and even millions of years underground. Matter that is sequestered is locked up for millions of years, such that it is taken out of the system.

An example of sequestration is an earth system box model of salt (NaCl), or sodium chloride. Rocks weather in the rain, resulting in the dissolution of sodium and chloride, which are transported to the ocean dissolved in water. The ocean is a reservoir of salt since salt will accumulate over time by the process of the continued weathering of the land. Edmund Halley (who predicted a comet’s return and was later posthumously named after him) proposed in 1715 that the amount of salt in the oceans is related to the age of the Earth, and he suggested that salt has been increasing in the world’s ocean over time, which will become saltier and saltier into the future. However, this idea was proved false when scientists determined that the world’s oceans have maintained a similar salt content over its history. There had to be a mechanism to remove salt from ocean water. The ocean loses salt through the evaporation of shallow seas and landlocked water. The salt left behind from the evaporation of the water in these regions is buried under sediments and becomes sequestered underground. The flux of incoming salt into the ocean from weathering is similar to the flux leaving the ocean by the process of evaporated salt being buried. This buried salt will remain underground for millions of years. The salt cycle is at an equilibrium, as the oceans maintain a fairly persistent rate of salinity. The sequestration of evaporated salt is an important mechanism that removes salt from the ocean. Scientists begin to wonder if Earth exhibits similar mechanisms that maintain an equilibrium through a process of feedbacks.

Equilibrium is a state in which opposing feedbacks are balanced and conditions remain stable. To illustrate this, imagine a classroom which is climate controlled with a thermostat. When the temperature in the room is above 75 degrees Fahrenheit, the air conditioner turns on, when the temperature in the room is below 65 degrees Fahrenheit, the heater turns on. The temperature within the classroom will most of the time be at equilibrium between 65 and 75 degrees Fahrenheit, as the heater and air conditioner are opposing forces that keep the room in a comfortable temperature range. Imagine now that the room becomes filled with students, which increase the temperature in the room when the room reaches 75 degrees Fahrenheit, the air conditioner turns on, cooling the room. The air conditioner is a negative feedback. A negative feedback is where there is an opposing force that reduces fluctuations in a system. In this example, the increase in the heat of the students in the room is opposed by the cooling of the air conditioner system turning on.

Imagine that a classmate plays a practical joke, and switches the thermostat. When the temperature in the room is above 75 degrees Fahrenheit, the heater turns on, when the temperature in the room is below 65 degrees Fahrenheit, the air conditioner will turn on. With this arrangement, when students enter the classroom and the temperature slowly reaches 75 degrees Fahrenheit, the heater turns on! The heater is a positive force in the same direction as the heat produced by the students entering the room. A positive feedback is where there are two forces that join together in the same direction, which leads to instability of a system over time. The classroom will get hotter and hotter, even if the students leave the room, the classroom will remain hot, since there is no opposing force to turn on the air conditioner. It likely will never drop down to 65 degrees Fahrenheit, with the heater turned on. Positive feedbacks are sometimes referred to as vicious cycles. The tipping point in our example is 75 degrees Fahrenheit, when the positive feedback (the heater) turned on, resulting in the instability of the system and leading to a very miserable hot classroom experience. Tipping points are to be avoided if there are systems in place with positive feedbacks.

Gaia or Medea?
One of the most important discussions within Earth system science is whether the Earth exhibits mostly negative feedbacks or positive feedbacks and how well regulated are these conditions that we find on Earth today. The two hypotheses are named after two figures in Greek mythology, Gaia the Goddess of Earth, and Medea lover of Jason, who murdered her own children. The Gaia hypothesis maintains that the global Earth system maintains an equilibrium or long-term stability through various negative feedbacks that oppose destabilization of the planet. The Medea hypothesis maintains that the global Earth system does not maintain a stable equilibrium resulting in frequent episodes of catastrophic events. From a geological point of view, the Gaia hypothesis predicts Uniformitarianism, that past geological processes through time have mostly remained continuous and uniform, while the Medea hypothesis predicts Catastrophism, where most past geological processes are the result of sudden, short-lived, and violent events.

This dichotomy of the optimistic view of the Gaia hypothesis and pessimistic Medea hypothesis is a simplification. In reality the true Earth system likely exhibits both types of negative feedbacks and positive feedbacks that interact in complex ways.

Imagine a classroom, now equipped with two thermostats, a normal negative feedback that turns on the air conditioner when the room gets above 75 degrees, and a malfunctioning positive feedback that turns on the heater when the room gets above 80 degrees. Assuming that the classroom begins with a temperature of 70 degrees, and each student who enters the classroom raises the temperature by 1 degree. The air conditioning will turn on when 5 students enter the classroom. This air conditioner is weak and only lowers the temperature by 1 degree every 10 minutes.

The classroom will maintain an equilibrium temperature as long as the rate of students entering the room is below 10 students per 100 minutes. For example, if 7 students entered the classroom at the same time, the temperature would rise by 7 degrees, to 77 degrees—turning on the air conditioner when it crossed 75 degrees and taking 20 minutes to lower the temperature back down to 75 degrees. Another 4 students could enter the classroom, raising the temperature to 79 degrees, turning on the air conditioner and lowering the temperature down to 75 degrees in 40 minutes.

However, if 12 students enter the room at the same time, the temperature will rise to 82 degrees, turning on both the air conditioner at 75 degrees and heater at 80 degrees, the heater warms the room faster (+2 degrees every 10 minutes) than the air conditioner is able to cool the room (−1 degree every 10 minutes). This positive feedback will cause the classroom to increase in temperature until it is a hot oven because the net temperature will increase +1 degree every 10 minutes. The tipping point was when the 12 students entered the room all at once, which set off this vicious cycle of a positive feedback. If the rate of students entering the classroom remains low, the room temperature will remain stable, and it appears to be governed by the Gaia hypothesis. However, if the rate of students entering the classroom is fast, the temperature could become unstable, as governed by the Medea hypothesis.

In determining the mechanisms of how Earth systems play out over time, we also need to be aware of the fallacy of the salient puddle at the beginning of this chapter—a puddle that believes it is perfectly tailored to the environment it finds itself in. The Gaia hypothesis views the Earth as a perfectly working system that is able to adjust to changes and maintain an equilibrium state. This is similar to the salient puddle believing that it fits perfectly within the environment that it finds itself within. In contrast, the Medea hypothesis views that there will inevitably be an event that dries up the puddle, and that the puddle is not at equilibrium under a warm sun.

Who Came Up with These Ideas?
The Gaia hypothesis, has a longer pedigree, and was first formulated in the 1970s by James Lovelock and Lynn Margulis. Initially called the Earth feedback hypothesis, the name Gaia was proposed by the writer William Golding, the author of Lord of the Flies and close neighborhood friend of Lovelock. Lovelock was an expert on air quality and respiratory diseases in England, but he took up the study of the Earth’s sulfur cycle, noting negative feedbacks that appeared to regulate cloud cover. In the 1970s, he had been invited to work on the Viking Missions to Mars by NASA to evaluate the Martian atmosphere for the possible presence of life. Lovelock suggested that if the Viking lander found significant oxygen in the Martian atmosphere it would be indicative of life existing there. Instead the Viking lander found that the Martian atmosphere was 96% carbon dioxide, similar to the atmosphere of Venus. Working with Lynn Margulis, the two formulated a hypothesis that there were natural negative feedback systems on Earth that kept both oxygen and carbon dioxide in the atmosphere within a low range, with photosynthesizing plants and microbes taking in carbon dioxide and producing oxygen while animals take in oxygen and produce carbon dioxide. Life on Earth appeared to keep the atmosphere stable in relation to these two types of gases in the atmosphere. Without life, carbon dioxide remained high in the atmosphere of Mars and Venus.

The Medea hypothesis is a newer and a more frightening idea, first proposed by Peter Ward, an American paleontologist in a book published in 2009 (The Medea Hypothesis: Is Life on Earth Ultimately Self-Destructive?). Ward started out as a marine biologist, but a traumatic diving accident that left his diving partner dead pushed him to study marine organisms found in rocks rather than deep underwater. Ward became interested in fossil ammonites along the coast of Europe, which flourished in the oceans of the Mesozoic Era, during the age of the dinosaurs, but had become extinct with the dinosaurs, 66 million years ago. Ward studied ammonites and other fossils, and he became fascinated with mass extinction events that have occurred in Earth’s history. He became keenly interested in the Permian-Triassic extinction event in South Africa, which divides the Paleozoic Era (ancient life) with the Mesozoic Era (middle life), the great time divisions of Earth’s history. This extinction event was one of the worst, called colloquially the Great Dying, and it appears to have be caused by an imbalance of too much carbon dioxide in the atmosphere. Ward saw these episodes of mass extinction events in the rock record as evidence of when the Earth system became out of balance and resulted in catastrophic change. Ward, with Donald Brownlee, postulated that Earth’s atmosphere millions of years in the future would lose all its carbon dioxide as tectonic and volcanic activity on Earth ceases, resulting in no new sources of carbon dioxide released into the atmosphere. Carbon dioxide would become sequestered underground as photosynthesizing plants and microbes die and are buried, and there would be no new carbon dioxide emitted from volcanoes. As a result, less and less carbon dioxide would be available in the atmosphere, ultimately dooming the planet with the inevitable extinction of all photosynthesizing life forms.

Neither of the advocates of the two hypotheses view the Earth system exclusively governed by either hypothesis, but a mix of both negative and positive feedbacks working on a global scale over long time intervals. Another way to frame the Gaia and Medea hypotheses is to ask whether the global Earth system behaves mostly under negative or positive feedbacks. Of course, the goal of this class is to determine how you can avoid positive feedback loops that would result in catastrophic change to your planet while keeping your planet balanced with negative feedback loops and remain a habitable planet for future generations.


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Web App Development with Google Apps Script/syntax

ajax
Ajax stands for "Asynchronous JavaScript and XML" and it's the ability for a web page to communicate with a server without reloading the page.

To do it in you use the   tool. This command, which has several varieties, runs JavaScript code on Google's servers and potentially returns information back to your web page.

For example, if you wanted to send a name to a server, then check if it's in your spreadsheet and return true if it is and false if it's not that you can set to a variable, you'd do this:

First the server side code:

Now the portion of "main.html" with the JavaScript you'd be doing

The flow is: main javascript -> function on server -> returns data to your SuccessHandler -> your successHandler does something.

Array numbering
A simple (and quite forgivable) mistake is to be off by a row or column when moving data back and forth between google sheets and javascript is the array numbering issuer. Javascript starts numbering at zero while google sheets starts numbering at 1. Here's an example:

filter
If you need to filter an array, use, um, :

As with find, forEach, and map you can also use the index if you want:

find
If you need to find an element in an array,  is super helpful:

forEach
If you need to loop through an array and do something with every element, you could of course do this:

But you could also do this, which I think reads better:

Note that you never have to initialize the "i" variable this way.

If you need to access the item (fruit) and the location (index) that it's in the array: Note the parentheses around  that makes that magic work. Also note that you still don't have to declare the i variable!

includes
If you're looking to see if an array includes a particular element, try :

map
Similar to forEach,  is a great way to create a new array based on an existing array. Let's say you wanted to add a tag around some text in an array:

You can also have easy access to the index of the item in the array:

sort
Sorting arrays is quite valuable and sometimes a pain. Assume you have a 2D array from a spreadsheet with columns of "name", "age", "email". If you want to sort them by age, do this:

really sort needs to return a true or false when determining who should go first, but by subtracting two numbers, if you get a zero or a negative number it'll be treated as false whereas a positive number will be treated as true.

If you wanted to sort by the age and the name (alphabetically) you could:

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
History of wireless telegraphy and broadcasting in Australia/Topical/Publications/Australasian Radio World/Issues/1936 12

Link to Issue PDF
| WorldRadioHistory.com's scan of Australasian Radio World - Vol. 01 No. 08 - December 1936 has been utilised to create the partial content for this page and can be downloaded at this link to further extend the content and enable further text correction of this issue: | ARW 1936 12

In general, only content which is required for other articles in this Wikibook has been entered here and text corrected. The material has been extensively used, inter alia, for compilation of biographical articles,  radio club articles and  station articles.

Front Cover
The Australasian Radio World

Dec. 1, 1936; Vol. 1 - No. 8; Price, 1/-

Registered at the G.P.O., Sydney, for transmission by post as a periodical

Cover Photo: Mullard valves in kitset (See Page 7)

Highlighted Contents:

SPECIAL XMAS ISSUE: PARTY RADIO MAGIC: SCOUT BATTERY THREE:

HIGH FIDELITY AT LOW COST: FIVE-METRE SCHEDULES DURING

XMAS: LATEST SHORTWAVE NEWS: THE STORY OF TELEVISION.

P.02 - Editorial Notes
'''Editorial Notes. . .'''

The Power Interference Problem

The main bugbear of radio reception today is power interference, and it is an annoyance that is becoming more acute with the rapid growth of shortwave listening. Also, as far as reception of Australian stations is concerned, the effects of man-made static are intensified by the comparatively low transmitting powers used. The problem is one that will have to be tackled sooner or later, even if it means the passing of legislation to compel everyone operating electrical equipment that is giving offence to fit suitable filters to silence the noise. In this respect, concerted action on the part of the tramway and power supply authorities, who are among the biggest offenders, is long overdue. In England, thousands of pounds have been spent in the past few years on research into the problem, mainly by the Electrical Research Association with the assistance of the B.B.C., the postal authorities, and the Radio Manufacturers' Association. The subject has also been exhaustively analysed by various countries on the Continent, particularly by Germany, and as a result of all this research the causes of interference are now generally understood and suitable remedies can usually be applied. In a few cases the remedy is expensive, and so further research is now being undertaken to try and find methods of achieving the desired result more cheaply. The goal that has been set in England will be attained when, by a combination of suppressing equipment at the receiver and the fitting of filters to interference-creating equipment, radio reception for 90 to 95 per cent. of the listening public will be free from man-made interference. The same goal should also be the aim of the Australian authorities, but so far next to nothing has been done. The first, simplest, and cheapest step would be to increase station power considerably, to 'provide a better signal-to-noise ratio. The next and most important move is to have legislation passed that will render owners of interfering equipment liable to prosecution. The radio inspectors appointed by the P.M.G.'s Department are already doing splendid work in quelling interference, but until they have legislative backing their task is a hopeless one.

P.02 - Contents Banner
The Australasian Radio World

Incorporating The

All-Wave All-World DX News

Managing Editor - A. Earl Read, B.Sc.

Vol. 1. - DECEMBER, 1936 - No.8.

P.02 - Contents
CONTENTS:

Radio Gifts For Christmas. . . . 3

"Scout Battery Three". . . . 6

Radio Ramblings. . . . 10

High Quality Reproduction At Low Cost. . . . 14

New £50,000 Factory Planned By Ever-Ready. . . . 16

The "Scout Superhet Four". . . . 17

This Pentode Business (2). . . . 22

The Story Of Television (1). . . . 24

Radio Step By Step (5). . . . 26

Party Radio Magic. . . . 28

The Cathode Ray Oscillograph At Work (4). . . . 30

Radio In India. . . . 33

The A.T.R.S. Bulletin. . . . 34

What's New In Radio. . . . 36

Radio Book Reviews. . . . 37

Round The N.Z. "B" Stations. . . . 38

VK Amateur Stations — Additions And Amendments. . . . 40

All-Wave All-World DX News. . . . 41

Radio VK4DO Is Model Amateur Station. . . . 42

Schedules Of Japan's New Shortwave Service. . . . 42

Shortwave Stations Of The World (4). . . . 43

DX Notes And News. . . . 45

P.02 - Publication Notes
The "Australasian Radio World" is published monthly by Trade Publications Proprietary, Ltd. Editorial offices, 214 George Street, Sydney, N.S.W. Telephone BW6577. Cable address: "Repress," Sydney. Advertisers please note that copy should reach office of publication by 15th of month preceding that specified for insertion.

Subscription rates: 1/- per copy, 10/6 per year (12 issues) post free to Australia and New Zealand. Subscribers in New Zealand can remit by Postal Note or Money Order.

Printed by Bridge Printery, 214 George Street, Sydney, N.S.W., for the proprietors of the "Australasian Radio World," 214 George St., Sydney (Footnote P.48)

P.38 - Round The N.Z. "B" Stations
'''Round the N.Z. "B" Stations. . .'''

'''1ZF. . . "Voice Of The Manawatu"'''

The first of a series of articles, written for the "Radio World" by a prominent New Zealand dxer. By "The SOUTHLANDER." ("The Southlander" was Merv Branks - Ed. Note) STATION 2ZF, Palmerston owned and operated by the Manawatu Radio Club. It has the distinction of being the oldest station on the air in New Zealand, and is now in its 13th year of broadcasting. The original call-sign was 2GI, and the first transmitter was battery-operated, a matter which caused considerable trouble and expense. Often listeners took turns to lend their "B" batteries, or there would have been no broadcast! Those were real pioneering days, when a wireless listener was looked upon as a crank. The Pick-up was not known in those days, and gramophone records were played on an ordinary gramophone into a telephone microphone, which was fitted with a cardboard horn to catch all the precious sounds. On The Air In 1924. The first listeners' meeting was held on July 9, 1924, at a time when there were only 38 radio sets in the city of Palmerston North. Everyone became a member of the Club, which was called the Palmerston North Radio Club. Today the receivers within a reasonable range of the transmitter number many thousands, as there is a population of 143,000 within a 50-mile radius. Immediately after this first meeting a five-watt transmitter was built and put into operation. About 1926 the licensing of broadcast stations came into force, and the club was allotted the call-sign of 2ZF, while its name was changed to the Manawatu Radio Club. Toward the end of the third year of operation one of the 50-watt valves blew out, which placed the club in serious financial difficulties. Debts totalled £97, but at the special meeting called to consider the position a committee was appointed to raise funds, as a result of its efforts £150 was collected in 10 days. Some Technical Details. The first transmitter in 1924 used a 210 as oscillator. Later a motor generator was purchased, and 50-watt De Forest valves were used. In 1928 a transmitter radiating 150 watts was built, and this did yeoman service for some years, when the whole transmitter was completely re-built, using crystal control, and a linear amplifier installed to give 250 watts unmodulated carrier. The present transmitter uses a 210 crystal oscillator, and two 865 screen-grid Radiotrons for buffers and amplifier stages, driving a 211 modulated oscillator. The first audio stage is also a 211, and the modulators are a pair of 845's in parallel, capable of modulating the class "C" oscillator to 100% without distortion. The final stage is a class "B" linear amplifier using four 250-watt Marconi valves in parallel pushpull, supplied from a three-phase full-wave mercury vapour rectifier outfit, giving 3,500 volts on the plates. This gives an unmodulated carrier of 250 watts to the aerial, which is an earthed Marconi "T" located on the roof of a two storey building. A cathode ray oscillograph is also part of the equipment at the station, and is used periodically to check adjustments of speech circuits, etc. The studio is large enough to accommodate an orchestra or band, while the announcers' room contains all the amplifiers, control panel, electric turn-tables, etc. The Club has a condenser microphone, one velocity type, and two double button carbon microphones. In addition to broadcasting their own programmes on Wednesdays, Fridays and Sundays at other times the Club relays the programmes from the National Station 2YA, Wellington, the programme being received over 100 miles of the Post and Telegraph Department landlines. First S.W. Re-Broadcast. One of the notable pioneering transmissions of 2ZF, occurred when PCJ was first picked up in New Zealand. The Club cabled the now famous station in Holland that their second transmission had been picked up, and they would like to hear England. In the next test transmission from PCJ, that station rebroadcast 2LO London and Daventry for the benefit of the Club, and then Paris and Langenberg, in Germany. The Club officials at 2ZF had secured the help of the local telegraph engineer for a relay line, picked up the transmission from PCJ, and rebroadcast the whole programme. The public was incredulous, and many considered it a first-class hoax! But it was a real thrill for the officials of 2ZF, as this was the first time that the public of New Zealand had heard England.

Photo Caption
(Start Photo Caption) The main studio of "B" class station 1ZF, Palmerston North — "Voice Of The Manawatu." (End Photo Caption)

Main Features Summary
MAIN FEATURES Call and Location: 2ZF, Messrs. Collinson and Cunninghame's Buildings, King Street, Palmerston North, New Zealand. Owners: Manawatu Radio Club. Frequency: 960 k.c., 312.3 metres. Power: 250 watts unmodulated carrier. Transmission Times: (E.A.S.T.), Mon., Thurs., Sat., 6.30-8.30 p.m.; Wed., Fri., Sun., 4.45-8.30 p.m. Area Served: 143,000 population within 50 mile radius. Longest Distance Verified Report: Honolulu, Los Angeles, New York.


text: Irish/Vocabulary/Kitchen

title: Irish/Vocabulary/Kitchen


text:
A-level Mathematics/CIE/Pure Mathematics 1/Differentiation

The Gradient of a Point on a Curve

The gradient of a point on a curve is equal to the gradient of the tangent line at that point. Since it is difficult to measure the gradient of the tangent directly, we use secant lines which approach the tangent. A secant line is a line that travels between two points on the curve.

The Formal Definition of a Derivative
Suppose we want to find the gradient of a point at coordinates $$(x,y)$$. We can approximate it using a secant line that travels through $$(x,y)$$ and $$(x+\Delta x,y+\Delta y)$$ where $$\Delta x$$ is a small change in $$x$$ and$$\Delta y$$ is the small change in $$y$$ that results from it.

The gradient of the secant line is $$\frac{rise}{run} = \frac{y+\Delta y - y}{x+\Delta x - x} = \frac{\Delta y}{\Delta x}$$. As $$\Delta x$$ gets smaller and smaller, $$\frac{\Delta y}{\Delta x}$$ approaches the gradient of the point $$(x,y)$$.

The limit of $$\frac{\Delta y}{\Delta x}$$ as $$\Delta x$$ approaches zero is the value that we are looking for: the gradient of the point $$(x,y)$$. For an arbitrary point $$(x,y)$$, the gradient can be given by the derivative.

The derivative $$f'$$ of a function $$f$$ is a function which provides the gradient of a point $$(x,f(x))$$ on the curve produced by $$f$$. The derivative is formally defined as $$\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}$$

Notation
There are two main ways to write the derivative of a function: $$f'(x)$$ and $$\dfrac{df}{dx}$$. These both mean the same thing: the derivative of $$f$$ with respect to $$x$$.

Simple Rules
These are a few simple rules that make differentiation of complicated expressions easier.

The Power Rule
$$\dfrac{d}{dx} x^n = nx^{n-1}$$

e.g. $$\dfrac{d}{dx} x^3 = 3x^{2}$$

The Sum Rule
$$\dfrac{d}{dx} \big(f(x) + g(x)\big) = f'(x) + g'(x)$$

e.g. $$\dfrac{d}{dx} (x^2 + x^4) = 2x + 4x^3$$

The Constant Rule
$$\dfrac{d}{dx} \big(kf(x)\big) = kf'(x)$$

e.g. $$\dfrac{d}{dx} (4x^2) = 4(2x) = 8x $$

The Chain Rule
$$\dfrac{d}{dx} f(g(x)) = f'(g(x))\times g'(x)$$

e.g. $$\dfrac{d}{dx} (2x+1)^3 = 3(2x+1)^2 \times 2 = 6(2x+1)^2$$

Tangents & Normals
A tangent is a line which travels through a given point and has a gradient which is equal to the gradient of the curve at that point. A normal is a line which travels through a given point and is perpendicular to the tangent of the curve at that point.

To find the equation of the tangent or normal of a curve at a particular point, we can use the equations of a line that we used in ../Coordinate Geometry/.

e.g. Find the equation of the tangent and normal of a curve $$x^3 - 5x^2 + 2x + 6$$ at the point $$(2,-2)$$

$$\begin{align} \dfrac{d}{dx} (x^3 - 5x^2 + 2x + 6) &= 3x^2 - 10x + 2 \\ \text{Gradient at }(2,-2) = 3(2)^2 - 10(2) + 2 &= 12 - 20 + 2 = -6 \\ \text{Tangent: } \frac{y+2}{x-2} = -6 \implies y+2 &= -6x+12 \implies y = 10-6x\\ \text{Normal: } \frac{y+2}{x-2} = \frac{-1}{-6} \implies y+2 &= \frac{x-2}{6} \implies y = \frac{x}{6} - \frac{7}{3} \end{align}$$

Increasing & Decreasing Functions
An increasing function is a function whose gradient is always greater than or equal to zero.

A decreasing function is a function whose gradient is always less than or equal to zero.

Rates of Change
The rate of change of a quantity is the derivative of a quantity with respect to time. For instance, if water were flowing into a bucket at a rate of 1 litre per second, the rate of change of the volume of water in the bucket would be 1 litre per second.

In some cases, the rates of change of two quantities are connected. For example, a circle with a changing radius will have a changing area that depends on the radius. The area of a circle is related to the radius by $$A = \pi r^2$$. If the radius is increasing at a rate of 3 cm/s, its rate of change $$\dfrac{dr}{dt} = 3$$. The rate of change of area can be found using the chain rule:

$$\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt} = \dfrac{d}{dr}(\pi r^2) \times 3 = 2 \pi r \times 3 = 6 \pi r$$

If the radius is increasing at a rate of 3 cm/s, and is currently 5cm, the rate of change of area is $$6 \pi (5) = 30 \pi$$ cm2/s.

Stationary Points

A stationary point is a point on the curve at which the gradient is zero. This means that the derivative of the function at that point is equal to zero.

Stationary points are either maxima or minima. Maxima are where the function reaches a maximum value. Minima are where the function reaches a minimum value. We can determine whether a stationary point is a maximum or minimum by looking at the second derivative.

If the second derivative is positive, the gradient is increasing with input, and thus the stationary point is a minimum.

If the second derivative is negative, the gradient is decreasing with input, and thus the stationary point is a maximum.

Curve Sketching
Stationary points are useful when sketching graphs of curves. By marking the stationary points, we can draw graphs more accurately than otherwise.