Non-harmonic tones

This'll just be a quick post, on music theory this time rather than the historical stuff.

Non-harmonic tones are basically the notes that aren't part of a chord per se, but are there to make the music more pretty and melodious. After all, most music isn't exactly comprised of entirely block chords, unless the composer's using them for a certain effect.

For example, in the example below, the A in the soprano note is a non-harmonic tone because it isn't a part of chord IVb.

The different kinds of non-harmonic tones also have different names. In the above example, the A is a passing note because the soprano line "passes" through there to get from the G to the B. This particular passing note is an unaccented passing note because it doesn't fall on the same beat as the rest of the chord. If it did- for example, if the G and the A were switched around- then it would be an accented passing note.

Now, let's look at the other kinds of non-harmonic tones!

In the above example, the A and the F# are neighbouring tones, so called because they are the "neighbour" of the notes on either side. Bear in mind that the two notes on either side must be the SAME note. If the note before is a step higher than the note in the middle, and the note after is a step lower than the note in the middle, or vice versa, then the middle note is a passing tone, not a neighbouring tone.

In the above example, the A is an anticipation tone. That is because it comes in earlier than it really needs to. The first chord is a IV chord... but wait! There's an A in the soprano! There's no A in C major chords... but there is one in D Major chords, and that's precisely what the second chord is! The A has simply come in too early, "anticipating" the next chord. That's why it's an anticipation tone.

(By the way, I've noticed that my cadences are pretty dodgy in some places. Apologies for that.)

The note after an anticipation tone can be either higher or lower than the anticipation tone. In the above example, we move from an A (anticipation tone in this case) to B.

Retardations and suspensions kind of work in the opposite way. Instead of being the same as the note in the next chord, retardations and suspensions carry over a note from a previous chord:

In the above examples, G is a note in the IV chord, but not in the ii chord. In the first example, G is a retardation tone because the note afterwards is higher. At the risk of being somewhat politically incorrect, you could say that the note is "retarded" because it has to wait longer until it can finally move up in pitch. In the second example, G is a suspension tone because the next note is lower. You could say that the G is "suspended in mid-air" until it finally falls on the F#.

An appogiatura is where you have a larger gap leading to the non-harmonic tone (normally an interval of a 3rd) and then a step downwards. Like this:

Except C is in an A minor chord, so I'm not entirely sure if it qualifies as a non-harmonic tone.

Escape tones are the opposite- a step and then a larger gap. Like it's trying to "escape" by stepping away but then realises that it can't.

(I just reversed the order of the chords from the last example. Original, aren't I?)

Those are pretty much all of the non-harmonic tones that you need to know for now. Isn't music theory just so incredibly fun? w00t.

Lit notes - The Turning by Tim Winton

I have a Lit exam on Wednesday next week and I sure as hell don't feel prepared. I've done a couple of close readings (or rather, I've done a couple of half close readings- an intro and a body paragraph on poetry and an intro and a body paragraph and a bit of another body paragraph on prose) and I started trying to organise my notes on the texts that we've done this year. (Note- this was at the time that I started writing this post, which I think was on Thursday.) My organisation is pretty shoddy though, just like my room where everything kind of has its place but really doesn't.

Anyway. Here are my notes for The Turning (the only reason I'm typing them up here is because I'm procrastinating over writing practise essays...). Bear in mind that I'm not that great a Lit student, so don't take my word as gospel. Or actually, maybe you can, since half my notes are taken from handouts and class powerpoints and whatnot.

ANYWAY...

The Turning by Tim Winton

Context
The Turning is set in Australia.

• Long, wide spaces- Australia is supposed to be one of the most sparsely populated countries so we get lots of space which is nice. There's a lot of times during the novel (or during the 17 short stories, if you prefer) where characters have to cross these long, wide spaces, like when Vic goes to find his dad in "Commission," when Vic is coming home after his cadet camp in "Immunity," or when Gail and Vic are going to visit Daisy and Fenn in "Defender." And then there's a chapter titled "Long, Clear View" which kind of adds to this because you wouldn't really be having a long, clear view if it wasn't for the long, wide spaces available. And then I guess you could also kind of argue that there's a long, wide space between Vic and Gail in terms of their relationship. Especially since Gail's continually visiting Angelus and even going out with some other guy there to boot.
• The outback- Well, it's pretty wild and interesting and unique to Australia, which is why there's always been lots of poems and stories and whatnot dedicated to it (bush rangers, anyone?). Sure enough, there's a few stories in here in which the wildness of Australia's outback is important- stories like "Aquifer," for instance, where Alan Mannering drowns in a swamp, as well as "Fog" where Bob Lang and Marie are stuck in the bush overnight.
• Connection between people and place- particularly important to Aboriginal culture, but it can still be seen in this book even though there aren't that many Aboriginals. For example, the swamp is a big part of the children's life in "Aquifer" and then there's the whole idea of returning to one's hometown to search for answers just as Frank does in "Family" and Gail kind of does in "Damaged Goods" (but not really since Gail's trying to find answers about Vic's life).
• Mainly secular society, but there are still many people who practise a religion of some description. The Turning has several characters who are Christians, such as Sherry and Dan in "The Turning" and Agnes' family. Christianity certainly helps Raelene in her time of need.
• There are a lot of immigrants but sometimes there's still a sense of who's "Australian" and who's an immigrant. Agnes' family is British, and Agnes refers to herself as "fresh off the boat."
• Notion of the "underdog" eventually gaining superiority: Frank was the "underdog" in his younger years- bullied by Max and never quite able to walk among the men just as Max could. However, he eventually becomes a professional footballer (though not for long).
• Archetypal Australian men and women: apparently men are meant to be all tough and independent and  masculine and whatnot and women are meant to be "stoic," which (according to Google) means "a person who can endure pain and hardship without showing their feelings or complaining." Now, there are characters that naturalise these ideas, as well as other characters who challenge them. There are whole bunch of characters who are, well, not quite as tough and independent as the "ideal" male figure. Max, for instance, is a violent alcoholic and we're pretty much set up to hate him through the point of view (we mainly see him from the point of view of the characters who he victimises- more on this later), Frank walks off the football pitch after a rather short career, Vic is so absorbed in his past that nothing in the present holds his attention, and so on. On the other hand, though, there are other characters that might be considered as upholding the ideals of dependability and so on such as Dan who recovers from alcoholism and I think Dyson from "Small Mercies" as well for being able to be relied on by Fay and her parents (if I remember correctly- it's been a while since I read that story. In fact it's been a while since I read the book). There are then also characters that represent the "stoic woman," such as Carol Lang, who doesn't buckle under that other woman's accusation that Carol stole her earrings, and Raelene, who puts up with Max's abuse for a while without complaining. I'm not so sure about which characters challenge this ideal. Maybe Fay? I don't know. As I said, I haven't read this book for a while, and I've only read this particular chapter/story once.

Genre

• The Turning can be considered to be Realist fiction, which is a less ornate style which uses verisimilitude (a very cool term which I think means that the story is made to be as realistic as possible) to illuminate truths about political and social elements of society, such as class differences. Realist fiction also often has individuals raging against the world rather than any massed effort to change the system. You can see a lot of individuals raging against society across the stories- Bob Lang leaves Angelus because there is something about the police force that makes him feel uncomfortable (potential corruption?), Rae is locked in an abusive relationship but undergoes spiritual growth to become internally stronger than Max, Frank is still struggling with being victimised by Max which results in him walking off the football pitch, and so on. You can also kind of see class differences being illuminated, such as in "On Her Knees" in the contrast between the unnamed home owner and Carol and Vic Lang who are left to clean it (after receiving empty accusations about stealing the home owner's earrings), though I don't recall seeing any huge differences.
• The Turning can also be considered to be Romantic fiction, using Romantic themes such as the power of nature as restorative, a lone man against society and freedom of the individual. (I wrote a little bit about the Romantic period on my post about music and society during the Romantic period.) As I said above, there are quite a few times where there are lone people against society. There are also quite a few instances throughout the novel where there are people alone in nature, using nature's restorative power to heal themselves. Raelene in "The Turning" often uses her walks along the beach to think, which leads her to come to revelations which eventually lead to her finding her own internal freedom from Max. Complementing this is language that revels in the beauty of nature through various kinds of imagery, such as in "Fresh-mown grass felt good beneath her feet and over the green smell of it you could almost taste the sea." Nature is green and fresh and feels good, contrasting with the horrors of the abusive relationship that Raelene is locked into. Another character who can be said to heal in nature is Bob Lang, because in his new home tucked away in the bush, he becomes the trustworthy, honest character known as "Honest Bob"- a character who he wasn't quite able to be when in society. As well as nature having the power to heal, nature is also powerful and dangerous as can be seen through Alan Mannering drowning in the swamp and Vic becoming injured on a hook when their fishing boat is overtaken by a wave.
• Finally, The Turning can be considered to be a series of parables- stories that are meant to teach us moral and/or spiritual lessons. For example, Max is a pretty mean character who eventually gets eaten by a shark. Also, before his death, Raelene became stronger than him through finding God. And then of course there's Carol Lang upholding the virtue of dignity by her actions in "On Her Knees." (I don't really have many more notes on how this book is a series of parables though.)

Language and Generic Conventions

I'm just going to put a few points down here because, well, a novel has lots of words and thus lots of conventions and stuff to write about. It really depends on how close you want to look. (For me- not very.)

• Possibly the one generic convention that will hit you the hardest in this book is the constantly changing point of view. The point of view moves between 1st person, 2nd person and 3rd person, the character in focus changing between stories. "Big World" is told through the 1st person perspective of an unnamed narrator who's just graduated from high school, "Abbreviation" is told through 3rd person limited focusing on Vic Lang, and so on. So what's so important about these points of view? Well, point of view in a novel really affects the way that we see characters and events in the story. It's just like how in class someone else might get a mark that doesn't seem so bad to you but it might seem devastating to the person in question. Now, let's look at Max's character. We're pretty much set up to hate Max through the point of view. The three stories in which Max is involved- "The Turning," "Sand" and "Family"- are through 3rd person limited perspective focusing on Raelene and Frank, two characters who are badly treated by Max, so we sympathise more with Raelene and Frank and the struggles that they have to endure against a seemingly tyrannical character. If one of the stories was told with a greater focus on Max, though, perhaps a slightly different story might be told. I'm not sure that a focus on Max would make us like him given all the other stuff that he's done, but if, say, Max actually felt remorseful for what he does, and the story was told from his point of view so that we could see his remorse, we might hate him less. Vic Lang is another character who is described from multiple points of view, but unlike Max, we can see both sides of the story. "Long, Clear View" allows us to see Vic's side of the story, but in 2nd person, which perhaps makes us closer to the raw emotion within the story. On the other hand, you have Gail talking about Vic's obsession with the past in "Damaged Goods," as well as 3rd person limited focusing on Gail in "Defender" in which Gail outright tells Vic about her frustration.
• Just like pretty much every other book, there are objects that can be read as symbols. Unfortunately I don't have many notes on the symbols in this book and I'm pretty bad at telling what can be considered as a symbol and what's best just left alone. Anyway, fire, water and sharks are all (possibly) symbols that recur throughout the novel. Fire is seen as damaging but cleansing, like the destruction of an old life and the start of a new one... oh wait! The novel title is, after all, "The Turning," "turning" also suggesting such radical changes! And I don't know about the cover of your copy, but the cover on mine has a burning car... okay, maybe I'm getting ahead of myself, because Lit doesn't really have visual analysis in it. But still. Now, where do we see fire in The Turning? Well, there's Agnes, who may or may not have burnt her "dead inside" house down at the end, causing the family to have to start anew. Then, at the end of "Big World," the narrator tells about all the changes and stuff that happens in the future after he talks about how the car "[smouldered] and [hissed]." As for water, well, pretty much all the people live near beaches, like a lot of people in Australia. Quite a few significant events happen in the water too, like Vic's injury in "Abbreviation," Alan Mannering's death in "Aquifer," Boner McPharlin is found with broken legs at Thunder Beach, and so on. And, where there are beaches, there are sharks. Boner McPharlin must have a thing about them because he catches and kills a shark at the bonfire party at Massacre Point and he gives a shark sticker to Jackie (well, according to my notes he does anyway). Then there are symbols only relevant to one story, like Raelene's snow globe in "The Turning" which reminds her of Jesus and gives her the strength to keep fighting Max.
• And, of course, like pretty much every other book under the sun (except for maybe some crazy postmodern book out there) has characters in it. These characters can then be read as representations of class, cultures, gender and so on. Like I guess you could say that Max is a representation of a violent drunk and that Carol Lang is a representation of an archetypal Australian stoic woman. Vic Lang's characterisation can be read as examining the effects of the past on the present. Etcetera. Oh, and don't pay too much attention to what I just said because I have no idea what the hell I'm doing.

Intertextuality

• The epigraph is part of a verse from T.S. Eliot's poem "Ash Wednesday." According to my notes, Ash Wednesday is the beginning of the Lent season, so it symbolises repentance, loss and waiting for salvation. And, sure enough, you see a hell of a lot of characters repenting stuff they've done before (like Bob Lang), going through some kind of loss (like Dyson in "Small Mercies) or waiting for salvation (like Raelene in "The Turning). In the epigraph, however, Winton removed the first and last line of the verse- the first line being "And pray to God to have mercy upon us" and the last line being "May the judgement not be too heavy upon us." Removing these lines means that Winton has, in essence, secularised the poem a bit, possibly to reflect that Australian society is considered to be a relatively secular society. He has also secularised concepts such as redemption and forgiveness suggesting that these concepts are universal to all, not just those who practise a religion. Another point of interest is the word "hope" in the line "Because I do not hope to turn again." "Hope" suggests a lack of control, because if you had control over doing something, then you would make it happen, rather than crossing your fingers and hoping that it might happen.
• Although not really intertextuality per se, the short stories that The Turning is comprised of are all interconnected. Obviously, there are characters that appear in multiple stories: Vic Lang, for instance, appears in "Abbreviation," "Damaged Goods," "On Her Knees," "Long, Clear View," "Reunion," "Commission," "Immunity" and "Defender." Max appears in the middle three stories: "The Turning," "Family" and "Sand." Apart from just characters, though, there are other themes that crop up constantly. Symbols such as fire, water and sharks are mentioned in multiple stories. Although the society is mainly secular, there are religious messages in some stories, like "Cockleshell" and "The Turning." The power of nature is also alluded to in many stories throughout the novel. There are also similarities between different characters in different stories, just like how Vic's infatuation with Strawberry Alison in "Damaged Goods" is somewhat mirrored by an unnamed girl's infatuation with Vic. Finally, there are the ideas of being stuck in the past- like Vic Lang, or like the unnamed narrator of "Aquifer"- and of needing to escape, like the many characters who seek to get out of Angelus and the several others who seek escape through alcohol like Max does.
• There are several other allusions to other texts. Biblical stories get mentioned a couple of times, as do some other books, such as Catcher in the Rye (Jackie briefly mentions it in "Boner McPharlin's Moll"). Unfortunately, even though I've actually read Catcher in the Rye, that was a while ago and to be honest I had a hard time getting my head around what was going on. (I'm a pretty poor reader. Get over it. Or maybe I'm not that bad a reader but I have to read something a few times for the words to sink in? I dunno.)
• Finally, the chapter "Family" starts with a poem by Chuang Tzu. I guess the main idea in this poem- that the need to win a prize drains an archer of his ability to shoot- is reflected in the story by how Frank's desire to prove people wrong eventually leads to him walking off the football pitch, which really doesn't help matters much.

Ideologies and Discourses

This is probably my shakiest point. Never mind the fact that Lit is already a shaky subject for me. Recipe for disaster much?

• Romantic ideas about escape via nature and the confinement of society are all throughout the novel. Although I've mentioned quite a few Romantic ideas back in the section about genre, I've got a few more to add on here. First of all, being in nature means that you don't have to worry about all the complex and cultural values that you have to encounter in society so you can act a bit more freely, like Max and Frank do in "Sand." Also, although this isn't really glorifying nature per se, small towns (which are a bit closer to nature due to their smaller size) are favoured over big cities, as can be seen through the way that the small towns like Angelus and White Point are named but the city is not. Society is also seen as confining through setting such as Raelene and Max's claustrophobic caravan home. Vic's dad also breaks down as a result of stuff happening in the police force, and his breakdown causes his relationship with his other family members to become more strained until at last he can't handle it any longer and he leaves.
• There are some ideas about masculinity and patriarchy. In "Sand," Max and Frank are becoming men, or rather, Max walks among the men while Frank hasn't quite become a man yet. In "Cockleshell," Brakey gets stung by a catfish but he refuses to tell his mother, believing that he is man enough to take it on by himself. Also, male figures are important to Raelene, so much so that in order to reject one man (Max), she has to accept another in his place (Jesus). Of course, some ideas about masculinity are challenged by some of the other characters, like Bob Lang who resorts to drinking for a while.
• Finally there are lots of ideas about family. Although there are lots of dysfunctional families in this story, like the Langs after Bob left, Raelene's family, and Frank and Max, there is still some sort of importance attached to family. For example, Frank and Max depend on each other.

I might expand this bit later. But I'm tired and I should probably get around to practising clarinet before my sister comes home (she's not a huge fan of Poulenc). TTFN.

Chemical Equilibrium

This is one topic which has quite a lot of new stuff to learn, so hold onto your hats!

The first two dot points, regarding enthalpy diagrams and collision theory, are explained in my post on Reaction Rates: http://year11misadventures.blogspot.com.au/2012/10/reaction-rates.html.

Now for the new stuff! Half this stuff probably needs diagrams but I can't be bothered doing them just yet, so I might get back to this post later (key word here is "might." Great blog writer I am).

Describe and explain the characteristics of a system in dynamic chemical and physical equilibrium

A lot of the reactions that we have dealt with thus far go to completion: that is, all of the reactants are fully consumed to form products (unless there is a limiting reagent, in which case only one of the reactants is fully consumed). However, there are a lot of reactions that do not go to completion. One such reaction which you would have probably encountered is the ionisation of acetic acid. Acetic acid only partially ionises into acetate/ethanoate ions and hydrogen ions: the rest remains as acetic acid molecules. Well, actually, that isn't totally true, because at any one time some acetic acid is dissociating into ions and some acetate and hydrogen ions are combining to form acetic acid molecules. The catch is, both of these processes are happening at the same rate, so the system is therefore at equilibrium.

Since the two reactions are proceeding at the same rate, the decrease in concentration of any one particular substance is counterbalanced by an equal increase due to the opposite reaction, and vice versa. Therefore, when a system is at equilibrium, the concentrations of the substances in the system are constant. This also means that other macroscopic properties ("macroscopic" being a word I only learned just then by glancing in my textbook), such as colour, also remain constant.

Write equilibrium law expressions for homogeneous and heterogeneous systems

Writing equilibrium law expressions is the easiest part of learning about equilibrium. Basically, on the top line, you have the concentrations of each of the products raised to the power of their coefficients, and on the bottom line you do the same thing for the reactants. For example, in the Haber process of producing ammonia, N₂ + 3H₂ ßà 2NH<sub>3, the equilibrium law expression is:

K = ([</sub>N₂] + [H₂]^3)/ ([NH₃]^2)

(K is the symbol used for equilibrium constants. Don't ask.)

Remember that solids and liquids do not get included in this expression, as they don't have concentrations. Only aqueous solutions and gases get included.

If your products are all solids and/or liquids, put 1 on the top line. One way of thinking about that is that the concentration of a solid or a liquid is equal to 1 (though obviously not strictly true as, as I just said, solids and liquids don't have concentrations).

Use K and equilibrium law expressions to explain the relative proportions of products and reactants in a system of dynamic chemical equilibrium

Since the products are the numerators and the reactants are the denominators, when there's a relatively large quantity of products formed, the equilibrium constant is relatively big. Similarly, a small equilibrium constant is indicative of the formation of a relatively small amount of products.

Explain, using the collision theory, the effect on the position of equilibrium when the following changes are made to a system initially at chemical equilibrium.

Changes in solution concentration: When the concentration of one of the aqueous solutions in the system gets changed, the system is no longer at equilibrium because the equilibrium law expression no longer equals whatever equilibrium constant it was equal to before (and the only time that an equilibrium constant changes for a solution is when the temperature is changed). Therefore, equilibrium shifts in favour of the reaction that will help to bring the concentration of said solution back to what it was before: if the concentration was increased, then the reaction that consumes that solution will be favoured, and if the concentration was decreased, then the reaction that forms that solution will be favoured.

Oh, wait, I was meant to use the collision theory. Oops. Well, increasing the concentration of an aqueous solution means that there are more particles of that solution, so that the chances of them colliding with other reactant particles are increased and, therefore, that reaction proceeds faster. If the concentration is decreased, then they collide less often and, therefore, that reaction proceeds slower.

Changes in partial pressures of gases: Partial pressures are kind of the same as solution concentrations (as far as I know, anyway). So just see what I've written for solution concentrations above. However, if the overall system pressure is changed, other stuff happens too, which I'll talk about later.

Addition of a catalyst: Catalysts do not affect equilibrium- they only affect the rate at which a system reaches equilibrium.

Predict, using Le Châtelier's principle, the impact of certain changes to a system initially at chemical equilibrium

Before I dive into all the prediction stuff, I first need to explain what Le Châtelier's principle is. Basically, Le Châtelier's principle states that if a change is made to a system at equilibrium, for example the concentration of one of the substances has been altered, or the temperature has been changed, then the system will do something to at least partially counteract this change. For example, if the pressure is increased, the rate of the reaction that produces fewer particles and thus partially counteracts the change in pressure will be favoured. And so on, and so forth. Let's take a closer look:

Addition of a catalyst- I'm starting with this one, because it's easy. As I said before, catalysts do not affect equilibrium- only the rate at which a system reaches equilibrium.

Changes in solution concentration/ partial pressure of gases: I accidentally touched on this in the previous section (the section where I was meant to talk about collision theory).

Change in pressure: When pressure is raised, all gases have a higher concentration, so both forward and reverse reactions proceed faster. However, one reaction will initially proceed faster than the other. The reaction favoured in this case is the one that produces fewer particles. For example, in the Haber process, N₂ + 3H₂ ßà 2NH₃, the production of ammonia would be increased if pressure was to increase as the forward reaction produces fewer particles than the reverse reaction (2 particles of ammonia for the forward reaction and 1 particle of nitrogen gas and 3 of hydrogen gas for the reverse reaction). The inverse is also true- if pressure is decreased, the reaction producing more particles is favoured.

Change in temperature: When temperature is raised, both reactions proceed more quickly (see my previous post on Reaction Rates). However, one will proceed more quickly than the other. To cool down the system, the endothermic reaction is favoured. The inverse is also true- if temperature is decreased, the exothermic reaction is favoured in order to heat up the system.

By the way, changing the temperature is the only way to change the equilibrium constant (K).

Interpret changes, such as colour changes, of physical and chemical systems at equilibrium

For changes like these, simply work out what colour (or whatever) each "side" of the reaction is meant to have- for example, the reactant ions might be yellow while the product ions are orange. As you do stuff to the system, work out whether the changes move more in favour of the product side or the reactant side, for example, is the solution becoming more yellow or more orange?

Conditions of Industrial Processes

Sometimes you get given some random industrial process and you have to predict the ideal conditions under which to perform these processes.

First of all, you should pretty much always suggest using a catalyst, as catalysts increase reaction rate without having any effect on the yield.

Secondly, use Le Châtelier's principle to work out whether an increase or decrease in temperature will favour the products. Then do the same for pressure. If an increase is required, that's great, since increasing temperature and pressure also increases reaction rate. If a decrease is required, then say that a "medium" temperature or pressure is required, since having a really low temperature or pressure will just make the reaction proceed at a snail's pace.

However, high temperatures and pressures cannot be indefinitely high, as there's economic and safety limits on stuff. Plus our technology is not infinitely good either.

Describe and explain the conjugate nature of buffer solutions

Buffer solutions? Never heard of these before... Oh, wait, my textbook has them in the next chapter. So maybe that's why I haven't heard of them before. Maybe this dot point got put in the wrong place or something. Or maybe we're meant to apply Le Châtelier's principle to buffer solutions once we learn about them. Stay tuned...

Exponentials and Logs

I'm going to very very briefly go over stuff that was covered last year, and then explain the new stuff.

Differentiating exponential equations: Multiply by the derivative of the power of e, keep everything else the same. (e.g. e^(2x) becomes 2e^(2x))
Integrating exponential equations: Divide by the derivative of the power of e, keep everything else the same (e.g. e^(2x) becomes (1/2)e^(2x))

ln (natural log) is a log to base e
A log with no base specified is a log to base 10

Converting logs to exponentials:

Remember, the bottom number (the a in this case) is the base.
According to Wikipedia, the b is called the argument in the logarithmic function (top) and the answer in the exponential function (bottom).
Also according to Wikipedia, the c is called the answer in the logarithmic function and the power in the exponential function.
Though Wikipedia also suggests that the terms "argument" and "power" are interchangeable.

Adding and subtracting logs: You know how when you multiply numbers with powers, you add the powers, and when you divide numbers with powers, you subtract the powers? Well, logs are the opposite. You multiply when you need to add and divide when you need to subtract. I know, I'm terrible at explaining. And there's the power law too, which involves bringing the power around to the front- also a pretty terrible explanation. So here's some formulae, courtesy of Microsoft Word:

Differentiating logarithmic equations: This method ONLY works for natural logs- logs with base e, or ln.

Derivative of the function ln (f(x)) is given by (f '(x))/ (f(x)) - that is, the derivative of the argument divided by the argument (see "converting logs to exponentials," two sections above).

If you don't have a natural log, you need to convert whatever you have into a natural log. To convert an exponential into a natural log, take a natural log of both sides. For example, 2^x = y can be rearranged to ln (2^x) = ln y and then x ln 2 = ln y and finally x = (ln y) / (ln 2). From there you can find dx/dy and then dy/dx.

To convert a log of another base into a natural log, first convert to exponential form and then use the above method.

Integrating to give logarithmic equations: The new stuff! Finally!

Remember how back in 3AB we were told that you can't find the integral of x^(-1) using the old "raise the power by one and divide by the new power" rule because then you'd end up dividing by 0? Well, fear no more, because now we have a new secret weapon! Natural logs!

You see, x^(-1) = 1/x. And the derivative of ln x = 1/x! Therefore, the integral of x^(-1) must be ln x! (EDIT: Actually, it's ln x + c.)

Well, actually, it's slightly more complicated than that. You see, if you want to find the area under the curve of y = x^(-1) you'd want to use integration, which would lead you to using ln x. The thing is, though, the curve of y = x^(-1) exists for negative values as well as positive values, while ln x is undefined for negative values. To get around this, we actually have to write the integral of x^(-1) as ln |x|. No, ln |x| + c. Oops.

Now for some more fancy stuff! Clearly, there are other logarithmic functions out there than ln x, and therefore lots more equations with logs as their integrals. Let's see how we can work this one out.

First of all, put the bit at the bottom into a natural log (probably not really correct terminology, but whatever). That is to say, if you have 2x + 3 at the bottom, make it ln (2x + 3) and move it to the top. Now, hopefully you also have a multiple of the derivative- in this case 2- sitting at the top with it. (If not, you can't integrate.) Then divide this multiple by the derivative.

Wow, today must be a day for terrible explanations. So here's an example that might shed some light on the matter.

(integral sign) (2 + cos 2x)/(4x + sin 2x) dx

Step 1: Put the bit at the bottom into a natural log and move it on top.

(2 + cos 2x) ln (4x + sin 2x) dx

Step 2: Divide by the derivative of the stuff that was originally at the bottom.

((2 + cos 2x) ln (4x + sin 2x))/(4 + 2 cos 2x)
= ((2 + cos 2x) ln (4x + sin 2x))/(2 (2 + cos 2x))
= (1/2) ln(4x + sin 2x)

Step 3: DON'T FORGET THE PLUS C!!!!

= (1/2) ln(4x + sin 2x) + c

(Please note: I don't know for sure if the above method is mathematically correct or whatever... but hey, it works. Just make sure you've got the derivative of the stuff at the bottom at the top, and you should be fine. Otherwise, use the substitution method as I outlined here: http://year11misadventures.blogspot.com.au/2013/05/integrating-by-substitution.html.)

Differential Equations

So what is a differential equation you might ask? Well, it's an equation that involves derivatives. For example,  dy/dx = 2 is a differential equation. To solve an equation like this, we just use integration, and integrate both sides with respect to x, giving us y = 2x + c. (Don't forget the +c!!!)

Solutions to differential equations involving a + c are called general solutions to the equation. If you're given more information, though, like what y is at a certain point, then you can find a particular solution.

My textbook's also reminding me that, technically, when you integrate both sides, you get a +c on both sides. The integral of (dy/dx) is actually y + c and the integral of 2 is 2x + c. You only need to leave in one of the cs, and this c is technically equal to the difference between the two cs.

Now, if you have something like y(dy/dx) = 2x + 3, you still integrate both sides with respect to x... (and yes, I'm sorry that I still don't know what to type in as a good substitute for the integral sign)

(integral sign) y(dy/dx) dx = (integral sign) 2x + 3 dx
(integral sign) y dy = (integral sign) 2x + 3 dx
(y^2)/2 + c = x^2 + 3x + c
(y^2)/2 = x^2 + 3x + c

Now, look closely at the second line. See how the y is with the dy and the x is with the dx? This is what you should aim to do when you rearrange the equation. Although dy/dx isn't really a fraction- it's the limit of a fraction- thinking of it as one might help you to rearrange the equation so that the ys are with the dy and the xs are with the dx.

Wow, that was a relatively short post (for me, anyway). I can't really be bothered doing area under a curve now because you've probably already done it before and I have other stuff to write about like integrating logarithmic functions, 3D vectors, the shapes of molecules, and Shakespeare's Hamlet. I might write about area under a curve once I've revised everything else that I need to (or maybe while I'm procrastinating over revising those things).

Oh, and by the way...

DON'T FORGET THE + C!!!!!!!!!!!!!!!!!!

Integrating by Substitution

This (hopefully) shouldn't take too long to explain. Basically, you generally use integrating by substitution when you've got something in brackets raised to a power and something else sitting outside those brackets that isn't a multiple of the derivative of the stuff in the brackets.

When you integrate by substitution, you set the stuff in the brackets equal to u. For example, if you want to find the integral of 20x(2x + 3)^3, you might then make 2x + 3 = u.

You then need to play with this equation to find two things: an equation for x and an equation for u.

Rearranging the equation, x = (u - 3)/2.
Also, du/dx = 2. Therefore, dx/du = (1/2).

I don't know how to type in integral signs on Blogger, and I don't know what the shorthand is either. So bear with me here.

(integral sign) 20x (2x + 3)^3 dx
= (integral sign) 20((u - 3)/2)(u)^3(dx/du) du
= (integral sign) 10 (u - 3) (u)^3(1/2) du
= (integral sign) (5u - 15)(u)^3 du
= (integral sign) 5u^4 - 15u^3 du
= u^5 - (15/4)u^4 + c
= u^4(u - (15/4)) + c
= (1/4)(u^4)(4u - 15) + c
= (1/4)(2x + 3)^4(8x + 12 - 15) + c
= (1/4)((2x + 3)^4)(8x - 3) + c

You have to make sure that you pop in your substitution for x (in this case x = (u - 3)/2) otherwise you'll end up trying to integrate two variables at once, which I'm told isn't allowed. This is also why you need to have a multiple of the derivative "sitting outside the brackets" in order to integrate. You don't really have to use the substitution method when you do have a multiple of the derivative present, but you still can.

(Well, I'm currently looking at another example which is to find the integral of 8 cos (2x) (sin 2x)^5... but I'm not really sure how to find x in terms of u using the substitution u = sin 2x. At least in this question a multiple of the derivative is present...)

Integrating Trig Functions

Okay. I HATE INTEGRATING TRIG FUNCTIONS. Absolutely hate doing it. But I have to for Spec. So I'm going to explain how to do it.

The Basics

Probably one of the main reasons why I hate integrating trig functions is because I keep mixing up the derivatives of the trig functions with their integrals. Plus you might get some that seem tricky to work out and then you realise that the trig function in question is simply a multiple of one of the three derivatives of tan x.

Here are the integrals that you need to know:

Integral of sin x = -cos x
Integral of cos x = sin x
Integral of (1/(cos x)^2), (sec x)^2 or 1 + (tan x)^2 = tan x

All the other rules work as before. If the trig function's raised to a power, raise the power by one and divide by the new power and the derivative of the trig function- but make sure that a multiple of said derivative is also present. So you can't differentiate (sin x)^3 directly because sin x's derivative, -cos x, isn't "sitting outside the brackets." (There are, however, other ways to differentiate this, which I will talk about later.) This is where I always screw up because I always divide by the integral by accident, which means that I get the positive/negative signs all wrong.

Oh, and don't forget the...

+ c

If you don't have the derivative sitting outside the brackets... (for trig functions only)

If you have sin x or cos x raised to an odd power, you can do a dirty trick as follows...

If you have, say, (sin x)^3, you can split that up into (sin x)(sin x)^2. Now, (sin x)^2 is also equal to 1 - (cos x)^2. This gives you (sin x)(1 - (cos x)^2), which then gives you sin x - sin x(cos x)^2 which can be integrated to -cos x + (1/3)(cos x)^3 + c!

(And yes, while I was working that one out I made my old mistakes of dividing by the integral instead of the derivative and forgetting the plus c... hopefully I don't make those mistakes in tests...)

If you have sin x or cos x raised to an even power, well, there's a dirty trick for that too, but it's not as convenient.

Here are two more trig identities to memorise (I hate memorisation, so that's just another reason why I hate integrating these things). (cos x)^2 = ((1 + cos 2x)/2) and (sin x)^2 = ((1 - cos 2x)/2). You can go ahead and prove them if you want to feel confident that they work. These functions can be integrated.

If you have (cos x)^6 or whatever, (cos x)^6 = ((cos x)^2)^3 = ((1 + cos 2x)/2)^3. Then you have to do the fun, fun work of expanding that and integrating it. And so on. Argh. I hate these things. I've pretty much said everything about them that I have to, though. Now for some more posts on integration by substitution, differential equations and maybe a quickie on area under a curve as well...