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Differentiation in HSC Mathematics: What You Need to Know!

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anotherworld2b:
I understand now :D
Thank you for your help jake

anotherworld2b:
I had a another question
What is the significance of finding whether a function is odd or not?


--- Quote from: jakesilove on January 21, 2017, 08:53:40 am ---A function is odd if



And even if



Let's use an example to make this clearer. We know that the standard parabola is even. This has equation



Now, let's prove it is even. We start by turning all positive x's into negative x's



But, remember that a negative squared is a positive



So, we've proved the above relation! Let's try now for a cubic





Now, remember that a negative cubed is still a negative



Therefore, the cubic is clearly odd.

I could by wrong, but in 2U I'm pretty sure you just need vertical asymptotes. These will occur when a denominator will be equal to zero. So, the function



will have an asymptote at x=2, as 2-2=0 (which is obviously not allowed).

Maybe horizontal asymptotes are also in the syllabus? In that case, we utilise limits. Say we had a function



Let's restrict it to the positive x for ease, but note that the function is even so the negative x's will have the same asymptote (can you see why?)
We want to know what happens for very large values of x. We start by dividing through by the highest power of x in the denominator (always do this).



Now, we let x approach infinity. What happens when a number is divided by infinity? Well, it turns into zero! So, we know that



And this is our asymptote

--- End quote ---

jamonwindeyer:

--- Quote from: anotherworld2b on January 24, 2017, 12:44:55 am ---I had a another question
What is the significance of finding whether a function is odd or not?

--- End quote ---

It basically gives us an easier way of determining the behaviour of a function for negative values of \(x\). Provided we know what happens for \(x>0\), if a function is either odd or even, we know what happens for \(x<0\) without doing any extra work. Very useful in curve sketching and other applications of differential (and integral) calculus :)

anotherworld2b:
jamon can you explain what behavior the function can have please?


--- Quote from: jamonwindeyer on January 24, 2017, 12:53:34 am ---It basically gives us an easier way of determining the behaviour of a function for negative values of \(x\). Provided we know what happens for \(x>0\), if a function is either odd or even, we know what happens for \(x<0\) without doing any extra work. Very useful in curve sketching and other applications of differential (and integral) calculus :)

--- End quote ---

Shadowxo:

--- Quote from: anotherworld2b on January 24, 2017, 11:18:49 am ---jamon can you explain what behavior the function can have please?


--- End quote ---

Hi,
If it's even, f(x) = f(-x)
This means the function is symmetrical at the y axis, so you only need to know the shape of half the graph, the other half is just a reflection of the first half.
If it's odd, f(x) = -f(-x)
This means that the values on one side of the graph are the negative values of the other side. eg a cubic x3 function, if it's odd then you just need to be able to graph one half and the other side is just the negative of those values

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