VCE Methods Question Thread!

[Special At Specialist]

Why does the domain of two functions added together become the intersection of the two?

It's easier to show by example:
Let's say f(x) = sqrt(x) and g(x) = sqrt(4 - x)
We can see that f(x) + g(x) = sqrt(x) + sqrt(4 - x)
But what is the domain of f(x) + g(x)?

For sqrt(x) to be defined, we need x to be 0 or more.
For sqrt(4 - x) to be defined, we need x to be 2 or less.
We need to satisfy both conditions for f(x) + g(x) to be defined.
So x must be between 0 and 2 (inclusive).
Thus the domain of f(x) + g(x) is [0, 2]

More generally, let's say f(x) has condition A imposed on it. And let's say g(x) has condition B imposed on it.
What conditions must h(x) have to be defined if h(x) = f(x) + g(x)?
It must have both conditions: A and B
ie. h(x) is only defined for the intersection of A and B (where both conditions are met simultaneously)

[kinslayer]

Why does the domain of two functions added together become the intersection of the two?

If h(x) = f(x) + g(x) then h(x) is defined only when both f(x) and g(x) are. The maximal domain for h(x), then, is all x for which both f(x) and g(x) are defined -- but this is nothing more than the intersection of the domains of f and g.

(An element is in the intersection of two sets if and only if it is in both sets).

[cosine]

Thank you guys, makes sense now!

Just wondering is the range the similar process? Or would I just have to sketch the new graph h(x) and restrict the domain and then determine the range?

[Apink!]

Hi,
just a quick question.
Are we expected to know integration by parts in the methods course?
Thanks!

[keltingmeith]

Hi,
just a quick question.
Are we expected to know integration by parts in the methods course?
Thanks!

Integration by parts is not in VCE, and as far as we can tell it won't be for a while (grrr, VCAA). However, a very similar technique often dubbed "integration by recognition" is in the methods curriculum, which is when they simply ask you to diff something, and use that result to integrate something. Eg, diff x*ln(x), and hence integrate ln(x).

[Apink!]

Oh thanks! I'm glad.

[Special At Specialist]

Hi,
just a quick question.
Are we expected to know integration by parts in the methods course?
Thanks!

No. Not even VCE Specialist Maths teaches integration by parts.

[keltingmeith]

Technically you shouldn't see anything as you can't 'see' a point in 2D

That was sorta the point.

[cosine]

What the hell is the point of integration? Like I see how derivatives can provide the gradient of certain functions but what does doing the opposite yield? Thanks

[keltingmeith]

What the hell is the point of integration? Like I see how derivatives can provide the gradient of certain functions but what does doing the opposite yield? Thanks

You'll later discover that the integral has many different applications to functions, such finding the area under a curve or the average value a function takes.

However, one really important use of the integral is its ability to solve what we call "differential equations". For example, you're often told that all objects fall at the same rate (more particularly, about 9.8 m/s^2). Well, we can use this to find the velocity and acceleration of an object as it descends. From our knowledge of derivatives, we know that velocity (m/s) is the rate of change of displacement (m), and so fits the differential equation:



Similarly, velocity is the rate of change of acceleration, and so:



So, we can use integration to solve this differential equation, and so we can then find where an object is at any particular time. This is great, because acceleration is easy to measure - but displacement isn't always very easy at all.

This particular example was quite simple, however they can get quite complicated very quickly.

[TheAspiringDoc]

What the hell is the point of integration? Like I see how derivatives can provide the gradient of certain functions but what does doing the opposite yield? Thanks

isn't there something cool like acceleration is the derivative of velocity? If this is true then velocity is the antiderivative of acceleration. Right?

[99.90 pls]

isn't there something cool like acceleration is the derivative of velocity? If this is true then velocity is the antiderivative of acceleration. Right?

Position - velocity - acceleration - jerk - snap - crackle - pop!

[keltingmeith]

isn't there something cool like acceleration is the derivative of velocity? If this is true then velocity is the antiderivative of acceleration. Right?

Refer to above your post.

[TheAspiringDoc]

Refer to above your post.

Fair point
Wikipedia's top result

Snap, Crackle, and Pop are the cartoon mascots of Kellogg's crisped-rice breakfast cereal Rice Krispies, known in Australia as Rice Bubbles.

Helolhagigglesnifflerofl, although further down:

"Snap", "Crackle", and "Pop" are terms sometimes facetiously used for the fourth, fifth, and sixth derivatives of position.

[Individu]

Thank you guys, makes sense now!

Just wondering is the range the similar process? Or would I just have to sketch the new graph h(x) and restrict the domain and then determine the range?

Anyone?