BEC'S methods questions

[Mao]

you wouldnt have to sketch these in non-calc. (maybe a cubic, but not this)
if you are in spec, you would use knowledge of addition of ordinates and reciprocol functions (but this type of rational function is not included either)

if the asymptotes get you (and I'm assuming its the non-verticle ones):

they can be generally found with and

in this case:

:

as x gets large,



that is, a slant asymptote of

but you REALLY dont need to know this for the methods course. =)

this stuff, limits, is maths with intuition without maths.

[/0]

They could make you use addition of ordinates, that's how they usually ask you to draw strange graphs, but not likely...

[bec]

nope i don't do spesh so that's good to know i don't have to waste time trying to learn that stuff...

another question though, when i'm asked a question like this:
Let f: [-1,1] , where
Find the values of x for which the function is decreasing.

....I would answer that as x E (0,1]  (with a square bracket) but the book uses a round bracket. Why is that? it's still increasing at 1 (ie, f'(1) isn't zero)

[Mao]

the end-point itself doesnt have a gradient, because you cannot draw an unique tangent to that point.

remember the definition of continuity:

[1]

[2]

in this case, we dont have both sides of the limit, so its not "continuous" at that point. (it ends)

and continuity is a criteria for differentiability, hence end-points are not differentiable.


it "is" still increasing, but you cannot include it.

given that is continuous and "smooth" for , then is defined for

[bec]

thanks mao, that makes sense.

what's the best way of finding the area enclosed by the y- or x-axis and a tangent and a normal to a curve?

first i just find the lengths of the lines individually using pythag, but it takes forever...is there a better way?

[Mao]

break into two triangles

say if you want to find the area enclosed by tangent/normal/x-axis:

tangent meets normal at (c1,c2)

tangent meets x at (x1,0)

normal meets x at (x2,0)

then triangle 1 is made up by (x1,0) -> (c1,0) -> (c1,c2)
triangle 2 is made up by (c1,c2) -> (c1,0) -> (x2,0)

here's a picture:


both are right-angled triangles (with x axis as base)

so the area is:





=)

the other case with y-axis will be

=)

[bec]

cool, thanks a lot

[bec]

I don't have solutions to these - can anyone tell me if I'm right (or where I went wrong)?

Find an approximate value for the following using the approximation formula:
1. , where p is small.

(I got


2. Find the approximate change in the volume of a sphere of radius 9cm when the radius increases by 0.01 cm.
(I got 324cm³

thanks

[Mao]

remember, the approximation formula goes like:



for 1:












2

[bec]

thanks mao

[bec]

how would i work this out?

[Glockmeister]

I can only think of expanding the term... there's got to be another way.

[Mao]

expanding will be the easiest.

i havent tried the definition that , but it'll become quite complicated (probably integration by parts).


just stick with what we know and love in methods: Binomial expansion =)

[Collin Li]

Mao, it is very easy because you end up with . A substitution from here for an easy kill.

[bec]

expanding it is!

what's h? do i need to know about that formula for methods?