math (1112) - Calc: tangents & normals

the last bit of introductory calculus O.O

time flies when you're having fun.

we know how to work out the tangent of a curve at a certain point already, correct?

its basically the derivative of the function, when subbing the x coordinate into where x is.

that will give us the tangent.

and to get the normal

we remember that

m1 x m2 = -1

where the m's are the gradients of the tangent and normal

that's the rule for perpendicular lines with COORDINATE GEOMETRY which i must.. re-go over

math (1110) - Calc: quotient rule

if y = v/u

then

y` = (vu` - uv`)/v^2

do yo ucatch that?

thats pretty much it

not necessary all the time to use it.. but it comes in handy

math (1105) - Calc: function of a function

went over something else first.. just revision but into differentiation.

square root x is like x ^ 1/2 basically.. so cube root is x ^ 1/3 etc. and of course x ^ -2 equals x/2

so yeah.. went by that with differentiation.

then.. we learned somehting new.. function of a function.. which was easier than i thought it would be. its for big powers. eg..

differentiate: (x+7)^9

you let u = the number, and y equal u to the pwoer of crap.. observe:

let u = x+7

(du)/(dx) (x+7)
= 1

let y = u^9

(dy)/(du) (u^9)
= 9u^8

then you multiply together.. and the two du cancel each other out.

(dy)/(dx) (1 x 9u^8)
= 9u^8

then you sub in the value of u that we let it equal coz it has to be x

so its 9(x+7)^8

so there ^^

math (1102) - Calc: short differentiation

talked about shortcuts for differentiation today

if f(x) = x^n then f`(x) = nx^(n-1)

chyeah boi.

its all written down in the booklet thingy

usually what we do is when we're asked to differentiate

we write it like this.. say the question was

differentiate x^2

we would write

d/dx (x^2) = 2x^(1)
=2

right?

the x on the d/dx means thats what we're doing it in relation to, if there was a t in the question isntead, the bottom would say t. the top, represents the other thing.. if the question was

y = 5x^4 - 3x

then we would put dy/dx in instead. catch it?

also, any number without the number we're focusing on (next to the d at the bottom) just turns into a 0 when working it out

math (1026) - Calc: secants

catching up for monday 26 0ct

a secant, is basically a straight line drawn in between two points in a curved graph. drawn on a straight line, its pretty much just on that line.

hmm.. there's stuff that i should probably know but i didnt exactly do the homework -.-

all ir emember is what it looksl ike ^^"

the thing about secants though, is that the answer (gradient) you get from them is only between those two points, it can change depending on what points you use.

math (1023) - Calc: continuity

continuity is basically just if the graph isnt detatched at say.. an asymptote or if it jumps in a piecemeal graph.

there was a rule, that if:

lim f(x) = f(a)
x->a

that means it would be continuous at x = a

i dunno it made sense before.. then i went braindead hahaha. i'll go over it again, didnt go over it much today coz we were gonna rush out to find out prefects.

oh and if the limit looks like

x->^+

then x approaches a from the right, aka values larger than a. same goes with ^- but from the left.

yeah.. thats kind of it for today's maths notes

math (1022) - Calc: limits 2

continued talking about limits today, did some exercises.. and now i understand whats going on.

the deal is to always sub the limit number into the limit letter thing.

eg.
lim
x -> 5

you sub the 5 into wherever x is.

but if its a 0, its undefined, you gotta factorize the top or bottom in order to cancel out some of the pronumerals.

thing about infinity thingies though.. you cant have them in fractions, gotta get rid of them. and as anything over x with the limit of x -> infinity, the anything over x will = 0

hope im not screwing upt hese ntoes coz im in a rush to play kh

cant have 0 on the denominator, so in order to get rid of it, divide everything (top and bottom) by the highest power of x (or whatever pronumeral that the limit is) on the denominator. otherwise, if its not 0 on the bottom anymore.. sub in man.

anyway.. hope i didnt miss anything.. im too excited to play.