math (0616) - E&LF: differentiation of expontentials

the important thing to remember, is that the derivative of e, is e. basically.

y = e^f(x)
y' = e^f(x) * f'(x)

notes are proably not the best way of studying maths

math (0422) - GeoDer: point of inflection

the point of inflection has a tangent with gradient = 0, ie. is a stationary point.

it occurs for x^3 graphs, where on both sides of the stationary point, y increases or decreases depending on the formula.

to test for this, we draw up a table and test for the number that results in f'(x) = 0, and then for a number higher than the x value that resulted in that, and lower than that.

the result will give us the shape of the graph

math (0421) - GeoDer: increase/decrease

this is essentially a continuation of yesterday's post.

just a few things to add.

maximum point is generally the staionary point or turning point tht is of highest y value. called "local maximum" if the graph is ongoing (ie. with arrows) but if it terminates then it's the absolute maximum.

same goes for minimum, which is the lowest.

interesting thing is that if one differentiates a straight line, the resulting graph would be a straight horizontal line, and it's y value would be equal to the gradient of the line, which was differentiated.

math (0420) - GeoDer: stationary point

GEOMETRY OF DERIVATIVE

or

GEOMETRICAL APPLICATIONS OF CALCULUS

yay..

at point where curve slopes upwards, tangent has positive gradient and y increases as x increases, therefore f(x) increases

at point whre curve slopes downwards, tangent has negative gradient and y decreases as x increases, therefore f(x) decreases

at the turning point or stationary point, the tangent's gradient = 0 and f(x) is stationary

also, differentiation a function twice gives the concavity of the graph at that point

math (0311) - Series: financial application

three formulas to remember

simple interest:

I = Prn

where:

I = interest
P = principal (original amount)
r = rate per unit
n = number of units

sort of related, learn:

A = P + I

compound interest

An = P ( 1 + r ) ^n

and the reverse..

Vn = P ( 1 - r ) ^n

Maths 4U - Transformations

Original

















Square Root

















Squared


















Absolute Valued
















Recipocated
















Cubed

First post... MY CRAPPY TIPS

First post ><

Okay firstly, my rule of studying is always looking at your syllabus dot points. Goes without saying but handy yeah.

Next practice paper questions... A lot of them... Like seriously... like 100x

[edit] Oh and bored of studies :) forgot them :)

I know the above isn't that useful but I'll wanted quote from Mr. GEE (baby baby baby)

'Don't SAVE'

Basically share notes and past papers...

I have a nice handy pile of physics and chem and 4U, that my parents brought. Anyone wants me to photocopy some questions. Especially 4U (I'm going thru on to check answers... I'll print out copy for our whole group doing 4U)


Oh Batehup, I'll will post some notes on Standard :)


Chris
(God Bless) =]

UEC HSC MATHS 2U

This was my UEC for my 2U HSC exam last year. It might not help but meh.

UEC (Ultimate Error Correction)

Golden Rules
- Look at how many marks and answer accordingly
- Change calculator to rads or deg accordingly
- Remember units
- Domain must always be in your head
- If unsure of answer, go back to it later
- In exam, WRITE ALL STEPS
- When about to round off, write exact answer first, then write the rounded off places in brackets

Arithmetic
- When you see 8 and 6, use CALCULATOR to find the fraction
- When you see 4 and 6, use CALCULATOR to find the fraction
- When forming quadratics, the sum of roots has a negative sign.

Integration:
- Remember +C
- When finding Area under the curve, Remember UNITS
- When asked to find a primitive, remember +C
- When finding volume, SQUARE before integrating
- When finding volume remember the π

Series and Applications:
- Nth term of GP given by: arn-1
- When finding i.e. x,4,y,9 when u want to find first term or common difference/ratio, use Tn=arn-1 or Tn=a+(n-1)d as opposed to simultaneous equations
- To prove limiting sum, absolute value of r must be less than 1
- To do ∑ questions, write out series then say it is an AP/GP with first term X, and common difference/ratio Y
- With the above case, the top is the last term and the bottom is the first term
- Read questions and do not assume it goes twice or once for a trip i.e. the truck going back and forth question
- When finding formula for Arithmetic Sequence, - Prove its arithmetic, with a=X and d=Y, Write out formula Tn=a+(n-1)d, sub in a and d, then expand and simplify

Locus and Parabola
- Sketch the locus before answering
- When its concave down or side left, then remember negative sign on the formula
- When squaring both sides, must SQUARE the CONSTANT

Absolute Values and Inequalities
- When dividing/multiplying by negatives in inequality, remember to change the direction of sign as well as the NEGATIVE SIGN
- When doing absolute values equality where one side has abs and the other doesn’t, MUST CHECK SOLUTION
- When throwing logs to other side, use calculator to check if + or – and change inequality sign accordingly
- Turning point only required in equality when there is a LONELY CONSTANT and in inequality when there is an ABSOLUTE VALUE ON ONE SIDE ONLY

Differentiation
- When u see acceleration-velocity , think differentiation
- When doing y d or y dd table, sub in points
- When doing y dd to y d questions, DO ALL STEPS
- For differentiation show questions, remember UNITS

Trigonometry
- For cosine rule: remember to square root answer when finding the side
- When doing area of a sector and finding the larger sector, consider reflex angle
- When domain has a π, answer must be in RADIANS
- When there is a quadratic, take into account acuteness to restrict domain
- When using sin rule to find an angle there are 2 cases, obtuse or acute.
- Area of sector: 1/2rrq
- Bearing must be in 3 digits
- Write exact ratios in non-rationalised form

Geometry
- “matching” not “corresponding”
- Because midpoints are the same for both diagonals, they bisect each other
- When giving radius of circle, remember units
- Similar triangles – NO SAS. Only : one angle equal and sides about equal angle are equal
- Dotted line when it is <>

Probability
- ALWAYS DRAW TREE DIAGRAMS WHEN UNSURE
- When it is a non-replacement question, remember to subtract from the total
- DRAW TABLE WHEN DOING DICE QUESTIONS

math (0302) - Series: sum of geometric sequence

contraversial formulas here..

Sn = a (r^n - 1)
........__________
................r - 1

now that's only if r > 1, if r < 1, then you have to switch the 1s and rs around in the formula.

quite the crazy gayness even though it ends up with the same answer.

math (0216) - Series: sigma notation

simple enough to understand. generally the question has that sum sign (sigma) that looks like what would happen if an E and a W had a baby.

underneath it says n = hwatever, and there's a number at the top. and its followed by an equation.

generally.. you sub in whatever numbers come between what n = and the number on the top, into the equation, and then add up the numbers you get.

sigma notation

math (1209) - Locus: parabolas

apparently, if you make a point (focus, let it = S) within a parabola, and the distance between that point (let it = a) is the distance between the parabola and a line (x = + or - a as the equation depending) then.. the distance from that focus point, to a point on the parabola (P) is equal to the distance from that point, down to the point on the x = a line that shares the same x-coordinate as P, and that would be B.

confusing?

diagrams help.

basically.

PS = PB

use distance formula, get rid of square root

we get:

(x - 0)^2 + (y - a)^2 = (x - x)^2 + (y - -a)^2

x^2 + y^2 - 2ay + a^2 = y^2 +2ay + a^2

cancel out the crap..

x^2 = 4ay

just saying, this was a parabola with vertex at the origin.

to change this, just sub in the different coordinates into the distance formula.

likewise..

if the formula was eg

-x^2 = 4ay

it would be concave down, coz rearranging to make y the subject would make the x^2 negative

math (1201) - Quad: + and x of roots

from now on im going to use * instead of x for multiplication, should be easier to understand without mixing it up with x-axis and all that crap, computer nerds would understand.. these are my notes anyway.

so the proof of this comes frooooommm..

ax^2 + bx + c = 0

what if we dont want the a there?

then its:

x^2 + (b/a)x + c/a = 0

coz that a is always annoying to work with, at least if its not 1

the deal is, we know the roots, lets call em A and B, multiply to get c and add to achieve b

so.. if you wanna get the sum of A and B, it'd be equal to the b value over the a value, aka -b/a, catch it?

i forget why its minus though, but oh wellers, i missed a fraction of the lesson

and to find the product, its c/a

math (1127) - Quad: quadratic identities

pretty straight forward, if two expressions are equivalent, all the corresponding coefficients must be equal.

you can use this to solve retarded versions of the quadratic function y = ax^2 + bx + c

i dont know what i can say.. its so weird

math (1126) - Quad: discriminant

got rather scared today when i found out our marks will always be compared to the rest of the grade from now on.

eep -.-

and theres so many more 3u people than us.

gay.

anyway.. besides seeing the proof of how the quadratic formula came to be, we worked with just one simple rule with discriminants.

let y = ax^2 + bx + c

if /\ < 0, y has no real roots, aka doenst collide witht he x-axis

now in terms of taht

if a > 0, and /\ <> 0 for all x values (aka the graph is concave up and above the x-axis)

if a < 0, and /\ < 0, the function is negative definite, and y < 0 for all x values (aka the graph is concave down and below the x-axis

math (1125) - Quad: quadratic inequalities

went over a few examples of quadratic inequalities today.

we highlight different parts depending on the sign

lets say

y = ax^2 + bx + c

if y = 0, highlights are on x-axis

if y > 0, highlights are above x-axis (arrows)

if y < 0, highlights are below x-axis (vertex)

concavity will change the graph, keep that in mind.

ooh i see how it goes. for negative numbers, y <> 0 is highlighted on the vertex.. so its still above/below

ok thats awesome.

notes finished for tonight ^^

math (1124) - Quad: roots in graphs

the discriminant (/\ - err.. this is meant to be at riangle), as we know is taken from the quadratic formula (its the bit under the root sign)

/\ = b^2 - 4ac

if /\ < 0
there will be no solution for ax^2 + bx + c = 0 so that means there will be no x-intercepts

if /\ = 0
there will only be one solution, and the graph will touch the x-axis once (at its vertex)

if /\ > 0
the graph would go through x-axis, intercepting at two points

math (1120) - Quad: y=a(x-u)^2

y=a(x+u)^2 or y=a(u-x)^2

with equations that look like this, we discovered thaat:

- a determines the concavity (if its negative it will point down)
- a also determines the steepness (wideness)
- u determines teh vertex (-u, 0)
solve bx + u = 0 for vertex (-u/b, 0)

math (1116) - Quad: quadratic polynomial

new topic started today: the quadratic polynomial

its pretty straight forward, today we just went through dot points and the basics, revision from all that parabola stuff we did earlier in the year.

man i did not miss them -.-

take forever to do.. so much space

anyway.. terms we have to get to know and use properly:

monic - quadratic where the value before x^2 is 1
coeffcient - the number before the pronumeral basically
leading term - the term with the highest power (x^2)
constant term - the one with no pronumeral
root - x co-ordinates where y = 0 (ie. the x-intercept)
zero - the same thing as root

we all know the quadratic funcion:

y = ax^2 + bx + c

got it memorized?

and if the coeffcient of x^2 is positive, the graph will be concave up, if negative it'll be concave down.

note that as the coeffcient of x^2 gets larger, the graph becomes narrower, and as the coefficient becomes smaller, the graph becomes wider

all basic stuff really.

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