Paper 1 shit Flashcards by Lewis Duggan

what is the factor theorem?

what numbers should you use when finding factors?

(x - a) is a factor of f(x) if and only if f(a) = 0

Numbers that multiply up to the non x term as this is the ultimate thing that decides the value

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What are?
f(x) + d
f(x) - d 
f(x + c)
f(x - c)
-f(x)
f(-x)
af(x)
f(ax)

vertical translation up d 
vertical translation down d 
Horizontal translation left c 
Horizontal translation right c 
Reflection over x axis 
Reflection over y axis 
Stretch parallel to the y axis factor a  
Stretch parallel to the x axis factor 1/a

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How do you convert a Cartesian equation to parametric?

what about when its annoying like a circle equation?

set x as = to T

Use common trig identities

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what is a Cartesian equation?

What a parametric equation?

Any standard equation that has x and y’s in it

2 lines of sperate equations where x and y = something separately, lots of t’s too

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what does tanx =?
what is the all mighty trig function?

how do we use the almighty trig to find 2 more identities?

How do we find sin2x and tan2x? cos2x?
how do you find the other 2 cos2x formulas?

tanx = sinx/cosx

sin^2 x + cos^2 x = 1

divide by sin^2x for 1 and then cos^2x for another

Use compound angle formula
Re arrange sin^2x + cos^2x = 1, for both sin^2x and cos^2x and sub in

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what is the perpendicular line of another line?

the negative reciprocal

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what is the distance between 2 points?

a squared plus b squared = c squared

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How do you find the equation of a circle when you are given numbers? how do you find the centre and radius?

re-arrange so x is next to x and y is next to y. Then you should complete the square. The centre is the numbers in the brackets. The radius is the number on the other side rooted.

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what is a chord in circles?

one point to another cutting a piece out

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what is the angle in a semi circle connecting to both ends of the circle?

an angle in a semi circle from a point connecting to both ends is a right angled triangle

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what information can we deduce from the perpendicular bisector of a circle? what can we find with 3 points?

The perpendicular bisector of a circle will always travel through the middle of a circle.
We can use this info with another point as the perpendicular bisector that also goes through 0. put them equal to each other to find the centre, then radius by finding distance between centre and one point on the circle.

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what is a normal in relation to circles? what is the tangent to the circle?

The normal is the line that passes through the point A on the circumference and passes through the centre of the circle. Not just radius, carries on further.
A Tangent is the line that travels through the curve of the line

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What is the normal?

passes through the point and is always perpendicular to the tangent

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How do you know if a line in relation to a circle is a tangent?

The line only crosses the circle at one point

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how do you find if lines intersect with circles? what about intersection between circle and circle?

substitution

expand, substitute

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How do you find tangent and normal to a curve?

How do you find the gradient when you have the gradient function term?

Find the co ordinates then find the gradient function. Sub in the x co ordinate into the dy by dx
To give the gradient of the tangent.
For normal, its the negative reciprocal of the gradient
Then use (y - y) = g( x - x)

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what is the nth term for a aruthemtic seires? - equation

Un = a + (n - 1)d

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what is the nth term for a geometric sequence?

Un = ar^n-1

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what does a hollow dot mean on a number line when dealing with inequalities?

not equal too

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what does a solid dot mean on a number line when dealing with inequalities?

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equal too

what is set notation?
what’s the process?
what about when there is multiple things when do you use union and intersection?

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notation in full.
squiggly line, x colon(such that)( dot dot), x exists in the, real numbers, equality…. close bracket

Use the union symbol when it is two inequalities that don’t overlap, when they do use intersection

what is the process to interval notation?
what does curved and square bracket mean?
how do you do interval notation when its just 1 term or uses infinity, how do you write -3 or equal too.

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x exists, bracket, number, comer, number, bracket.
curved bracket means not equal too
square bracket means equal too

JUST X E (not X e R)

Use curved bracket (as cannot equal too infinity), -infinity, comer, -3, square bracket.
(between - infinity and -3)

what is a sum to infinity? (geometric sequences)

when does it work?

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when a sequence is done an infinite amount of times what that upper limit number is.
Only works when modulus of r is less than 1

When drawing inequalities, how do you find out weather its above or bellow the line

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Sub in 0,0 assuming the line does not pass through this. if the inequality holds then 0, 0 is in the region you want and if the inequality is false than 0,0 is in the region you don’t want

``` what is a constant sequence? what is converging sequence? what is a diverging sequence? what is a periodic sequence? what is a oscillating sequence? ```

when the same number is repeated when a number gets closer and closer to a value a non converging sequence when multiple numbers are repeated with peroid n same numbers repeating over a central value (perodici)

how do you find exact trig values? | How do you find if its postive or neagative?

draw a right angle triangle, find the missing side. using knowledge of trig graphs decide if its negative of positive in terms of which point on the respective curve it is at

What are the translation order rules?

Bid maths backwards inside brackets, Bid maths outside If 1 in and 1 out then don't matter

what does the graph of sinx look like? what does the graph of cosx look like? what does the graph of tanx look like? Whats the method to remebering there shape?

sinx starts from 0, 90' is at 1, 180' is 0, 270 is -1. 360 is 0, cosx starts from 1, 90 is 0, 180 is -1, 270 is 0, 360 is 1 tanx starts from 0, at 90 goes up, after 90 starts from low to 0 at 180 then up, then down to 0 at 360 Starts where it ends

what are the two need to learn triangles?

1, 1 root 2 | 1, 2 root 3

How do you find co effficants of x in double bracket equations?(binomial expansion)

find co efficants of all possible ways to multiply the powers together to equal what your looking for. add these co efficants together

how to find approximation using binomial expansion? | e.g whats (1.03)^8?

Turn the inside of the bracket = to the number ur approximating. solve for X, sub this value of X into the expansion

what is the differentiation formulae?

times and minus

what is the integration formulae?

add then divide

how do u find a stationary point?

when dy/dx = 0

how do you find if the stationary point is local maximum or minimum?

find the second derivative, sub in the stationary point and if it is less then 0 it is maximum, more then 0 then minimum

what is the binomial expansion equation? (to find full expansions of (a +bx)^n?. what is the n? what do we do with the coefficient of x when expanding? when doing (16a + bx)^n, what do we do? what cant we forget? what is the validity rule?

On the equation sheet. 1 + nx + n(n-1)x/2! + n(n-1(n-2)x^2/3! etc the power put it where there's and x and square or cube etc it too. take a factor of 16 out, don't forget this is still affected by the power (16^n(16a/16 + bx?16)^n) Take mod x and its coefficient n put it < 1 and solve |aX| < 1

When implicititly differentiating what is dy/dx of Y? y^2? y^3?

``` Y = dy/dx y^2 = 2y dy/dx y^3 = 3y^2 dy/dx y^4 = 4y^3 dy/dx ```

What is an explicit function? What is an implicit function? How do we change from explicit to implicit?

explicit is when y = somin Explicit is not whem y = somin differeitate in respect to Y

what is differiotnal formulea for cosx? what goes to what?

``` sinkx = kcoskx sinx to cosx to - sinx to - cosx back to sinx ```

``` what's the differential of e^kx? 2^x? 3^x? lnx? sinx ```

``` ke^kx 2^xln2 3^xln3 1/x cosx ```

What is the area of a trapezium? What is the trapezium rule?

1/2 h(a + b) 1/2 the width times height 1 and height 2 1/2 h (width) times (first + the last) + 2times the rest

what does h stand for? (Priziam rule) How do you work it out? - using equation on booklet How do we then find are y ordinates?

strip width strip width = top number above integrate sign - bottom number over how many strips we want do a table of value from the bottom x ordinates value to top value going up in the h intervals. Sub these numbers into equation and work out the y ordinates.

``` What is the intergal of 1/x? what is the intergal of sin ax? what is the intergral of e^ax? what is the intergral of sec^2 ax? what is the intergral of f'(x)/ f(x)? ```

``` ln|x| + c -1/a cos ax + c 1/a e^ax + c 1/a tan ax + c ln|f(x)| + c ```

What is the process of intergration by subsitution? When intergrating with x outside the bracket what do you do? When intergating 2 brackets at same time what you do? What do u do to the bounds when intergating by subsitution?

``` Let u = the inside bracket differeitate du/dx make it dx = simplify, then intergate new term sub in u and + c ``` use intergation by subsitution as normal, x will cancell Do it as normal, change the other bracket into "in terms of u" and sinplify n solve Change them to in terms of u aswell

When intergating by parts what is the acronym for the order of selceting u and dv/dx ? When do we intergate like this? how many times max? What is the process?

``` Logs Algebra Trig Exponentials When there is a prodcut of functions and we need to make it simpler. twice Let u = differeinate Let dv/dx = interate for v Sub into equation + C ```

what is the change of sign method? (numerical methods)

when f(a) > 0 and f(b) < 0, there is a change in sign so if the function is continuous so there is at least 1 root between a and b

What is the x = g(x) method? | when does this work to converge on the point?

set x = to a function of x then use ans = and then un + 1 thing, every x is un when the gradient at the intersection point is between -1 and 1

How do you draw a staircase or spider diagram? what does the type of diagram depend on?

curve then line depends on the gradient at the roots as when it is negative it is cobweb if the gradient at the root is positive it is a staircase diagram

how do you use the newton Raphson method?

use the formulae in the booklet

Paper 1 shit Flashcards

(49 cards)