Pure stuff Flashcards
(86 cards)
1
Q
what is the nth term for a aruthemtic seires? - equation
A
Un = a + (n - 1)d
2
Q
what is the nth term for a geometric sequence?
A
Un = ar^n-1
3
Q
What is the circumference of a sector?
A
theatre r (radians)
4
Q
what is the area of a sector?
A
1/2 theatre r^2 (radians)
5
Q
what is the perpendicular line of another line?
A
- the negative reciprocal
6
Q
what is the midpoint of a line?
A
add the values and divide by 2
7
Q
what is the distance between 2 points?
A
a squared plus b squared = c squared
8
Q
what is the differentiation formulae?
A
nx^n-1 (times and minus
9
Q
what is the integration formulae?
A
add then divide
10
Q
how do u find a stationary point?
A
when dy/dx = 0
11
Q
how do you find if the stationary point is local maximum or minimum?
A
find the second derivative, sub in the stationary point and if it is less then 0 it is maximum, more then 0 then minimum
12
Q
What are the translation order rules?
A
Bid maths backwards inside brackets,
Bid maths outside
If 1 in and 1 out then don’t matter
13
Q
what does f(2x) mean? and f2(x) mean? why?
A
strecth factors 1/3rd in x direction - bc x is inside bracket when x = 0 it flips
strecth factor 2 in y direction
14
Q
what does (x + 2) mean? why?
A
left 2 flips as inside bracket
15
Q
what doea (x) - 2 mean? why?
A
goes down 2 as outside bracket
16
Q
how do u differentiate from first principles?
A
use the formulae, sub stuff. At the end h = 0
17
Q
what does a factorial number have to be?
A
An integer
18
Q
where is the factorial button on the calculator?
A
above x^-1
19
Q
what’s 0!
A
1
20
Q
what is n and r in nCr? where is nCr on calc?
A
n is the number of items
r is how many your choosing of them
nCr is above division symbol
21
Q
what does the n and r in (n/r) represent? in relation to pascals triangle
A
n is the row in pascals triangle and r is the position so how far inwards on pascals triangle.
22
Q
how do you work out factorials?
A
use the equation on sheet
23
Q
How do you simplify n in an algebraic factorial question?
A
Simplify n! to n( n -1)! and then + (n - 2)! etc
24
Q
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?
A
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
25
how do you find co efficients of x using smaller equation?
power of the bracket is the n and the number of coefficient power is the r. so use nCr like that for column 1. The last x value must be to the power of x it asks you to find. which must add up to the power of the bracket with the middle column. Therefore equation power - last bracket power = middle column power
26
How do you find co effficants of x in double bracket equations?
find co efficants of all possible ways to multiply the powers together to equal what your looking for. add these co efficants together
27
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
28
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.
29
what is a chord in circles?
one point to another cutting a piece out
30
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
31
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.
32
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
33
What is the normal?
passes through the point and is always perpendicular to the tangent
34
How do you know if a line in relation to a circle is a tangent?
The line only crosses the circle at one point
35
how do you find if lines intersect with circles? what about intersection between circle and circle?
substitution
| expand, substitute
36
How do you find tangent and normal to a curve?
```
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)
```
37
How do you know if a curve is increasing, decreasing and stationary? how do you find these values?
increasing if dy by dx > 0
stationary if dy by dx = 0
Negative if dy by dx < 0
Solve each one to get the ranges
38
How do you find if it is convex or concave
Find the second derivative
Find using the second derivative if it is postiov3e or negative
Negative is concave, positive is convex
39
How do you find points of inflexion that are non stationary? how do you double check this?
Find dy by dx, find stationary points so dy by dx = 0 to check the inflection point does not match this.
Find second derivative, put this equal to 0, this is your general inflection points
We need to check it is positive and negative so the inflection point is going through convex to concave. so choose points close to the value either side of the inflexion point.
40
How do you find local maximum and minimum
Sub the stationary point into the 2nd derivative, if < 0 it is maximum so a negative number
If > 0 it is minimum so a positive number
41
What is the product rule? when do we use this?
The first times the derrivitae of the second plus the 2nd times the derivitive of the 1st.
The product rule is used to differentiate a product of functions.
42
When do we use the quiotent rule?
When there is a fraction that u cant change into a standard equation.
43
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
```
44
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
45
What is differetional of Y = e^x formulea?
Ke^kx
46
what is differiotnal formulea for cosx? what goes to what?
```
sinkx = kcoskx
sinx to
cosx to
- sinx to
- cosx back to
sinx
```
47
what is the differetional of lnkx?
1/x
48
When should we use the chain rule? what is the process in using it?
when diferntiating a function of a function
Let u = g(x), numbers in the bracket
then differeiate du/dx and dy/dx then times this together to get dy/dx and sub back out for u
49
What is the intergal of 1/x?
ln|x| + c
50
what is the intergal of sin ax?
-1/a cos ax + c
51
what is the intergral of e^ax?
1/a e^ax + c
52
what is the intergral of sec^2 ax?
1/a tan ax + c
53
what is the intergral of f'(x)/ f(x)?
ln|f(x)| + c
54
What is the process of intergration by subsitution?
```
Let u = the inside bracket
differeitate du/dx
make it dx =
simplify, then intergate new term
sub in u and + c
```
55
When intergrating with x outside the bracket what do you do?
use intergation by subsitution as normal, x will cancell
56
When intergating 2 brackets at same time what you do?
Do it as normal, change the other bracket into "in terms of u" and sinplify n solve
57
What do u do to the bounds when intergating by subsitution?
Change them to in terms of u aswell
58
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
```
59
When do we use partial fractions? ehat is the process?
When we have a fraction that is difficult to intergrate we split it up.
Let fraction = A/one part + B/other part
Times everything by demonitator of the fraction
Sub in so x is 0 to find one
Then other for other.
60
What do you do when there is 3rd bracket when doing partial fractions?
Use C aswell
61
What do you do when it is x^2 when doing partial fractions?
do first part of demnoitator as normal, then do b/ the term to ^1 + c/ the term to ^2
62
What are the limits rules when differentiating cosx and sinx?
as h goes to 0
sinh/h goes to 1
cosh-1/h goes to 0
63
what is the sine rule?
| when do we use this?
sina/A = sinb/B = sinc/C
| a) two angles and one side, or b) two sides and one angle
64
what is the cosine rule?
when do we use this?
how can we change the equation for different sides?
a^2 = b^2 + c^2 - 2bc cosA
a) three sides or b) two sides and a angle
whatver the little first letter is make it equal to big letter itself on the end
65
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
66
How do you find the area of a triangle w the angle?
1/2ab SinC
67
how do you find the radius when given the rest of the equation of the circle?
the numbers inside brackets squared = radius squared
68
If a cubic graph is positive where does it start and end?
| what about negative
bottom left to top right
| top left to bottom right
69
```
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
```
70
what is the factor theorem?
| what do we need to be careful about when substituting in a?
(x - a) is a factor of f(x) only if f(a) = 0
Same in the other direction
when a is negative in the bracket substitute in positive a, vise versa.
71
what does a hollow dot mean on a number line when dealing with inequalities?
not equal too
72
what does a solid dot mean on a number line when dealing with inequalities?
equal too
73
what is set notation?
what's the process?
what about when there are two numbers where x is not between them its outside them? ( x>6 and x is less than and equal to -3)
notation in full.
squiggly line, x exists in the, real numbers, colon(such that)( dot dot), equality.... close bracket
Use the union symbol and place it between the two notations
74
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.
what about when there is two numbers where x is not between them?
x exists, bracket, number, comer, number, bracket.
curved bracket means not equal too
square bracket means equal too
Use curved bracket (as cannot equal too infinity), -infinity, comer, -3, square bracket.
(between - infinity and -3)
like set notation just use the union symbol between the two terms, no need to re write x exists in
75
what is a sum to infinity? (geometric sequences)
| when does it work?
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
76
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
77
How do you convert a Cartesian equation to parametric?
what about hwen its annoying like a circle equation?
set x as = to T
Use common trig identities
78
when drawing an inequality what kind of line should be drawn when the inequity is < or >. What about or >/ ?
a dotted line
| a filled in line
79
When drawing inequalities, how do you find out weather its above or bellow the line
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
80
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 times (first + the last + 2times the rest)
81
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.
82
what is the change of sign method? (numerical methods)
when f(a) > 0 and f(b) < 0, there is a change in sign and so if the function is continuous then there is at least 1 root between a and b
83
```
what's the differential of e^kx?
2^x?
3^x?
lnx?
sinx
```
```
ke^kx
2^xln2
3^xln3
1/x
cosx
```
84
when theatre is small what does costhearte, sinthearte, and tanthearte approximate too?
```
costhearte = 1 - 1/2thearte^2
sinthearte = thearte
tantharte = thearte
```
85
using first principles how do you differentiate sinx(x)?
write it out,
double angle formulea
factor and then separate the fractions
use small angle aproximation
86
what is the intergation of parametric equations formulea?
intergal of y dx = intergal of y dx/dt dt
