pure Flashcards
how to find coefficient eg of 4 when (3-x)^8
8C4 a(in this case 3)^4 b(in this case -x)^4
when is binomial theorem valid
when x lies between -1 and 1
what to do if you have a surd as a denominator in a fraction
rationalise the denominator by using a complete the square form eg times ur fraction by 1-root2/1-root2
the chain rule
define g(x) as thing u do first, f(x) as thing you do second and then f’(g(x))g’(x)
integrate
Add 1 to the power and divide by the power +1 and then substitute in values to find what you ahem to add(c) to get correct equation
product rule
use when u and v are both functions of x
u x dv/dx+ v x du/dx
quotient rule
use for a fraction with functions of x as top and bottom
u= numerator, v= denominator
dy/dx=(vu’-uv’)/v^2
remainder theorem/ find the remainder
to find the remainder when f(x) is divided by (x-b) do f(b)/ substitute b into the equation and see what the answer is
the factor theorem
the factors must be factors of the +c at the end of the equation
the discriminant
b^2-4ac- if =less than 0 there are no solutions, if =0 there is 1, if =more than 0 there are 2
stationary points
where dy/dx=0
classifying stationary points
- make a table and sub in the x values into dy/dx- if +, 0, - its a max point, if -, 0, + its a minimum point
- differentiate again- less than zero= max point, if =0 use the grid method, more than 0= minimum point
f(x) moved a upwards
f(x)+a
F(x)-a
moved a downwards
f(x) moved a leftwards
f(x+a)
F(x-a)
a rightwards
stretch by scale factor a parallel to the y axis
af(x)
f(ax)
f(x) stretched by scale factor 1/a parallel to the x axis
reflection on x axis
-f(x)
f(-x)
reflection on y axis
y=e^x reflected on y=x
y=loge^x
e^x differentiated
e^x
trapezium rule
1/2h(y1+last y coordinate)+2(all rest added together)
area under speed time graph
distance
gradient of distance time graph
speed
gradient of speed time graph
acceleration
area under acceleration time graph
change in velocity
prove k^3-k is divisible by 6
=k(k^2-1)
=k(k-1)(k+1)
=(k-1)k(k+1) 3 consecutive integers
at least one is even so will have a multiple of 2 and as there are 3 consecutive numbers one must be a multiple of 3, thus it must be divisible by both 3 and 2 and so 6
prove root 2 is irrational
assume it is rational and therefore can be represented as a/b where a and b are integers and a/b is in its simplest form
so 2=a^2/b^2
2b^2=2a^2- a must be even as its square is divisible by 2 so a=2k
hence b^2=k^2 so b must also be even and therefore a/b is not in its simplest form as both can be divided by 2
prove there are an infinite number of primes
assume there are a finite number of primes and make p the product of all prime numbers +1 (so p. is larger than any prime)
- if p is prime we have found another prime exists
- if p is not prime then p has a prime factor which must be a factor of 1
- this is not possible as they are all whole numbers, hence there is a contradiction
prove there exists no integers A,b for which 5a+15b= 1
so a+3b=1/5
a and b are integers so a+3b must = integer- contradiction exists
position to term rule of an arithmetic sequence
un=u1+(n-1)d
position to term rule of a geometric sequence
un=ar^n-1
term to term rule of an arithmetic sequence
un=un-1+d
term to term rule of a geometric sequence
un+1=run
sum of arithmetic sequence
- add 1st and last term in the sequence
- divide that sum by 2
- multiply by the total number of terms in the sequence
partial sum of geometric sequence
sn=a(1-r^n)/1-r
sum to infinity of geometric sequence
=a/1-r
if a geometric sequence converges, what is its limit
1-a/r
cosec*
1/sin*
sec*
1/cos*
cot*
cos/sin or 1/tan*
tan*
sin/cos
sin^2+cos^2
=1
sin(A+B)
=sinAcosB+cosAsinB
cos(A+B)
=cosAcosB-sinAsinB
tan(A+B)
=tanA+tanB/1-tanAtanB
derivative of sinx
cosx
derivative of cosx
-sinx
derivative of tanx
sec^2x
cosine rule
- cosA=(b^2+c^2-a^2)/2bc
2. a^2=b^2+c^2-2bcCosA
to find a bearing
- draw diagram with North/South line
- join the points you want to find the distance between and use the cosine rule to do so
- work out the angle using the sine rule
- bearing should be the angle between north and the point so probably won’t just be internal angle
base^power=number- logarithm
logbase^number=power
arc length in °
x/360 x 2πr
area of sector in °
x/360 x πr^2
arc length in radians
Xr
area of sector in radians
(Xr^2)/2
what’s a mapping
a rule that connects input to output
what’s a domain
set of all possible inputs
codomain
set of all numbers from which you can choose your outputs
range
all outputs from the domain
well defined mapping
everything in the domain has an output and only 1 output in the codomain
function
well-defined map
how do you work out the graph of the inverse function
graph of function reflected on y=x
for an inverse to be a well defined map
f(x) must be 1-1 and range=codomain
volume of prism
area of base x h
divide by 2 if triangular prism
volume of cylinder
πr^2h
SA of cylinder
2πrh+2πr^2
volume of a sphere
4/3∏r^3
volume of a cone
1/3∏r^2h
volume of a pyramid
1/3 w x l x h
SA of a sphere
4∏r^2
SA of a cone
∏r^2+∏rl where l is slant height
