Mistakes Flashcards
When implicit differntiwjton, and don’t have numbers, what to do
Sub back into equation to find them
Don’t assume the thing they drew is the x and y axis, what to do
Don’t assume, so if you want max height find the difference between LOWEST point and highest, which could be neagtive…
How to find min max range from domain
Find normal max range see if it lies
Find the smallest part by taking the bounds of the domain, draw it
How to show change of sign exact wording
FUNCTION IS CONITNOUS between,
Change of sign
Don’t lack, when integrating after so,ging something, must read q
Might have changed it, take some cknstsnt or divide andwer as a result
Finding Cartesian equation of the plane of trajectory
Must take it from the GROUND
So whatever height it was , say 5, the trajector would be 5 - this , where x is the height ABOVE THE GROUND
model height abive as x, and find the difference in height, and work out from there
DONT LACK
Proof it’s a sqaure
Angles, must do ALL OF THE ANGLES , so use gradients to prove perpendicualr
Then sides, two adjacent sides must be the same, say tangent, or radius
How to explain why binomial expansion valid again
You find the values of x for which it’s valid, set the WHOLE x expression modded to be less than x
Work out the values of x
And show bith limits in there
Explain why the value of the accerlwtion ( they said a so need to find a) immdiestly before velocity becomes constant is likely to mean that model B is a better model than model A?
Find the accerlwtion using differntiation
Find accerlwtion at that time, should be 0
Which means the accerlwtion was decreasing, and so the car reaches maximum speed WITHOUT sudden change in speed that was seen before
How to show two linewr liens never intersect
If the grsdient is the same this means they’re parallel so jever intersect!
How to show how SPEED increases
Find accerlwtion
If negative and already newgtive, then velocity more negative so speed increases etc
Remember about series arithmetic geometric discrete CONTINOUS
They are DISCRETE
Hence if you tryna find the last value for which n is positive, set Un to be >0
Then go to the NEXT TERM
Then find the sum
Don’t lack for inverse transformations
Write it out first, then perform transformation, might have to do 1 / instead
Just write it out fully
Cartesian equation
I’m terms of y
Btw you can express any point not in a parametric curve in terms font too
How to find the angle easily suvat
Just set new variables again to max range equation to form a suvat ewuation and can solve for theta hella easilly
Remember check for degrees rwdians , draw out the graph hi cal to find range
Convert back to TIME
How to set things in I and j to 0
Remember equilibrium make i equal rn j equal
And equilbroum not matter directions resuktsnt forces are 0
When a sequence repeats what it called
Periodic, with period etc
Remember CINVERGENT means converges at a point , but divergent?
Divergent means DOESNT converge, so do all the values out of the mod r is less than 1
When does a series /sequence convergent when r =1
The sequence is convergent at r =1, but the SERIES is divergent , hende have to ude less than 1
So be CAREFUL !
Why did they not find the roots
Why is statement invalid if you just say change of sign
Change of signs were missed as intervals were too far away from them
Did not mention it’s CONTINOUS
Why does Newton raphsom fail to give correct root?
How to get rhe proper root
Because rue start value has grsdient TOO CLOSE TO A STATIONARY point, and so tangent meets x axis really far away from the root
It does converge to a root but not the correct one
Start closer to the root, hence in between rhide values
Remember ball in ther air v time graoh
V t represents grsdient which si CONSTSNT so CONSTSNT until it reaches the ground at - initsl velocity again!
Why would a balloon keep increasing volume be a problem
It would burst lmao
Might need to use differential equation if need a new expression dint lack
Always give answers as numbers
Don’t leave exact unless they tell you
BRUH displace rn time graph, DONT COUNT AREA FOR DISTANCE
DOMT BE STUPIDDD
Remember POINTS OF INTERSECTION are quire what
Not only x but Y too!
Partial fraction technique for binomial
Is not invalid, just rememebr 1 /something can be expanded too
This means you are adding two things indtead of multiplying which could get lengthy
What does modelled as a particle mean !
What does uniform Planck mean
IMPORTANT
Have negligible size so contact force acts at a point
Size and shape does not karrer
2; this means the centre of mass lies in the centre due to uniform mass DISTRIbution !
Bruh remember thst things like bricks are discrete, so if you have 4.48 bricks at distance x we know that it must be 4 bricks so take moment sand find new distance
Don’t mess up silly mistakes here olease
READ q , about magentiude, or if it JUST SAYS TIME
Then need ONLY ONE TIME , don’t lack there
When particle at origin means 0, remember velocity or displacement,emf or whatever is not positve or negative at 0, similarly it wouldn’t be north east at 0, has to be somewhere else
0 is 0
FINAL RATE ANSWER, MAKE SURE TI DO WHAT
ADD UNITS
Proof by contradiction of prime number other than 3 which is 1 less than a sqaure number
Set up contradiction, assume there is a prime number p
Let p = n2 -1
SEE DOTS, always factorise
And now show the case for when a factor could be one, as that’s the only way to be prime, remember 1 and itself
This only occurs when n is 2 and p is 3p
But any time above n it won’t create a prime, hence contradicts, and thus n is only like this at 2
Remember every time agaisnt slope
There is component of the weight, which reduces accerlwtion, so max distance is less 😭
How to prove by contradiction irrational number root 2 ,3
Assume root 2 rational hence can be in the form a/b where a and b are whole and a /b is in its simplest form
Therefore a and b can not be multiples of each other
Let 2 = A2/ B2
So a2 = 2b2 , hence a2 is even
Hence a is even and written as 2p
Sub back in, show what b is
B2 is even, b is even, 2Q
Now multiples of Esch other hence contradiction
Prove infinitely many primes contradiction
Assume finite list prime p1 p2 to pn
Let P be a number multiply all + 1
P is either prime or not confirmed
If prime, we found nee number not in list = contradiction
Id mit Prime, P Must be able to be divided by a prime fsctor ( by prime division theory).
However Any factor must also be a factor of 1, which is not possible
Hence contradiction
Remember transformations
Replace the x all totgeht
Or the whole function y
So need to do different things if you first
