Paper 2 shit Flashcards
(43 cards)
what is proof by exhaustion?
working through all of the possibilities to show the original conjecture must be true
what is proof by counter example?
finding an example to disprove the statement
What is a differential equation?
when there is a dy/dx term in it
how do you solve a differential equation by separating the variables?
What do you do when you have e to the power ox x + c?
what do you do when c changes?
when finding the particular solution what do you do?
put all the y terms and all the x terms onto sperate sides.
Intergate
Put + c on the right side end
Seperate it into eto the x + e to c
e to the c simplifies to A so = Ae^x
turn it into another constant
sub in the co ordinates to find the constant C then sub constant in for the answer
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
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 is the perpendicular line of another line?
- the negative reciprocal
what is the distance between 2 points?
a squared plus b squared = c squared
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.
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
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
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
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
How do you find concave and convex?
second deritive greater than 0 its convex
second derivite less than 0 is concave
(solve inequlaity)
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
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
whatever the little first letter is make it equal to big letter itself on the end
(log laws)
what happens when you add logs
what happens when you minus logs
what can you do if something is to the power?
what can you do if the bottom term is same as middle?
what does logb of 1 =?
times them divide them bring the power down = 1 0
using first principles how do you differentiate sinx or cos(x)?
write it out,
double angle formulea
factor and then separate the fractions
use small angle aproximation
how do you differeinate by first principles?
what do we need to remember?
use the equation,
at the end h goes to 0
When do we use partial fractions? what 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.
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
How do you do suvat in 2D using vectors? what’s the process?
what if you need to find the magnitude of anything?
use the suvat normally but use the vectors in the form i and j or in translation form and solve as normal
Use phytahgros as vector triangle
