Other Flashcards
What is proof by contradiction?
Assuming the opposite of a statement is true, and then showing that causes a contradiction so the original statement must be true
What is proof by deduction?
The traditional way of proving a statement
What is opportunity sampling?
When members from a given population are willing to participate in the investigation. Examples include radio or television phone-ins. It is easy to set up but can be biased.
What is random sampling?
Each member of the sample frame has an equal chance of being selected. It is generally non-biased but hard to set up for very large samples.
What is stratified sampling?
If children make up 20% of the population, we would make sure that children make up 20% of the total sample.
What is quota sampling?
This type of sampling requires the sampler or interviewer to complete their investigation according to a set of instructions. The instructions will usually specify which quotas are to be met. It is non-biased but hard to set up
What is systematic sampling?
Systematic sampling uses a simple rule to choose people. For example, every 10th member of the sample frame could be selected.
Coefficient of restitution formula
e = (v2 - v1)/(u1 - u2)
Separation speed divided by approach speed
v1/v2 relationship test formula
v2 = ( u/(m1+m2) )( em1 + m1u1 + m2u2)
Mnemonic: u over m1+m2 times sum of em1, mu1, mu2
Polar to cartesian conversion rules
y = rsin(θ) x = rcos(θ) θ = arctan(y/x) r = x^2 + y^2
Polar differentiation
r = f(θ)
y = sin(θ)f(θ)
Now find dy/dθ as normal
Differential equation in the form (dy/dx) + yf(x) = g(x)
Multiply both sides by integrating factor of e^∫f(x)dx. The left side then becomes d/dx(yF), where F is the integrating factor. Remember to multiply the right side by F too.
sinh(x) =
(e^x - e^-x)/2
tanh(x) =
(e^2x - 1)/(e^2x + 1)
cosh(x) + sinh(x) =
e^x
cosh^2(x) - sinh^2(x) =
1
What is Osborn’s rule?
When converting a trig formula to a hyperbolic one, replace all the functions with their hyp versions, but replace sin^2(x) with -sinh^2(x)
Differential equation in the form ay’’ + by’ + cy = 0
Make a quadratic and factorise
if (m - k)(m - j): y = Ae^kx + Be^jx
if (m - k)^2: y = e^kx(A + Bx)
if p +/- qi: y = e^px( Acos(qx) + Bsin(qx) )
How do you decide which method to use to solve a differential equation?
[1O] 1. If possible, get all the variables on different sides, nearer maths style
[1O] 2. Check if it’s in the form (dy/dx) + yf(x) = g(x), and use integrating factors
[2O] 3. Check if it’s in the form ay’’ + by’ + cy = f(x)
[2O] 4. If in the form d²y/dx² = f(y)
Multiply both sides by 2(dy/dx), and turn the left side into 2y’y’’
[2O] 5. Use p substitution (put p = dy/dx), and then do Step 1 and remove p at the end. If you end up with 3 variables, substitute dp/dx = dp/dy * P
Variance
Average of the squared distances from the mean (measure of how spread out the data is (MSMSM). It’s the square of standard deviation
Normal distribution standardisation
Z = (X - m)/s X = point on distribution Z = equivalent point on Z~N(0, 1)
Binomial approximation
X~B(n, p) => X~N( np, np(1-p) )
Remember to account for rounding
Distribution of means
X~N(m, s^2/n)
n = number of items used to calculate the mean
P(X > a | X > b) =
P(X > a)/P(X > b)
P(X > a) given X > b
Polynomial fraction with quadratic factor in denominator
2x-1)/(x+1)(x^2 + 1
A/(x+1) + (Bx+C)/(x^2 + 1)
Polynomial fraction with repeated factor in denominator
(x-1)/(x+1)(x-2)^2
A/(x+1) + B/(x-2) + C/(x-2)^2
The repeated factor splits into two
