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
ax^2 + bx + c with roots α and β
α + β = -b/a
αβ = c/a
ax^3 + bx^2 + cx + d with roots α, β and γ
α + β + γ = -b/a
αβ + βγ + γα = c/a
αβγ = -d/a
ax^4 + bx^3 + cx^2 + dx + e with roots α, β, γ and δ
α + β + γ + δ = -b/a
αβ + αγ + αδ + βγ + βδ + γδ = c/a
αβγ + αβδ + αγδ + βγδ = -d/a
αβγδ = e/a
Work done =
Change in KE
Fd =
0.5m(v^2 - u^2)
Work done against friction =
μRd (Fr * distance)
Power =
Forwards force * velocity
Driving force/resistance formula
Work in + KE1 + GPE1 + EPE1 = KE2 + GPE2 + EPE2 + Rd
R = non-gravitational resistance force, d = distance
Work in = Power * time
Tension in a stretched string
T = λx/l λ = modulus of elasticity (N) x = extension l = original length
Elastic potential energy
E = (λx^2)/2l
α^3 + β^3 + γ^3 =
(α + β + γ)^3 - 3(α + β + γ)(αβ + αγ + γβ) + 3αβγ
-b/a)^3 - 3(-b/a)(c/a) - 3(d/a
Mean value of f(x) in the interval [a, b]
(1/b-a)*[ ∫f(x)dx from b to a ]
[PPQ] What is meant by extrapolation?
Extrapolation is making predictions outside the original data range
[PPQ] Dangers of extrapolation
Unreliable as the trend may not continue
For what values of x does the binomial expansion of (1 + kx)^n make sense?
|x| < |1/k|
Particular integral guess for sin(kx) or cos(kx)
asin(kx) + bcos(kx)
Particular integral guess for a constant
a
Particular integral guess for kx
ax + b
Particular integral guess for kx^2
ax^2 + bx + c
Particular integral guess for ke^px
ke^px
kxe^px if that doesn’t work
Particular integral guess for ke^x
kxe^x
Integration using inverse chain rule
∫ g’(x)*f’(g(x)) dx = f(g(x))
Matrix for the reflection of a point in one of the axes
Draw a 3x3 matrix of zeroes with a diagonal, left to right line of ones, and then add a negative sign to the one in the row corresponding to the axis you want to reflect in
3D rotation around x axis
(1 0 0)
(0 c -s)
(0 s c)
3D rotation around y axis
(c 0 s)
(0 1 0)
(-s 0 c)
3D rotation around z axis
(c -s 0)
(s c 0)
(0 0 1)
Determining the area scale factor from the transformation matrix
The determinant represents the area scale factor for the transformation
Enlargement by “a” horizontally and “b” vertically
(a 0)
0 b
Reflection in the line y = x
(0 1)
1 0
Reflection in the line y = -x
(0 -1)
-1 0
Matrix multiplication
Dot product the top row of A with the left column of B to get the top-left cell of the result. Matrix multiplication is not commutative.
Determinant of:
a b
(c d)
ad - bc
Determinant of:
a b c
(d e f)
(g h i)
a(ei - fh) - b(df - gi) + c(de - gh)
Multiply each item on the top row by the determinant of the 2x2 matrix you’re left with when you remove that item’s row and column
Inverse of:
a b
(c d)
(d -b)
(-c a)
Divided by determinant
Inverse of a 3x3 matrix
Find determinant
Make a matrix of the minors (determinants of the the 2x2 matrix you’re left with when you remove each item’s row and column)
Flip the signs of that matrix according to the pattern (+-+, -+-, +-+)
Flip this matrix 90 degrees (turn all rows into columns)
Divide by the determinant of original
Angle of deflection when changing velocity from ai + bj to ci + dj
arctan(b/a) - arctan(d/c)
The quartic equation x^4 - 3x^3 + 15x + 1 = 0 has roots a, b, c, and d. Find the equation with the roots 2a+1, 2b+1, 2c+1, and 2d+1
Let w = 2x + 1, so x = (w - 1)/2
Substitute this into the equation and simplify
2x – y + z = 1
3x – 5y + 4z = q
3x + 2y – z = 0
Find q
Choose a random x value (such as 0)
Solve for y and z
Use your 3 values to calculate q
Impulse-momentum formula
I = m(v - u) I = impulse m = mass v = new velocity u = old velocity
What does “maximum velocity” mean?
The acceleration is 0
Coefficient of restitution
e = (b - a)/app app = approach speed
