Mathematics II

What is the formula for the limit of a composite function?

lim (x -> a) f(g(x)) = f(lim (x -> a) g(x))

When is a function continuous?

lim (x -> a-) f(x) = lim (x -> a+) f(x) = f(a)

When does a limit exist?

lim (x -> a-) f(x) = lim (x -> a+) f(x)

What is the formula for the ceiling function?

⌈x⌉ = the smallest integer greater than or equal to x

What is the formula for the floor function?

⌊x⌋ = the greatest integer less than or equal to x

What is the formula for L'Hôpital's Rule?

If lim (x -> c) f(x)/g(x) is indeterminate, then lim (x -> c) f(x)/g(x) = lim (x -> c) f'(x)/g'(x)

If f(x) is differentiable at c, f is [?] at c?

continuous

What is the formula for the Riemann Sum?

Σ f($x_i$*) Δx

What is the process when solving a Riemann Sum problem?

1. Calculate Δx

2. $x_i$ = a + iΔx

3. plug $x_i$ into f($x_i$*)

4. isolate i sum and replace with n

5. Simplify n expression

6. Calculate answer when n -> infinity

Determine an appropriate inequality that bounds A using the Comparison Theorem and write a general Min-Max Bounding Method

Min-Max Bounding Method:

S = ∫[a to b] f(x) dx with f(x) contiunous on [a,b]

1. Identify the integrand i.e. f(x) = (the integrand)

2. Find where f(x) is largest and smallest on the interval

   • Check if f(x) increasing or decreasing

   • OR evaluate f(a) and f(b) if monotonic (always increasing/decreasing)

3. Use the bounding inequality
   min f(x) (b - a) ≤ ∫[a to b] f(x) dx ≤ max f(x) (b - a)

4. Substitute in the values for min/max f(x) - compute the bounds

What is the formula for the average value of a function?

Average = (1/(b-a)) ∫[a to b] f(x) dx

What is the formula for the area between two curves?

Area = ∫[a to b] |top curve - bottom curve| dx

Area = ∫[a to b] |right curve - left curve| dy

How to differentiate partial derivates?

Differentiate as usual but pretend y is a constant for ∂f/∂x and vice versa for ∂f/∂y

$$∂^2f/∂x∂y$$

Read R -> L

ie ∂f/∂y first

then ∂f/∂x

What is Newton-Raphson Method?

$x_{n+1}$ = $x_n$ - $f(x_n)/f'(x_n)$

Critical Points

a point on a graph where the derivative is either zero or undefined.

What is the formula for the instantaneous rate of change?

f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h

f'(x) < 0 = f(x) ?

f'(x) > 0 = f(x) ?

f'(x) < 0 = f(x) decreasing

f'(x) > 0 = f(x) increasing

f'(x) = 0, f''(x) < 0 = ?

f'(x) = 0, f''(x) > 0 = ?

f'(x) = 0, f''(x) < 0 = maximum

f'(x) = 0, f''(x) > 0 = minimum

When is there a:

Vertical asymptote?

Horizontal asymptote?

Vertical asymptote = lim (x -> a+) f(x) or lim (x -> a-) = +/- infinity

Horizontal asymptote = y = L if lim (x -> +/- infinity) f(x) = L

Σr

$$Σr^2$$

$$Σr^3$$

between r =1 to n

Σr = n(n+1)/2

$$Σr^2 = (n(n+1)(2n+1))/6$$

$$Σr^3 = (n^2(n+1)^2)/4$$

What is the formula for the Fundamental Theorem of Calculus?

If F is an antiderivative of f on [a, b], then ∫[a to b] f(x) dx = F(b) - F(a)

What is the formula for the [?] of a function?

1. gradient

2. magnitude?

3. direction?

1. ∇f = (∂f/∂x, ∂f/∂y)

2. |∇f| = √((∂f/∂x)^2 + (∂f/∂y)^2)

3. ∇f/ |∇f|

1. Product rule

2. Quotient rule

3. integration by parts

1. uv' + vu'

2. (vu' - uv')/v^2

3. ∫uv' dx = uv - ∫vu' dx

a.b = ?

alternative notation (2 forms) = ?

a.b = |a| |b| cosθ

alternative = <a,b> or (a^T)b

What is the:

1. Cauchy-Schwarz Inequality?

2. Triangle inequality

1. |<x,y>| $\le$ | x | | y|

2. | x + y | $\le$ | x | + | y |

$v_1^→$ = (2, 1)$^T$

$v_2^{\to}$ = (4, 1)$^T$

How do you show span(v₁, v₂) long and short method?

Long Method:

1. $v^→ =$ $(x, y)^T = α₁v₁^→ + α₂v₂^→$

2. Split into x and y
   x = 2α₁ + 4α₂
   y = α₁ + α₂

3. Solve for α₁ and α₂
   α₁ = 2y - x/2
   α₂ = x/2 - y

Short Method: There are 2 vectors of dimension 2 ∴ span R²

<a, b> = ? for x in [0, 1]

∫(1 to 0) a(x)b(x) dx

What is the formula for the determinant of a [?] matrix?

1. 2x2

2. 3x3

3. 4x4

1. det(A) = ad - bc

2. det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

3. Turn matrix into upper triangular form

det(A) = (product of diagonal entries) x (correction factors)

Correction factors:

- swap 2 rows = x -1

- multiply a row by k = x k

- adding a multiple of one row to another = no change

Linear Map = ?

T(ax + by) = aT(x) + bT(y)

What is the formula for the [?] of a matrix?

1. eigenvalues?

2. eigenvectors?

1. eigenvalues = Solve det(A - λI) = 0

2. eigenvectors = Solve (A - λI)v = 0 where v is the eigenvector

What is the formula for the sum of an

1. arithmetic series?

2. geometric series?

1. S_n = n/2 * (a + l) where l is the last term

2. S_n = a(1 - r^n) / (1 - r) for r ≠ 1

How do you find:

1. PDF (if you have e.g. f(x) = {a if 0 ≤ x < 0.5, a/3 if 0.5 ≤ x ≤ 1})

2. CDF from PDF

3. PDF from CDF

4. Probability

   1. Using PDF

   2. Using CDF

1. ∫[a to b] f(x) dx + ∫[c to d] g(x) dx +… = 1 (solve sum of integrals = 1)

2. F(x) = ∫[-∞ to x] f(t) dt - piecewise i.e. based on bounds (might not be -∞)

3. differentiate - d/dx F(x)

4. Probability:

   1. PDF: ∫[a to b] f(x) dx + ∫[c to d] g(x) dx using bounds from PDF

   2. CDF: P(a < X < b) = F(b) - F(a)

PDF = ?

CDF = ?

PDF = P(a <= X <= b) = ∫(b to a) f(x) dx where f(x) = PDF

CDF = F_x(x) = F(x) = P(X <= x) =

∫(x to - infinity) f(x) dx

How to convert to standard z?

z = (x - μ)/σ

1. E[aX + bY] = ?

2. V[aX + bY] if X and Y independent = ?

3. V[aX + bY + c] = ?

1. E[aX + bY] = aE[X] + bE[Y]

2. V[aX + bY] if X and Y independent =

a^2V[X] + b^2V[Y]

3. V[aX + bY + c] = a^2V[X] + b^2V[Y] + 2ab*cov(X)

What is the formula for the expected value of a random variable?

E[X] = Σ [x P(X=x)] for discrete variables or E[X] = ∫ x f(x) dx for continuous variables

What is the formula for variance?

Var(X) = E[X^2] - (E[X])^2

1. E[X_sm] = ?

2. V[X_sm] = ?

3. What is V_sm and how does it differ from $s^2$?

1. E[X_sm] = E[X]

2. V[X_sm] = V[X]/n

3. V_sm = characterise the variance of a given distribution
   $s²$ = specific sample variance (estimate = /n - 1) - only use if not given true variance
   

std error = ?

σ[X_sm] ∝ 1/√n

therefore

σ[X_sm] = σ[X]/√n

What is the formula for the margin of error?

margin of error = z-score * σ[X_sm]

where z-score is for the specific confidence interval and takes into account the extra left tail

e.g. 95% confidence interval → z = 0.975 as 5% left over / 2 (as two tailed) = 2.5%
∴ z-score = P(Z ≤ z₂) where z₂ = 0.95 + 0.025

Is:

d₉₅ > d₉₀ or d₉₀ > d₉₅ and why?

d₉₅ > d₉₀ as greater confidence requires larger interval

μ = ? in regards to margin of error

μ = μ_sm +/- d where d = margin of error

$s^2$ = ?

$$s^2=\sum\frac{\left(x_{i}-\mu_{sm}\right)^2}{n-1}$$

Normal ↔ Binomial

µ = ?

σ = ?

µ = np

σ = √(np(1-p))

What are the 2 ways to write a confidence interval of 95%

P($X_{sm}-d$ ≤ µ ≤ $X_{sm}+d$ ) = P(µ - d ≤ $X_{sm}$ ≤ µ + d) = 0.95

What is the chain rule (conditional probability)?

P(A $\cap$ B) = P(A | B) x P(B)

What is Bayes’ Theorem?

P(B | A) = $\frac{P(A\vert B)P(B)}{P(A)}$