Math: Probability

Probability calculates the likelihood of an Outcome occuring A Priori.

Probability Space

A Probability Space is defined as the Triple (Ω, E, P):

1. A Probability Function:
   • P mapped to the interval [0,1]
   • P : E → [0,1]
2. A Set of Outcomes Ω.
   • Say, a 6-sided dice is being represented: Ω = {1,2,3,4,5,6}
3. An Event Space E of all Events such that for every ∀(e ∈ E)(e ⊆ Ω).
   • E.g. - e₀ ∈ E and e₀ = {2, 5} (Complex Event where it lands on any of 2, 5.)
   • E.g. - e₁ ∈ E and e₁ = {1} (Lands on 1.)

Events

• Indepdendent
  • Two events A, B are are Indepdendent if the liklihood of A doesn't influence the liklihood of B occurring.
• Dependent
  • Two events A, B are are Dependent if the liklihood of A occurring influences the liklihood of B occurring.
• Mutually Exclusive
  • Two events A, B are are Mutually Exclusive if they can never occur simultaneously.
  • Two events A, B are are Mutually Exclusive if A obtaining entails that B cannot also obtain.

https://en.wikipedia.org/wiki/Probability_space

Probability Rules

Definitions:

• P(A) = 1:
  • A is Certain to occur.
  • Certainty
• P(A) = 0:
  • A Certainly Won't occur.
  • Impossibility
• P(A ∪ B)
  • The likelihood of either A or B occuring.
• P(A ∩ B)
  • The likelihood of both A and B occuring.
• P(A′)
  • Complement of A.
  • The Outcome(s) where A doesn't occur.
• P(A|B)
  • The likelihood of A given B.

General Rules:

1. Addition Rule:
   • (When A, B are Independent Events:) P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
2. Multiplication Rule:
   • (When A, B are Independent Events:) P(A ∩ B) = P(A) × P(B)
3. Complement Rule:
   • P(A′) = 1 - P(A)
   • The likelihood of A′ occuring is 1 minus A.
4. Conditional Probability:
   • P(A|B) = P(A ∩ B) / P(B)
   • Bayes' Theorem
   • Note that: P(A|B) ≠ P(B|A) (not Symmetric).

https://www.geeksforgeeks.org/probability-rules/

https://www.nasa.gov/wp-content/uploads/2023/11/210624-probability-formulas.pdf

Resources and Links

1. https://en.wikipedia.org/wiki/Probability_space
2. https://www.geeksforgeeks.org/probability-rules/
3. https://www.nasa.gov/wp-content/uploads/2023/11/210624-probability-formulas.pdf

Math: Permutations

Definitions

1. Permutations: ₙPₐ = n! / (n - a)!
   • The number of Permutations of n items taken a-many at a time.
   • Consider Heap's Algorithm.
     • Given (the digit String) 012 the generated Permutations are [ '012', '102', '201', '021', '120', '210' ]
   • An arrangement of elements (with or without repetition) where order matters.
     • Order matters: 012 ≠ 102.
     • Counts String Permutations like 0012, 1102, etc. when repetition is allowed.
2. Combinations: ₙCₐ = n! / (a! × (n - a)!)
   • The number of Combinations of n items taken a-many at a time.
   • An arrangement of elements (with or without repetition) where order doesn't matter.
     • Order doesn't matter: 012 = 102.
       • 012, 102 are elements of the same Equivalence Class using the generated Strings (from above).
       • Are considered the same String Combination despite the ordering of the digits 0, 1, 2.
     • Counts String Combinations like 0012, 1102, etc. when repetition is allowed.
3. Variations
   • Where order matters: ₙVₐ = ₙPₐ
   • Where order doesn't matter: ₙVₐ = nᵃ
   • The number of Variations of n items taken a-many at a time.
   • An arrangement of elements (with or without repetition) where order matters.
     • Order matters: 012 ≠ 102.
     • Counts String Variations like 0012, 1102, etc. when repetition is allowed.

Relationships

• Since ₙCₐ = n! / (a! × (n - a)!), it follows that ₙCₐ = ₙPₐ / a! (from substitution).
• Also, ₙPₐ = ₙCₐ × a! (from the above).
• Where order matters: ₙVₐ = ₙPₐ (Variation and Permutation count can be computed in the same manner).

Resources and Links

1. https://www.geeksforgeeks.org/permutations-and-combinations/
2. http://webpages.charlotte.edu/ghetyei/courses/old/S23.3166/Permutations_Combinations_Variations.pdf
3. https://ds.johnpospisil.com/probability/combinatorics/

Math: Geometry

Trigonometric Ratios

Used to determine certain properties of a Right Triange.

Where φ is the Angle, O the Opposite Length, A the Adjacent Length, and H the Hypotenuse:

1. Sine: sin(φ) = O ÷ H
2. Cosine: cos(φ) = A ÷ H
3. Tangent tan(φ) = O ÷ A

A helpful mnemonic device I was taught in High School: SOH, CAH, TOA.

4. Cosecant:
   • csc(φ) = H ÷ O
   • The Inverse of Sine: csc(φ) = 1 ÷ sin(φ)
   • sin(φ) = cos(90° - φ)
5. Secant:
   • sec(φ) = H ÷ A
   • The Inverse of Cosine: sec(φ) = 1 ÷ cos(φ)
6. Cotangent:
   • cot(φ) = A ÷ O
   • The Inverse of Tangent: cot(φ) = 1 ÷ tan(φ)

Radians Conversion

Where r expresses the number of Radians and d the number of Degrees:

1. Degrees to Radians: r = d × (π ÷ 180)
2. Radians to Degrees:
   • d = r / (π ÷ 180)
   • d = (r × 180) ÷ π
3. 90° = π ÷ 2 Radians

Acute Angles

1. Any two Acute Angles of Right Triangle will add up to 90°

Math: Statistics

Statistics calculates the likelihood of an Outcome occuring by observed Frequency, trend, or pattern.

Median, Mean, Mode

Given a sum S of L-many values of some (sorted) Set of events E:

• Median
  • If L is even, take the middle two events and divide by 2.
  • If L is odd, take the middle event and divide by 1.
• Mode
  • Most common or frequently encountered value in the Set.
• Mean
  • = S / L

Standard Normal Distribution

• π
  • Pi
  • π = 3.14159...
  • https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/PI
• e
  • Euler's Constant
  • e = 2.71828...
  • https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/E

https://www.cse.wustl.edu/~garnett/cse515t/fall_2019/files/lecture_notes/5.pdf

const VARIANCE_U = 1
const MEAN_O_2 =  1
const X = 1
const NORMALIZATION_CONSTANT = 1 / Math.sqrt(VARIANCE_U * 2 * Math.PI)
const EXP = -1 * Math.pow(X - VARIANCE_U, 2) / (2 * Math.pow(MEAN_O_2, 2))
const GUASS = 1 / NORMALIZATION_CONSTANT * Math.pow(Math.E, EXP)

The Math 142 course I took delimits course content to the subtopics subheaded below. (We didn't cover calculating Z Scores from scratch.)

• Computation of the Distribution itself is given by this table.

Standard Deviation

Of a Normal Distribution.

Given a set of values {X₀, ..., Xₙ}:

1. Calculate the Mean (μ):
   • In Normal Distribution, the Mean, Mode, and Median are the same.
   • μ = (X₀ + ... + Xₙ)/n
2. Calculate the Standard Deviation (σ) like so:
   • σ = √(((X₀-μ)² + ... + (Xₙ-μ)²) / n - 1)
   • Note that both the numerator and denominator are squared!
3. It's also helpful to remember the following:
   • μ ± σ (the range -σ <= μ <= σ or one Standard Deviation) covers 68%
   • μ ± 2σ (the range -2σ <= μ <= 2σ or two Standard Deviations) covers 95%
   • μ ± 3σ (the range -3σ <= μ <= 3σ or three Standard Deviations) covers 99.7%

Calculate Z Score

General approach:

1. Calculate Z (the number of Standard Deviations) independently.
   • This can be calculated from a specified Percentage (say 10% of the total Area)
   • Or from a Score using the same equation: Z = (x - μ) / σ where x (lowercase) is the specified Score.
2. Then use that value to solve for some X (some Score within the Standard Deviation) Z = (X - μ) / σ or Z * σ + μ = X.
   • Example: find the Score (X) for the top 10% of some Normal Distribution with Mean 140 (μ = 140) and a Standard Deviation of 19 (σ = 19).
   • Z.₁₀ * σ + μ = X or Z.₁₀ * 19 + 140 = X.

Calculating Interquartile Ranges

Given the sorted sequence 51, 52, 54, 55, 63, 70, 90, 100:

• Minimum: 51
• Lower Quartile: 52 - find the Median from index 0 to the Absolute Median (inclusive, exclusive)
• Mean: (51, 52, 54, 55, 63, 70, 90, 100) / 8 = 66.875
• Median: (55 + 63) / 2 = 59
• Upper Quartile: 90 - find the Median from the Absolute Median to the end (exclusive, inclusive)
• Maximum: 100
• Interquartile Range: 90 - 52 = 38

Resources and Links

1. https://www.geeksforgeeks.org/statistics/?ref=shm
2. https://www.geeksforgeeks.org/standard-normal-distribution/#normal-distribution-definition
3. https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/PI
4. https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/E
5. https://www.cse.wustl.edu/~garnett/cse515t/fall_2019/files/lecture_notes/5.pdf
6. https://mathblog.com/statistics/definitions/z-score/percentile-to-z-table/

Code samples:

1. https://leetcode.com/problems/median-of-two-sorted-arrays/description/

Math: Finance Topics

Math and Finance topics often encountered in Programming/Software Engineering.

Quick Reference

Amortization and Interest

1. Non-Amortizing Loans are computed in a number of ways.
   • One way involves Basic Compound Interest divided by Total Months: (P * (1 + r) ^ (12 * t)) / (t * 12)
   • For example, if Monthly Payment Amounts (M) are known the (Compounded) Future Value can be computed:
     • A = (M * ((1 + r / n) ^ (n * t) - 1)) / (r / n) where r is the Interest Rate (Annual), n the number of months, and t the number of years.
     • Note: This equation can be used to compute Future Amortized Values and many other Annuities!
2. Amortization is a standard technique to calculate Monthly or Total Payments (over time) as the Principle (of a Loan) is paid down:
   • Computations vary for Monthly or Total Amortization Payments/Schedules/Table.
   • There are also different kinds of Amortization (and each is computed a bit differently):
     • Full amortization with a fixed rate (Straight-Line)
     • Full amortization with a variable rate
     • Full amortization with deferred interest
     • Partial amortization with a balloon payment (Balloon)
     • Negative amortization
     • Bullet
     • Declining Balance
     • etc.
   • Computed long-hand here
   • Monthly Payments: M = (P * (r / 12)) / (1 - (1 + r / 12) ^ (-12 * t)) where P is the Principal, r is the Interest Rate (Annual), and t the number of years.

Principle and Interest Payments

As found when calculating most kinds of Mortgages:

• $250,000 with 12% down $250,000 - ($250,000 * .12) = $250,000 - $30,000 or $220,000 the Mortgage Principal lent.
• Suppose the Annual Interest Rate is 6%.
• Total (first) Monthly Interest Payment is calculated: $220,000 * (.06 / 12) = $1,100.
• Given a Monthly Payment of say $1,928 then Mortgage Principal is reduced by: $1,928 - $1,100 = $828 (the first month).
• (The process then repeats for subsequent Monthly Payments.)

General method to calculate Simple Interest: I = P * r * t where P is the Principal, r is the Interest Rate (Annual or Monthly), and t the relevant duration of time.

Resources and Links

1. https://corporatefinanceinstitute.com/resources/commercial-lending/non-amortizing-loan/
2. https://www.investopedia.com/terms/a/amortization.asp
3. https://www.accountria.com/blog/what-is-amortization-10-different-types-of-amortized-loans/
4. https://www.public.asu.edu/~kamman/notes/finance/amortized-loan-notes.pdf
5. https://www.investopedia.com/terms/f/future-value-annuity.asp

Math: Financial Programming

Amortization and Interest

1. Non-Amortizing Loans are computed in a number of ways.

   • One way involves Basic Compound Interest divided by Total Months:

     const PRINCIPLE = 1000
     const ANNUAL_INTEREST_RATE = .07
     const YEARS = 5
     
     const TOTAL_LOAN_PAYMENTS = PRINCIPLE * Math.pow(1 + ANNUAL_INTEREST_RATE, YEARS)
     console.log(TOTAL_LOAN_PAYMENTS) // 1402.5517307000005
     
     const MONTHLY_PAYMENTS = TOTAL_LOAN_PAYMENTS / (YEARS * 12)
     console.log(MONTHLY_PAYMENTS) // 23.375862178333342
     

2. Amortization is a standard technique to calculate Monthly or Total Payments (over time) as the Principle (of a Loan) is paid down:

   • A standard approach to calculate Monthly Payments:

     const ANNUAL_INTEREST_RATE = .07
     const PRINCIPLE = 1000
     const MONTHLY_INTEREST_RATE = ANNUAL_INTEREST_RATE / 12
     const NUMER = MONTHLY_INTEREST_RATE * Math.pow(1 + MONTHLY_INTEREST_RATE, 60)
     const DENOM = Math.pow(1 + MONTHLY_INTEREST_RATE, 60) - 1
     const MONTHLY_PAYMENTS = PRINCIPLE * NUMER / DENOM
     console.log(MONTHLY_PAYMENTS) // 19.801198540349468
     

Precision and Rounding

Rounding and Precision effects in JavaScript (and languages with equivalent inbuilt Math functions).

1. Such effects are often encountered since quantities, amounts, etc. don't cleanly divide (and must therefore be Rounded or Truncated).
2. Most currencies and denominations (globally) are represented with Decimal (Floating Point) Precision (e.g. - $1.41).

https://www.public.asu.edu/~kamman/notes/finance/amortized-loan-notes.pdf helpfully described Amorization, Rounding, and Precision.

Verbiage

"To precision" generally means to the number of significant numbers or integers after the decimal point (the Character or sign .):

from decimal import *
getcontext().prec = 6
Decimal(1) / Decimal(7)
Decimal('0.142857')

To Cents

Additionally, it's common to represent "Dollar" amounts as "Cents" to avoid some of these issues.

Advantages:

1. Don't have to worrry about post-decimal point sign Precision.
2. Can help avoid unintentional duplicate Rounding Effects.

Disadvantages:

1. Cents must still be Rounded.

Inbuilt Functions

Note that toPrecision() will unacceptably Truncate in some cases:

let num = 5.123456;

console.log(num.toPrecision()); // '5.123456'
console.log(num.toPrecision(5)); // '5.1235'

And, if the numbers are too big it'll return scientific notation. For example:

const N = 100000 / 24
console.log(N)
console.log(N.toPrecision(3)) // "4.17e+3"
console.log(N.toPrecision(6)) // "4166.67"

Mathematical Approaches and Techniques

JavaScript's Math.round(), Math.ceil(), and Math.floor() methods, combined with a multiplier, offer versatile solutions for computing to a specific Precision.

This is likely the most commonly encountered approach to Round to Precision 2 (from personal work experience in Finance and from widely available blogs like: https://www.zipy.ai/blog/how-to-round-to-at-most-two-decimal-places-in-javascript):

Round to nearest:

let number = 2.12556;
let rounded = Math.round(100 * number) / 100
console.log(rounded) // Output: 2.13

Always rounding up:

let number = 2.12556
let rounded = Math.ceil(number * 100) / 100
console.log(rounded) // Output: 2.13

Always rounding down:

let number = 2.12556
let rounded = Math.floor(number * 100) / 100
console.log(rounded) // Output: 2.12

Resources and Links

1. https://www.zipy.ai/blog/how-to-round-to-at-most-two-decimal-places-in-javascript

Code samples:

1. https://www.codewars.com/kata/59c68ea2aeb2843e18000109

Math: Exponents

Review. Last time I took coursework about these was in highschool.

Quick Reference

Resources and Links

1. https://assets.ea.asu.edu/ulc/MAT117/117_Formula_Sheet_11-23.pdf

Math: Logarithms

Review. Last time I took coursework about these topics was in highschool.

Quick Reference

Graphing a Logarithmic Function

Given an equation of the form:

Find some easy Powers of a:

1. 0th Power:

   $$\displaystyle a^0 = 1$$
   $$\displaystyle g(1) = \log_a1 = 0$$

   Coordinate:

   $$\displaystyle (1, 0)$$

2. 1st Power:

   $$\displaystyle a^1 = a$$
   $$\displaystyle g(a) = \log_aa = 1$$

   Coordinate:

   $$\displaystyle (a, 1)$$

3. Generally

   $$\displaystyle a^y = (x+m)$$
   $$\displaystyle g(x) = k + n\log_a(x+m) = k + ny$$

   Coordinate:

   $$\displaystyle (x-m, k + ny)$$

   Think of the above as x, y transformations of the more basic:

   $$\displaystyle a^y = x$$
   $$\displaystyle g(x) = \log_ax = y$$

   Coordinate:

   $$\displaystyle (x, y)$$

Resources and Links

1. https://assets.ea.asu.edu/ulc/MAT117/117_Formula_Sheet_11-23.pdf
2. https://www.cmor-faculty.rice.edu/~heinken/latex/symbols.pdf

Math: Functions

Review. Last time I took coursework about these was in highschool.

Quick Reference

W.R.T. Parabolic Graphs.

Inverse Function

Given:

To find the Inverse Function of f:

1. Replace f(x) with y:

   $$\displaystyle y = 2x - 4$$

2. Swap y and x:

   $$\displaystyle x = 2y - 4$$

3. Then solve for y:

   $$\displaystyle \frac{x + 4}{2} = y$$
   $$\displaystyle y = \frac{x + 4}{2}$$

4. Replace y with the Inverse Function of f:

   $$\displaystyle f^{-1}(x) = \frac{x + 4}{2}$$

Inverse Functions satisfy the following conditions:

Domains

W.R.T. Composed and Combined Functions Graphs.

The Domains of Combined and Composed Functions are determined and restricted by f and g (the Outer and Inner Functions, respectively).

Suppose:

The intersection of the Domain of f and the Domain of g is (0, ∞) (exclusive, exclusive):

1. Or, x > 0.
2. The Domain of a Combined or Composite Function (of f, g) is (0, ∞) or x > 0.
3. Neither the specific Operation nor the order of f or g matters in these specific cases.

Combined Function Example

The Domains of Combined Functions:

will exclude:

1. 0 since f does.
2. Any n < 0 since g does.
3. So, x > 0 is the resultant Domain of the Combined Function.

Composed Function Example

Similarly for the Composed (Composite) Function:

the following are again excluded from the Domain:

1. 0 since f does.
2. Any n < 0 since g does.
3. So, x > 0 is the resultant Domain of the Composed Function.

Further Comments

Generally, the order of f and g do result in differing Combined or Composite Domains:

1. Note that the Domain of f ◌ g is distinct from seeing the Domain of f, g as "standalone" (like a function in JavaScript) and called successively.

2. Suppose that the above functions f were defined instead like so:

   $$\displaystyle f(x) = \frac{1}{x - 4}$$

3. Although 4 is absent from the Domain of "standalone" Function f it's not necessarily excluded from f ◌ g.

1. The above is a valid Rational Fraction afterall - f ◌ g is in some sense treated as only taking g values (and is no longer seen as "standalone").
2. However, 0 remains excluded (as g is treated as a "standalone" Function in some sense prior to f ◌ g).

https://qcenter.hartford.uconn.edu/wp-content/uploads/sites/629/2022/02/Algebra-Workshops-WS-4.pdf

The easiest way to determine the Domain is as follows:

1. Find the Domain of the Inner Function.
2. Substitute g(x) into each x in f.
3. Multiply the Numerator and the Denominator by a constant (for example by: √x in the example above) to remove the second Denominator like so:

4. Take the Intersection of the result with the Domain of the Inner Function.

Parabolic Graphs

Transform Quadratic Functions into the form:

1. This can then be used with Quadratic Formula (above).
2. Or, to find the Vertex and other properties of the depicted Function (which forms a Parabola):

3. If a > 0, the depicted Parabola opens upwards and the Function will have a minimum.
4. If a < 0, the depicted Parabola opens downwards and the Function will have a maximum.
5. If the full Vertex coordinate is known at (h, k), the Function f(x) will satisfy:

Range

To find the Range of a Parabola given a Function of the form:

1. Transform the f into:

2. Complete the Square of x² - 6x. Take half the coefficient of x and Square it: (-6/2)².
3. x² - 6x + 9 which can be simplified into: (x - 3)².
4. The original equation must be modified by -(a * 9) to keep the equality (5 * 9) - (5 * 9) = 0:
   $$\displaystyle f(x) = 5(x^{2} - 6x + 9) + 50 + -(5 \cdot 9)$$ $$\displaystyle f(x) = 5(x - 3)^{2} + 50 + -(5 \cdot 9)$$ $$\displaystyle f(x) = 5(x - 3)^{2} + 50 - 45$$

The result:

5. since a > 0, the Parabola opens upwards.
6. The Vertex (h, k) is the minimum.
7. The Range is then [k, ∞), [5, ∞).

System of Linear Factors

1. Find the Zeros of the Function.
2. If f(n) = 0 then (x-n) is a Factor of the Function (by the Factor Theorem).

Resources and Links

1. https://assets.ea.asu.edu/ulc/MAT117/117_Formula_Sheet_11-23.pdf
2. https://qcenter.hartford.uconn.edu/wp-content/uploads/sites/629/2022/02/Algebra-Workshops-WS-4.pdf

Math: Limits, Intercepts, and Asymptotes

Limits

1. Right Limit
   • Usually signified with a superscript +
   • Going/Moving from Right to the Limit.
2. Left Limit
   • Usually signified with a superscript -
   • Going/Moving from Left to the Limit.
3. Two-Sided Limit
   • f has a Two-Sided Limit iff it has Right Limit, a Left Limit, and they are equivalent.

Intercepts

1. X-Intercept of a Function:
   • To find the X-Intercept, set y = 0
   • This is properly expressed as a Coordinate Pair.
2. Y-Intercept of a Function:
   • To find the Y-Intercept, set x = 0
   • This is properly expressed as a Coordinate Pair.

Asymptotes

1. A Line the Graph of a Function approaches (as a Limit) that it never touches, intersects, or crosses.
2. Vertical Asymptote
   • An Asymptote (Line) that crosses the X-Axis .
   • A vertical Line approached but never crossed by the Graph of a Function.
   • Example: x = -4
3. Horizontal Asymptote
   • An Asymptote (Line) that crosses the Y-Axis.
   • A horizontal Line approached but never crossed by the Graph of a Function.
   • Example: y = -4

Resources and Links

1. https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(LibreTexts)/03%3A_Graphing_Lines/3.03%3A_Graph_Using_Intercepts

Math: Calculus - Limits

Rules of Thumb:

1. Generally, directly substitute the Limit in and solve (if such a Limit exists).
   • Determine the appropriate Laws to solve.
2. Otherwise, one can inspect the graphed Function.

Continuity

Is a Local property. Holds of a Function f at a point x.

Continuity of f at a point x in its Domain fails if and only if, any one of the following holds:

1. The Left-Sided Limit Does Not Exist (DNE) or is Infinite.

   $$\displaystyle \lim_{x \to a^-} f(x) = \pm \infty $$ $$\displaystyle \lim_{x \to a^-} f(x) = \text{DNE} $$

2. The Right-Sided Limit Does Not Exist (DNE) or is Infinite.

   $$\displaystyle \lim_{x \to a^+} f(x) = \pm \infty $$ $$\displaystyle \lim_{x \to a^+} f(x) = \text{DNE} $$

3. Both the Left-Sided Limit and Right-Sided Limit exist and are Finite but:

   • Don't equal each other.
   • Either of the two don't equal f(x).

Discontinuities

1. Jumps:

• Typically encountered with Piecewise Functions.
• Point oscillation within a Function that leaves a Gap whilst still remaining defined.

2. Holes:

• Where a value isn't in the Domain of f and is Undefined, Does Not Exist.

3. Gaps:

• Where a continuous Interval isn't Defined or produces nothing but Holes.
• An Interval such that every value in the Interval isn't in the Domain of f.

Limits

We often begin with DNE or undefined values as interesting places to find Limits.

• That a Function f is undefined at a doesn't imply it doesn't have a Limit.
• Indeed, a might be the Limit.

A (Two-Sided) Limit exists if and only if:

1. It's Left-Sided Limit exists.

2. It's Right-Sided Limit exists.

3. And:

Importantly, when the Limit goes to ±∞ and/or equals ±∞ we say it approaches a (Horizontal or Vertical) Asymptote of the Function.

Laws

Resources and Links

1. https://math.asu.edu/mat265
2. https://openstax.org/books/calculus-volume-1/pages/2-introduction

Math: Calculus - Derivatives

Rules of Thumb:

1. Determine which notation is being used.
2. Use the appropriate combination of rules and equivalences to eliminate d/dx and Limits from the equation.

Derivatives

As the Instantaneous Rate of Change of a Function f at point c.

1. Forward is on the Interval [c,c+h] computed from the Right. If the Right-Sided Limit exists:

   $$\displaystyle \lim_{h \to 0^+} \frac{f(c+h)-f(c)}{h}$$

2. Backward is on the Interval [c-h,c] computed from the Left. If the Limit exists:

   $$\displaystyle \lim_{h \to 0^-} \frac{f(c+h)-f(c)}{h} = \lim_{s \to 0^+} \frac{f(c)-f(c-s)}{s}$$

3. Given a Function f, defined on an Interval (a,b), its Instantaneous Rate of Change at c in (a,b) is the value of the Forward and Backward Instantaneous Rates of Change at c when they both exist and are the same.

4. Given a Function f defined on an Interval (a,b) and a c in (a,b) the Derivative of f at c (f'(c)) is its Instantaneous Rate of Change at c:

To compute the Instantaneous Rate of Change it's sometimes necessary to divide a Function into a Piecewise Functions and evaluate each part of the Piecewise Function definition with its corresponding Left- or Right-Sided Limit.

It may also be useful to break a Function f into two non-Piecewise parts (to reduce the simplification of terms):

The following are therefore interchangeable ways to rewrite the Derivative:

g' and k' can be solved independently then recombined to solve f'.

Notation

Leibniz Notation

He had many and the exact meaning of this notation varies quite a bit.

Perhaps the most relevant disambiguation here is given by:

This treatment of y mirrors the other approaches defined or described here.

Also:

Trigonometric Functions

Inverse Trigonometric Functions:

1. arctan = tan¯¹
2. arcsin = sin¯¹
3. arccos = cos¯¹
4. arccot = cot¯¹
5. arcsec = sec¯¹
6. arccsc = csc¯¹

Power Rule

Given:

What happens to the 2? Recall that f'(c) where c is a Constant will be 0. Thus, 2 is omitted from the resultant Derivative.

Sum Rule

Where f and g are both Differentiable:

Difference Rule

Where f and g are both Differentiable:

Product Rule

Where f and g are both Differentiable:

Quotient Rule

Where f and g are both Differentiable:

Chain Rule

The above are used to evaluate Derivatives for Combined Functions.

The Chain Rule is the primary tool for evaluating the Derivative of a Composite Functions (Composition of two or more Functions).

Given two Functions f and g where both are Differentiable and such that f is Differentiable at g(x):

The Derivative of the Composite Function is given by:

And, where y is the Function of u and u is the Function of x:

Also note the following relationship:

dy/dx can be obtain through a combination of the Chain and Power Rules.

Inverse Functions

Where f¯¹ is the Inverse Function of f:

The Derivative of the Inverse Function of f is then given by:

Exponential Functions

The following equations hold:

As do:

Logarithmic Differentiation

Example. Given a Function f:

1. Taken the Natural Logarithm of both sides before Differentiation:

2. This allows simplification before any complicated Differentiation attempt:

3. Now find the Derivative:

4. Now solve. Eventually you'll be left with a dangling y:

5. Replace with the original equation above to eliminate y and only leave x:

Resources and Links

1. https://math.asu.edu/mat265
2. https://openstax.org/books/calculus-volume-1/pages/3-3-differentiation-rules
3. https://cdn.blot.im/folder/v-fe12c195/blog_8fcd421159a44b99bdecdde569644473/_spivak/ch10.pdf

Math: Calculus - Advanced Topics

Implicit Differentiation

Implicit Functions allow for Variables (y, x, etc.) to be defined dependently (1 = y + x) as opposed to independently (y = -x + 1)) of each other.

Implicit Differentiation:

• To find the Derivative of a Function y Implicitly in terms of x.
• Allows for tbe Derivatives of Graphs or curves that aren't Functions to be expressed (e.g. - a graphed Circle).

Example

1. Consider the following Implicit Function (to be more precise a Function where y is implicitly defined by the following equation):

   $$\displaystyle x^2 + y^2 = 25$$

2. Differentiate both sides of the equation:

   $$\displaystyle \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)$$

3. Given that the Function's a Combined Function, use the Combined Function Sum Rule (from above):

   $$\displaystyle \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)$$

4. The Derivative of a Constant is 0 so replace the right-hand side:

   $$\displaystyle \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 0$$

5. Use the Power Rule on the left-hand side:

   $$\displaystyle 2x^1 + \frac{d}{dx}(y^2) = 0$$ $$\displaystyle 2x^1 = 2x$$ $$\displaystyle 2x + \frac{d}{dx}(y^2) = 0$$

6. Simplify the second left-hand term:

   $$\displaystyle \frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$$ $$\displaystyle 2x + 2y\frac{dy}{dx} = 0$$

This is through the Chain Rule.

7. Simplify and express the result in dy/dx form:
   $$\displaystyle 2y\frac{dy}{dx} = -2x$$ $$\displaystyle y\frac{dy}{dx} = -x$$ $$\displaystyle \frac{dy}{dx} = -x/y$$

Marginal Cost

Given a Cost Function C such as:

1. The cost of producing the n-th unit is given by the following equation (producing $ cost):

   $$\displaystyle C(X_{n}) - C(X_{n-1})$$

2. The average rate of change of total cost for the n-th unit is given by the following equation (producing: $/unit):

   $$\displaystyle \frac{C(X_{n}) - C(X_{n-1})}{X_{n} - X_{n-1}}$$

3. The instantaneous rate of change of total cost when n units have been produced is given by the following equation (producting $/unit):

   $$\displaystyle \lim_{h \to 0} \frac{C(X_{n}-h) - C(X_{n})}{h}$$

Tangent Line Problem

Given: a Function y = f(a) and point (a, f(a)) within that Function, a Tangent Line can be computed using the following equation:

This can also be thought of as (where m is the Slope):

One can also think of this as finding the Derivative of f then plugging in a to calculate b in the equation:

Given an equation f(x) = 2x, to find the Tangent Line at (2, 4):

1. Compute the Derivative of f (f'(x)).
2. Solve for b: 4 = f'(2) * 2 + b.
3. The resultant Tangent Line: y = f'(2) * x + b.

Normal Line Problem

To find the Normal Line from points (a, f(a)), take the Negative Reciprocal of the Tangent Line's Slope and plug in (a, f(a)).

Negative Reciprocals can be obtain by swapping Numerators and Denominators, then the Sign:

Linear Approximation

This technique involves two steps:

1. To define a Tangent Line at a.
2. Then to use that Tangent Line to approximate some other number (called the Approximate).

Definitions

1. Linear Approximation - calculate a Tangent Line at a that will be used to Approximate other values once defined:

   $$\displaystyle y = f(a) + f'(a)(x-a)$$ $$\displaystyle L(x) = f'(a)(x-a) + f(a)$$

2. Y-Differential - the distance up to the Tangent Line:

   $$\displaystyle dy = f'(x) \cdot dx$$

3. Calculating the Amount of Error - calculating how accurate the Approximation is:

   $$\displaystyle \Delta y = f(a + dx) - f(a)$$

Example

1. Given a Function:

   $$\displaystyle f(x) = (6 + x)^{\frac{3}{2}}$$

2. Approximate that Function, f, at a = 2:

   $$\displaystyle L(x) = f'(a)(x-a)+f(a)$$ $$\displaystyle L(x) = f'(2)(x-2)+f(2)$$

3. Solve for f'(a) (f'(2)) and f(a) (f(2)).

   $$\displaystyle f'(2) = \frac{1}{12}$$ $$\displaystyle f(2) = 2$$

4. Resulting in:

   $$\displaystyle L(x) =\frac{1}{12}(x-2)+2$$

5. Now, Approximate the number 7.95^(1/3) by plugging it into the Tangent Line defined above:

   $$\displaystyle L(x) =\frac{1}{12}(7.95^{\frac{1}{3}}-2)+2$$

Forms and Ratios of Functions

Recollect Limit behavior:

• 0 represents a number arbitrarily close to zero.
• C some other numeric Constant.
• +∞ represents a number that grows indefinitely/arbitrarily.
• -∞ represents a number that grows negatively indefinitely/arbitrarily.

W.R.T. to L'Hôpital's Rules 0, +/-∞ are relevant evaluations. We will compare two Functions f and g.

What about scenarios like the following?

If f, g evaluated to ∞ and 0, how would we approach finding a solution? The prior Rules apply to Constants and other defined operations.

Determinate Forms

Evaluates to a single 0, C or +/-∞ (and is thus directly Differentiable, defined, and solved using the prior Rules).

Determinate Forms (where the left-hand side is the Limit of f, the middle operator the relevant infix operation between f and g, and the right g):

• 0+0
• 0-0
• 0×0
• +/-∞ × +/-∞
• 0/∞,c/∞
• ∞/0,∞/c
• c × ∞, c > 0
• 0^∞
• n^∞, n ≠ 0
• ∞^∞

Indeterminate Forms

Does not evaluate to 0 or +/-∞ and is undefined.

Indeterminate Forms (where the left-hand side is the Limit of f, the middle operator the relevant infix operation between f and g, and the right g):

• ∞-∞
• 0/0
• +/-∞/+/-∞
• 0 × ∞
• 0^0
• ∞^0
• 1^∞

L'Hôpital's Rule

L'Hôpital's Rules apply when two Functions f and g are Diiferentiable, the right-hand Limit exists, they are equal to each other, and they both equal 0, -∞, ∞ (Indeterminate Form).

L'Hôpital's Rule concerns Limit Ratios:

0/0 Rule

Where f and g are Diiferentiable over an Open Interval including or excluding a:

If:

Then:

if the Limit on the right exists.

This is called L'Hôpital's 0/0 Rule.

∞/∞ Rule

Where f and g are Diiferentiable over an Open Interval including or excluding a:

If:

Then:

if the Limit on the right exists. Note that the Sign must match.

This is called L'Hôpital's ∞/∞ Rule.

Extensions

Sometimes, f and g may appear to be unsuitable for L'Hôpital's 0/0 or ∞/∞ Rule (although f, g are Indeterminate Ratio Forms).

However, upon closer inspection f, g might be assuaged into an acceptable Form. So, check before you conclude otherwise.

Helpful to rewrite a Function into Ratio Form (converts f into a Fraction):

Maxima and Minima

1. A Function f has a Local Maximum at c if there exists an Open Interval I containing c such that I is contained in the Domain of f and f(c) ≥ f(x) for all x ∈ I.
2. A Function f has a Local Minimum at c if there exists an Open Interval I containing c such that I is contained in the Domain of f and f(c) ≤ f(x) for all x ∈ I.
3. A function f has a Local Extremum at c if f has a Local Maximum at c or f has a Local Minimum at c.

Critical Number

We say that c is a Critical Number of f if f'(c) = 0 or f'(c) is undefined.

• E.g. - The Zeroes of the First Derivative of f.

Fermat's Theorem

If f has a Local Extremum at c and f is Differentiable at c, then f'(c) = 0.

Mean Value Theorem

1. Let f be a Continuous Function over the Closed Interval [a,b] and Differentiable over the Open Interval (a,b).
2. There then exists at least one Point c where c ∈ (a,b) such that:

Rolle's Theorem

1. Observe that the following is a special case of the Mean Value Theorem.
2. Let f be a Continuous Function over the Closed Interval [a,b] and Differentiable over the Open Interval (a,b) and f(a) = f(b).
3. There then exists at least one Point c where c ∈ (a,b) such that f'(c) = 0.

Concavity

Test for Concavity

1. Let f be a Function that is twice Differentiable over an Interval I.
2. If f′′(x) > 0 for all x ∈ I, then f is Concave Up over I.
3. If f′′(x) < 0 for all x ∈ I, then f is Concave Down over I.

First Derivative Test

Sppose that f is a Continuous Function over an Interval I containing a Critical Point (Critical Number) c. If f is Differentiable over I, except possibly at c, then f(c) satisfies one of the following descriptions:

1. If f′ changes sign from positive when x < c to negative when x > c, then f(c) is a Local Maximum of f.
2. If f′ changes sign from negative when x < c to positive when x > c, then f(c) is a Local Minimum of f.
3. If f' has the same sign for x < c and x > c, then f(c) is neither a Local Maximum nor a Local Minimum of f.

Second Derivative Test

Suppose f′(c) = 0, f′′ is Continuous over an Interval containing c.

1. If f′′(c) > 0, then f has a Local Minimum at c.
2. If f′′(c ) < 0, then f has a Local Maximum at c.
3. If f′′(c) = 0, then the test is inconclusive.

Resources and Links

1. https://math.asu.edu/mat265
2. https://openstax.org/books/calculus-volume-1/pages/3-3-differentiation-rules
3. https://cdn.blot.im/folder/v-fe12c195/blog_8fcd421159a44b99bdecdde569644473/_spivak/ch10.pdf
4. https://www.sfu.ca/math-coursenotes/Math%20157%20Course%20Notes/sec_Hopital.html
5. https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem

Math: Calculus - Anti-Derivatives

Indefinite Integrals (Anti-Derivatives) are often used for approximating calculated Areas.

Anti-Derivatives

For a Function f, find a Function f' such that:

For example:

So, the left-hand side (x^3) is the Anti-Derivatives for 3x^2. And:

Is the Family of Anti-Derivatives for 3x^2.

Worked Example

Notes

Riemann Sums and Approximating Accumulation

Left, Right, Middle Riemann Sums divide an Interval into n Rectangles allowing of the Area under a Curved Line to be Approximately calculated.

The more the better (the more accurate) and that's the key intution is that as this number goes to Infinity, the more accurate the Approximation will be.

• Left - we use the left-most Coordinate to compute each Y Value and Rectangle.
• Right - we use the right-most Coordinate to compute each Y Value and Rectangle.
• Middle - we use the middle Coordinate (Left - Right / 2) to compute each Y Value and Rectangle.

1. Given a partition Δx that divides an Interval [a,b] into x0, ..., xn.
2. We can Approximate the Area under a boundary defined by a Function f (where x0 is some initial value) using the following method:

If some X Value isn't an Ordinal, we can multiply or subtract to the relevant Left, Right, or Middle Coordinate.

Resources and Links

1. https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives

Math: Calculus - Definite Integrals

Definite Integrals are often used to exactly calculate Area (using Geometry).

Definition

if f(x) is a Function defined Interval [a,b], the Definite Integral of f from a to b is given by:

provided the Limit exists. If the Limit exists, the Function f(x) is said to be Integrable on [a,b], or is an Integrable Function.

Equivalences

Properties

Zero Interval:

Reversal of Limits Rule:

Sum Rule:

Difference Rule:

Constant Multiple Rule:

Additive Rule:

Comparison Theorem

If f(x) ≥ 0 for a ≤ x ≤ b, then:

If a ≤ x ≤ b, then:

If m and M are constants where m ≤ f(x) ≤ M for a ≤ x ≤ b, then:

Geometric Solutions

1. When using Geometric approaches, Areas y <= 0 are substracted from Areas y >= 0.
2. When the Function and Interval don't touch or intersect the Y-Axis:
   • Add the (gap) Area below to the Area of the compute Function.

For example:

1/4 the Area of the computed Circle plus the 4 by 2 Region below the Semi-Circle.

Fundamental Theorem of Calculus

In simple cases the Fundamental Theorem of Calculus can be expressed as:

1. The Derivative of the Definite Integral of f(x) is just the original Function with the Upper Limit term (b, above) replacing the Variable of Integration (dt, above).
2. If the Upper Limit is an expression, use the FTC Chain Rule (below).
3. If both the Upper Limit and the Lower Limit are expressions, use Part Two (below).
   • Solve using the FTC Chain Rule for the Upper Limit and the Lower Limit then substract.

Part One

If f(x) is Continuous over an Interval [a,b] and the Function F(x) is defined by:

then the following holds over (a,b):

Part Two

If f is Continuous over an Interval [a,b] and the Function F(x) is any Anti-Derivative of f(x) then:

FTC Chain Rule

If the Upper Limit is itself an expression (say 2+5x), we must apply the FTC Chain Rule:

where u is the expression 2+5x.

Note: one does NOT subtract F(a) from F(b) when using this approach (only F(u) * du/dx).

Worked Example

By the Reversal of Limits Rule:

By Substitution:

By Substitution:

Mean Value Theorem of Derivatives

Resources and Links

1. https://openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral
2. https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus

Math: Calculus - Additional Study Notes

Some Equivalences

By the Chain Rule:

Logarithms and the Exponential Function Limit:

Techniques

Recapitulating and summarizing some prior notes.

Left Limit of a Function With Absolute Values

1. Example: f(x) = |x-5|.
2. The Left Limit of f as it goes to 5 is -(x-5) or -x+5.

Tangent Line Problem

1. Find the Derivative of f. The Slope, m, of the Tangent Line equals f'.
2. Plug the X and Y values into y = mx + b then solve for b to get the Slope Intercept Form.
3. Slope Intercept Form: y = f'(x)x + b.

Related Rates - Volume

1. V = 4/3 * pi * r^3:
   • d/dt of V (dV/dt).
   • dV/dt is given (say 5 ft/s).
   • Solve this down to dr/dt.
2. Then, calculate the actual V (not the Derivative). Say 10 Minutes in or 10 * 60 * 5.
   • Solve for r at V (not the Derivative): 10 * 60 * 5 = 4/3 * pi * r^3.
3. One can then solve dr/dt by substituting r.

Related Rates - Particle

1. Solve in three pieces: dx/dt, dD/dt (D = r), dy/dt.
2. D = sqrt(x^2 + y^2):
   • dD/dt = 1/2(x^2 + y^2)^(-1/2) * (2x * dx/dt + 2y * dy/dt).
3. Some equation will be supplied to define the curve the Particle is moving on: say y = 3*sqrt(3x + 1).
   • Find the Derivative dy/dx.
4. dx/dt will be given - it's the Velocity of the Particle at the instant in question.
5. Plug dx/dt and dy/dt into dD/dt to obtain the answer.

Linear Approximations

1. For some say x = 1, find f(1) and f'(1).
2. Or, L(x) = f'(1)(1-1) + f(1) (L(x) = f'(x)(x-a) + f(a)).

Mean Value Theorem Estimations

1. Given say Continuous Function f.
2. One estimates the Mean Value Theorem for say: f(5) and f(1) like so:
   • Suppose -5 <= f'(x) <= 1.
   • One knows 5-1=4.
   • So, the Lower Bound is (5-1) * -5.
   • The Upper Bound is (5-1) * 1.
3. So, -20 <= f(5)-f(1) <= 4.

Anti-Derivative Fraction Expansion

It's generally better to expand the Fraction like so:

When solving an Anti-Derivative (rather than attempting to reverse the Quotient Rule).

Once one's expanded the Divisor out, remove Quotients with Exponents by rewriting them as Negative Powers like so:

Integral of Euler's Constant to a Power

Since:

Upper Limit Evaluation

Remember that using the Fundamental Theorem of Calculus on an Upper Limit equation doesn't involve subtracting F(a) from F(b) (only F(u) * du/dx).

Upper and Lower Limit Evaluation

Where u is the Upper Limit expression and l the Lower Limit expression: F(u) * du/dx - F(l) * dl/dx.

Displacement and Distance

Displacement - net distance traveled:

Distance - total distance traveled. Use the Absolute Value:

Resources and Links

1. https://openstax.org/books/calculus-volume-1/pages/5-4-integration-formulas-and-the-net-change-theorem?query=average