Discrete Mathematics: Functions, Groups, and Graph Theory
Functions and Mapping Properties
Analysis of the Sine Function
a) Let f:R→Rf:\mathbb{R}\rightarrow\mathbb{R}f:R→R be defined by f(x)=sinxf(x)=\sin xf(x)=sinx.
(i) Image Set and Surjectivity
Given f(x)=sinxf(x)=\sin xf(x)=sinx. For every real number xxx, −1≤sinx≤1-1\leq \sin x\leq 1−1≤sinx≤1.
Hence the image set is f(R)={y:y=sinx, x∈R}f(\mathbb{R})=\{y:y=\sin x,\;x\in\mathbb{R}\}f(R)={y:y=sinx,x∈R} ={y:−1≤y≤1}=\{y:-1\leq y\leq1\}={y:−1≤y≤1}. f(R)=[−1,
Read MoreStatistical Analysis and Machine Learning Fundamentals
Measures of Central Tendency and Dispersion
a) Measures of Central Tendency
These metrics identify the central point of a data distribution.
- Mean (Average): The sum of all values divided by the count.
- Example: A retail store counts daily customers over 5 days: [10, 15, 20, 25, 30]. The Mean is (10+15+20+25+30)/5 = 20.
- Median (Middle Value): The middle number in a sorted list. It is highly resistant to outliers.
- Example: If software engineer salaries are [$60k, $65k, $70k, $80k, $250k], the Median is $
Trigonometry, Polar Coordinates, and Vector Analysis
Trigonometric Functions and Properties
Even Functions: cos(-t) = cos(t), sec(-t) = sec(t)
Odd Functions: sin(-t) = -sin(t), tan(-t) = -tan(t), csc(-t) = -csc(t), cot(-t) = -cot(t)
Example: A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
A) P(-15/17, 8/17):
- sin(t) = 8/17
- cos(t) = -15/17
- tan(t) = -8/15
- csc(t) = 17/8
- sec(t) = -17/15
- cot(t) = -15/8
Graphs of Trigonometric Functions
Amplitude, Period, and Phase Shift:
For the equation
Read MoreEssential Algebra Formulas and Graphing Techniques
Algebraic Formulas and Equations
Quadratic and Polynomial Functions
- AOS: x = -b/2a
- Quadratic Formula: x = (-b ± √b² – 4ac) / 2a
- Discriminant (b² – 4ac):
- > 0: Two solutions
- < 0: No real solutions
- = 0: One solution
- Completing the Square: Take 1/2 of b, square it, and add to both sides. Rewrite as (x ± 1/2b)².
Sequences and Financial Math
- Arithmetic Sequence: aₙ = a₁ + d(n – 1)
- Geometric Sequence: aₙ = a₁ * rⁿ⁻¹
- Compound Interest (Annually): P(1 + r)ᵗ
- Compound Interest (Quarterly): P(
Algebra, Geometry, and Financial Mathematics Essentials
Algebra, Linear Graphs & Simultaneous Equations
1. Solving & Simplifying Equations
Algebraic Expressions
- Ex 1 (Simplify): 3x – 5y + 2x + 8y = 5x + 3y
- Ex 2 (Expand & Solve): 3(x-4) = 2x+5 → 3x-12 = 2x+5 → x = 17
- Tips:
- Only add/subtract exact like terms (same variables).
- Expand brackets by multiplying the outside term by everything inside.
- Move the smaller variable first to keep values positive.
Linear Inequalities
- Ex 1 (Solve): -3x + 4 ≤ 10 → -3x ≤ 6 → x ≥ -2
- Tips:
- Flip the sign (<
Applied Mathematics and Statistical Methods Solutions
1. Partial Differential Equations and Probability
(a) Solve the PDE (D² − 3DDʹ)z = 0
The auxiliary equation is obtained by replacing D with m and Dʹ with 1:
m² − 3m = 0 ⇒ m(m − 3) = 0 ⇒ m = 0, 3
Solution: z = f₁(y) + f₂(y + 3x)
(b) Classify the PDE: 4∂²u/∂x² − 4∂²u/∂x∂t + ∂²u/∂t² = 0
Comparing with A·u_xx + B·u_xt + C·u_tt = 0: A = 4, B = −4, C = 1
Discriminant = B² − 4AC = 16 − 16 = 0
Since B² − 4AC = 0, the equation is Parabolic.