Notes, summaries, assignments, exams, and exercises for Mathematics for Baccalaureate

Analyzing Algorithm Efficiency

Exponential Recursive Calls

return f(n-1) + f(n-1)

❌ Very bad algorithm
Why: It results in an exponential number of calls and massive recomputation.
Better: Use iterative multiplication or fast exponentiation.

Recursive vs. Iterative Practice

S(n) = S(n-1) + n*n*n

⚠️ Same asymptotic cost as iterative, but worse in practice
Why: Recursion adds significant stack overhead.
Better: Use a loop or a closed-form formula.

Efficient Linear Algorithms

return Q(n-1) + 2*n - 1

✅ Efficient linear algorithm

Multiplications: Θ(n), Additions: Θ(n).
Why: There is no redundant computation.

Optimal Element Inspection

temp = recursive call
if temp <= A[n-1]

✅ Optimal
Why: You must look at every element; therefore, Θ(n) is unavoidable....

AdaBoost: Adaptive Boosting Explained

AdaBoost is one of the simplest and earliest boosting algorithms. The main idea behind AdaBoost is to combine many weak learners (models that do slightly better than random guessing) into one strong learner.

It works by training multiple models one after another. After each model, the algorithm checks which data points were predicted wrong. It then gives more importance (weight) to those wrongly predicted samples so that the next model focuses more on correcting those mistakes.

Each new model tries to fix the errors made by the previous ones. At the end, all models are combined using weighted voting to make the final prediction. This helps improve accuracy and reduces errors.

Key Characteristics of AdaBoost

• Combines

Sample Size Calculation

To find what n must be:

n ≥ (Z_1-α/2 / MOE)² * p(1-p)

Wald Confidence Interval

SE^ = sqrt((p^ * (1 – p)) / n)

Odds Ratio Analysis

Calculated by criss-cross of 2x2 table.

Example: The odds for someone to smoke ≥ 5 cigarettes and have lung cancer is 3.40 times that of someone who smokes < 5 cigarettes daily to have lung cancer.

Fisher's Exact Test

Used when cell frequencies are < 5 in a 2x2 table. Tests for independence.

• H₀: Odds ratio = 1
• H_A: Odds ratio ≠ 1

Chi-Square Test of Independence

• H₀: The two variables are independent.
• H_A: The two variables are not independent.

Logit Model

To find the number of odds used in the model, find all combinations of levels (e.g., a 2x3 model has 6 combinations).

Exp(estimate) = odds for...

Key Principles of Financial Budgeting

Entity Principle

Consider the company as a functional entity.

Leadership Principle

The Financial Planning Manager develops and coordinates the budget.

Authority Principle

The Finance Manager supervises and presents the budget to the Board of Directors (Partners).

Participation Principle

Involve all key participants in the budget's preparation.

Commitment Principle

All managers undertake to follow the budget, notifying any deviation in a timely manner.

Goal Principle

The budget is based on the strategic planning objectives.

Accounting Principle

A budget system should mirror the current accounting system.

Measurement Principle

All estimates must be in currency units.

Predictability Principle

The predictions generated must...

Taylor Polynomials and Derivatives

Start by finding the derivatives of f(x) as needed. In the example question, it wants the degree 3, so we calculate up to the 3rd derivative, f'''(x), and then use the formula.

Solving Differential Equations

The equation ds/dt = sin(t) shows us that it is a pure time differential, meaning taking the integral of ds/dt will get us s(t). Since we know s(0) = 0, we use this to find the constant C. Once found, substitute it into s(3), and you have solved the problem.

Euler's Method for Numerical Solutions

To use Euler's Method, create your table with x moving at the rate given as Δ. Use the initial points and the formula provided. If given a function f(x), find the derivative. If already given a derivative or dy/dx,...

1. Probability

• P(A) = (number of favorable outcomes) / (total number of possible outcomes)
• P(A or B) = P(A) + P(B) - P(A and B)
• P(A|B) = P(A and B) / P(B)
• Expected Value = Σ(x * P(x))

2. Statistical and Practical Significance

• Statistical significance: The likelihood of getting the observed results by chance is very small (usually p < 0.05).
• Practical significance: The results are large enough to be meaningful in real-world applications.

3. Levels of Measurement

• Nominal: Categories with no order.
• Ordinal: Categories with order, but differences are not meaningful.
• Interval: Ordered categories with meaningful differences, no true zero.
• Ratio: Ordered categories with meaningful differences and a true zero.

4. Types of Sampling

• Random: Each member of the

Data Collection Definitions

• Primary data: Data personally collected by you. Examples include traffic counts, pedestrian counts, environmental indexes, questionnaires, or land use surveys.
• Secondary data: Data collected by someone else. This can be found in books, on the internet, or in academic journals.
• Census: A detailed, compulsory survey carried out by nearly all countries every 10 years.

Primary Data Analysis

Advantages

• It is up to date (current).
• You know how the data has been collected.
• Includes data relevant to coursework.
• Only covers your study area.
• Collected in the format that you want.

Disadvantages

• Data may include personal bias.
• Collection can be time-consuming.
• Can be expensive to collect.
• Hard to study temporal changes.
• Some data might be unavailable

Queueing Models

M/M/s: Exponential arrival and service with more than one server. These models are needed in the economic analysis of the system.

M/G/1: Exponential arrival with general service time.

M/D/1: Exponential arrival with deterministic (fixed) service time. Because of lower variability in service time, performance measures are better than M/M/1.

Queues with priority: Performance measures will improve for high-priority customers and worsen for low-priority customers. However, performance measures averaged across all customers (high AND low priority) will remain the same as before.

Distinguish between the number of servers and the number of queues:

1. An M/M/2 queue with two servers but a single queue of waiting customers.
2. Two M/M/1 queues where

Limits of Sequences

We will often need to specify that a number x is close to another number a. However, this doesn’t mean anything unless we specify how close. If ε is a positive number, then the statement “x is within ε of a” does have meaning. It means that the distance between x and a is less than ε – that is, |x − a| ε.

Theorem 2.1.1

If x, y, a, and ε are real numbers with ε > 0, then (a) |y|.

Theorem 2.1.2 (Triangle Inequality)

If a and b are real numbers, then (a) |a + b| ≤ |a| + |b|; and (b) ||a| − |b| ≤ |a − b|. Proof: For part (a), we observe that −|a| ≤ a ≤ |a| and −|b| ≤ b ≤ |b|. If we add these inequalities, the result is
−(|a| + |b|) ≤ a + b ≤ |a| + |b|. By the preceding theorem (with “&...

Process of Contouring and Isolines

Isolines connect points of equal measurements. Contours indicate elevation and changes in elevation, such as slopes and hills. Isotherms connect points of equal temperature and tell us the current state of the atmosphere.

Rules for Drawing Isolines

• Connect points of equal value.
• Never cross one another.
• Never touch.
• The interval for isolines is the same for the whole map.

Contours show us slopes, valleys, and hills.

Topographic Profiles and Vertical Exaggeration

A profile is a topographic profile representing the lay of the land from the side. To draw a profile, a transect (straight line) is drawn between two points to show hills, valleys, and slopes.

Vertical Exaggeration is a mathematical representation used to ensure...