Notes, summaries, assignments, exams, and exercises for Mathematics for Other courses

Understanding Imaginary Numbers

√-25 has no real solution because (-5)² = 25 and (5)² = 25, never -25.

Definition of the Imaginary Unit

√-1 = i (imaginary unit). Therefore, √-25 = √25 * √-1 = 5i.

In general: √-any = √any * i. Example: √-200 = i√200 = 10i√2.

Powers of i

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

This pattern repeats every four powers. Example: i¹⁵ = i¹² * i³ = -i.

Complex Numbers

Complex numbers are a combination of real and imaginary numbers. Standard form: a + bi (Real part + Imaginary part).

Operations with Complex Numbers

• Absolute Value: |4+3i| = √4²+3² = √16+9 = √25 = 5
• Conjugate: The conjugate of (4+3i) is (4-3i).
• Addition: (4+3i) + (2-i) = 6+2i
• Subtraction: (4+3i) - (2-i) = 2+4i
• Multiplication: (4+3i)*(2-i)

Statistical Attributes and Variables

Attributes are qualitative characters. The different categories of an attribute are called modalities. These conditions should be exhaustive and mutually exclusive.

Variables are quantitative traits. Variables can also be expressed as attributes. The measurement of a variable results in a value, which can be discrete or continuous.

Defining Population and Samples

• Population: A collective of individuals or elements that have certain common characteristics and are the object of observation and statistical study.
• Subpopulation: A subset of elements of the population that has a certain characteristic which differentiates them from others.
• Sample: A subset of elements of the population that aims to represent the entire

Trigonometric Functions

Trig functions (in radians): sin(x), cos(x), tan(x), acos(x), asin(x), atan(x), cosh(x), sinh(x), csc(x), sec(x), and more.

Logarithm and Exponential Functions

• exp(x): e^x
• log(x): natural logarithm (log_e(x))
• log10(x): base-10 logarithm
• log2(x): base-2 logarithm
• sqrt(x): square root

Complex Numbers

• i or j: reserved for √(-1) unless otherwise designated
• abs(x): absolute value
• real(x): real part of a number
• imag(x): imaginary part of a number

Rounding Functions

• fix(x): round towards zero
• floor(x): round towards -∞
• ceil(x): round towards +∞
• round(x): round to the nearest integer

Predefined Constants

• pi = 3.14159...
• i, j = −1 (imaginary unit)
• Inf = infinity

Arrays and Matrices

• x = [1 2 3 4 5 6 7 8 9 10];: row vector with 10 columns
• x = [1;

✅ Unit 3: Perceptron, Backpropagation & Its Variants

⸻

🔹 A. Single Layer Perceptron Network

📌 1. Gradient Optimization Procedures

• Gradient descent is a fundamental technique for optimizing weight updates in neural networks.
• Goal: minimize the error/loss function.
• Update rule:

w_new = w_old − η ∂E/∂w

where:

• η: learning rate
• ∂E/∂w: gradient of the error

📌 2. Hebbian Learning Rule

• Based on the principle: “Neurons that fire together wire together.”
• Weight update rule:

ΔW_ij = η x_i y_j

• x_i: input
• y_j: output
• No error term involved (unsupervised)

📌 3. Perceptron Learning Algorithm

• Used in binary classification (outputs either 0 or 1).
• Algorithm:
  • Initialize weights randomly.
  • For each input:
    • Compute output: y = step(w · x + b)
    • Update weights

CME 315 Midterm 1 Cheat Sheet

MATLAB General Setup and Output

f = @(x) <expression>;

x = linspace(startValue, endValue, numberOfPoints);
x = startValue:spacing:endValue;
y = f(x);

.*    ./    .^

plot(x,y,'LineWidth',1.5)
fplot(f,[a b])
hold on
yline(0,'--')
plot(xPoint,f(xPoint),'ro')
xlabel('Variable (units)')
ylabel('Function (units)')
legend('Curve','Location','best')
grid on

fprintf('Result = %.6f units\n',result)
fprintf('Residual = %.3e\n',residual)
fprintf('Iterations = %d\n',iter)

lim x->0 sinx/x = 1 | H.A.: compare degrees | V.A.: denom = 0 | Continuous if: f(a), lim, equal

DERIVATIVES: (x^n)'=nx^(n-1), (e^x)'=e^x, (a^x)'=a^x ln a, (lnx)'=1/x | (uv)'=u'v+uv', (u/v)'=(u'v-uv')/v^2 |chain: (f(g(x)))'=f'(g(x))g'(x)

TRIG DERIVATIVES: (sin)'=cos, (cos)'=-sin, (tan)'=sec^2 | (sec)'=sec·tan, (csc)'=-csc·cot, (cot)'=-csc^2

CRITICAL POINTS:f'=0 or DNE ⇒ crit pt | f'>0 inc | f'<0 dec | f''>0 conc up | f''<0 conc down | inflec = f'' signchange

INTEGRATION: ∫x^n dx = x^(n+1)/(n+1)+C | ∫e^x dx = e^x+C | ∫a^x dx = a^x/ln a+C | ∫1/x dx = ln|x|+C | ∫sin x dx

= -cos x+C | ∫cos x dx = sin x+C

FTC: Part 1: d/dx ∫_a^x f(t) dt = f(x) | Part 2: ∫_a^b f(x) dx = F(b)-F(a)

AREA & VOLUME: A = ∫_a^b (top - bot)...

Working Capital Needs: Calculation Methodology

1. Determine Working Capital (WC)

   Formula: Current Assets (CA) - Current Liabilities (CL)

   This calculation determines if the company can meet its short-term obligations (liquidity).

2. Determine Accounts Receivable Turnover Period

   Formula: (Accounts Receivable / Annual Net Sales) X 360

   This estimates the period (in days) in which the company recovers its credit sales.

3. Determine Inventory Turnover Period

   Formula: (Inventory / Cost of Goods Sold (COGS)) X 360

   This estimates the time (in days) required for the company to move its inventory within a year.

4. Determine Accounts Payable Turnover Period

   Formula: (Accounts Payable / COGS) X 360

5. Determine Financial Series (Cash Collection Period)

   Formula: Step 2 (Accounts Receivable

Criterion-Oriented Validation According to Crocker and Algina

TMA 10: Crocker and Algina: Study D. Criterion-oriented validation is carried out through representative samples.

Steps for Criterion-Oriented Validation

1. Identify the appropriate criterion.
2. Identify the test and the criterion to be scored.
3. Administer the test.
4. Obtain the measure of the external criterion.
5. Determine the intensity of the relationship between the test scores and the criterion.

Typical Estimation Failure

Typical failure NO ESTIMATE: Allows major scores in the prognosis of statistics to be used through our knowledge of statistical inference.

External Criterion

External criterion: Interest

1. A variable or characteristic of real interest that the test aims to predict or infer.
2. The need