Limits and Continuity

1. Limits and Continuity

• Limits describe the behavior of function values f(x) as x approaches a specific value c.
• The notation $\lim_{x \to c} f(x) = L$ means f(x) approaches L as x approaches c.
• If f(x) is well-behaved at x = c, then $\lim_{x \to c} f(x) = f(c)$.
• Examples of basic limits include: $\lim_{x \to c} k = k$, $\lim_{x \to c} x = c$, $\lim_{x \to c} x^n = c^n$, and $\lim_{x \to c} \sqrt[n]{x} = \sqrt[n]{c}$.
• Techniques for evaluating limits may involve factoring and canceling terms, or rationalizing expressions.
• Right-handed limit: $\lim_{x \to c^+} f(x)$, Left-handed limit: $\lim_{x \to c^-} f(x)$
• $\lim_{x \to c} f(x) = L$ if and only if both one-sided limits equal L.
• $\lim_{x \to c} f(x) = \infty$ indicates f(x) increases without bound as x approaches c.
• A function f(x) is continuous at x = c if f(c) is defined, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$.
• Polynomials, rational functions, trig functions, exponential functions, and logarithmic functions are continuous on their domains.

2. Differentiation

• The slope of the tangent line to a curve at a point is given by $m = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$.
• The derivative of a function is defined as $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$.
• Basic differentiation rules: $\frac{d}{dx} [c] = 0$, $\frac{d}{dx} [x^n] = nx^{n-1}$, $\frac{d}{dx} [cf(x)] = cf'(x)$, $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$, $\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$
• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
• Quotient Rule: $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
• Chain Rule: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$
• Implicit differentiation is a technique used to find the derivative of a function defined implicitly.
• Related rates problems involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known.
• Higher-order derivatives are derivatives of derivatives.

3. Applications of Differentiation

• $f(c)$ is a minimum of $f$ on $I$ if $f(c) \le f(x)$ for all $x$ in $I$.
• $f(c)$ is a maximum of $f$ on $I$ if $f(c) \ge f(x)$ for all $x$ in $I$.
• Extreme values are also called absolute maximum or absolute minimum.
• A continuous function on a closed interval has both a minimum and a maximum on that interval.
• Mean Value Theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$
• If $f'(x) \gt 0$ for all $x$ in $(a, b)$, then $f$ is increasing on $[a, b]$.
• If $f'(x) \lt 0$ for all $x$ in $(a, b)$, then $f$ is decreasing on $[a, b]$.
• If $f'(x) = 0$ for all $x$ in $(a, b)$, then $f$ is constant on $[a, b]$.
• If $f''(x) \gt 0$ for all $x$ in $I$, then the graph of $f$ is concave upward on $I$.
• If $f''(x) \lt 0$ for all $x$ in $I$, then the graph of $f$ is concave downward on $I$.
• Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.
• Newton's Method is an iterative process for approximating the roots of a real-valued function: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

4. Integration

• $\int f(x) dx = F(x) + C$ where $F'(x) = f(x)$
• Area $= \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$
• $\int_a^b f(x) dx = \lim_{\Delta x \to 0} \sum_{i=1}^{n} f(x_i) \Delta x$
• Fundamental Theorem of Calculus: If $f$ is continuous on $[a, b]$, then $\frac{d}{dx} [\int_a^x f(t) dt] = f(x)$.
• Fundamental Theorem of Calculus: If $f$ is continuous on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x) = f(x)$.
• Integration by Substitution: $\int f(g(x))g'(x) dx = \int f(u) du$ where $u = g(x)$.
• Numerical integration techniques approximate the value of a definite integral.
• $\int \frac{1}{x} dx = ln |x| + C$

Design for Testability

• Design for Testability (DFT) is needed to address manufacturing defects, reduce test costs, improve yield, and enhance reliability.
• Internal nodes of a chip are hard to control and observe, making testing complex circuits difficult.
• Testing a chip involves applying test vectors to inputs, observing outputs, and comparing them with expected values.
• Circuits with high controllability and observability are easier to test.
• Observability: Ease of observing an internal node from the primary outputs.
• Controllability: Ease of setting an internal node to 0 or 1.
• Fault models like stuck-at faults simplifies testing by modeling physical defects as nodes being stuck at 0 or 1.
• The stuck-at fault model is technology-independent and represents many common failures, but has limitations.
• Test generation involves creating test vectors, either manually or using Automatic Test Pattern Generation (ATPG), to achieve high fault coverage with minimum vectors.
• ATPG is an NP-complete problem that may require significant time for large circuits.
• Ad-hoc DFT: Simple techniques like partitioning circuits, adding test points, and avoiding asynchronous or redundant logic.
• Scan-Based DFT: Makes all flip-flops controllable and observable.
• Replaces flip-flops with scan flip-flops, which have normal mode and scan mode.
• Consists of Scan flip-flops forming a scan chain
• Scan Flip-Flop Multiplexer selects between Data In (DI, in normal mode) and Scan In (SI, in scan mode), with a master-slave flip-flop controlled by the clock signal (CLK).
• Scan utilizes test patterns that are scanned in, and test results that are scanned out.
• Built-In Self-Test (BIST): Tests the circuit using on-chip test circuitry.
• Uses an on-chip test pattern generator (TPG) and output response analyzer (ORA).
• Benefits: At-speed testing and reduced test cost
• Drawbacks: Higher overhead, may not achieve high fault coverage

Matrisekalkylregler (Matrix Calculation Rules - Translated from Norwegian)

• Multiplication with a Scalar:
  • If $A$ is an $m \times n$ matrix and $k$ is a scalar, then $kA$ is an $m \times n$ matrix where each element is multiplied by $k$.
• Addition:
  • If $A$ and $B$ are $m \times n$ matrices, then $A + B$ is an $m \times n$ matrix where the elements are the sum of the corresponding elements in $A$ and $B$.
• Subtraction:
  • If $A$ and $B$ are $m \times n$ matrices, then $A - B$ is an $m \times n$ matrix where the elements are the difference of the corresponding elements in $A$ and $B$.
• Matrix Multiplication:
  • If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then $AB$ is an $m \times p$ matrix.
  • The element in row $i$ and column $j$ in $AB$ is given by: $(AB){ij} = \sum{k=1}^{n} A_{ik}B_{kj}$
• Transposition:
  • If $A$ is an $m \times n$ matrix, then $A^T$ is an $n \times m$ matrix where the rows in $A$ are the columns in $A^T$ and vice versa.
• Inverse Matrix:
  • If $A$ is an $n \times n$ matrix, and there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix, then $A^{-1}$ is the inverse matrix of $A$.

Matriz de Risco (Risk Matrix - Translated from Portuguese)

• The risk matrix categorizes risks based on impact and probability, providing recommended actions, responsible parties, and deadlines for mitigation.
• Risk Categories: Aquisição (Acquisition), Operacional (Operational), Segurança (Security), Pessoal (Personnel), Legal (Legal), Qualidade (Quality), Financeiro (Financial), Tecnologia (Technology), Imagem (Image), Comunicação (Communication)
• Examples of Risks and Recommended Actions:
  • Atraso na entrega de equipamentos (Delay in equipment delivery): Acompanhamento rigoroso dos prazos de entrega (Rigorous monitoring of delivery deadlines), communication with suppliers, contingency plan.
  • Falha de energia (Power outage): Instalação de gerador de energia (Install energy generator), no-breaks for critical equipment, regular inspection of electrical installations.
  • Ataque cibernético (Cyber attack): Implementar firewalls (Implement firewalls), anti-virus, awareness training.
  • Absenteísmo da equipe (Staff absenteeism): Vaccination campaigns, workplace exercises.
  • Rompimento de contrato com principal fornecedor (Breach of contract with main supplier): Long term contracts, alternative suppliers.
  • Perda de dados confidenciais (Loss of confidential data): Data encryption, restricted access, regular backups.
  • Descumprimento de normas e legislação ambiental (Non-compliance with environmental laws and regulations): Legal advice, environmental training, internal audits.
  • Acidente de trabalho (Workplace accident): Safety programs, training, equipment (EPI).
  • Reclamações de clientes (Customer complaints): Improving processes, efficient communication, customer services training.
  • Variação cambial (Exchange rate variation): Hedging, currency diversification.
  • Obsolescência tecnológica (Technological obsolescence): Tech updates, monitoring tendencies.
  • Dificuldade em reter talentos (Difficulty in retaining talent): career plans, program of recognition, organizational research.
  • Falhas em equipamentos críticos (Critical Equipment Failures): Preventive maintenance, stock of replacement parts.
  • Impacto negativo na imagem da empresa (Negative impacts on company image transparent communication, social- responsibility actions and crises management.
  • Aumento de custos não previstos (Unforeseen cost increases): Contingnecy financial planners monitoring financial indicators and deviation analysis.
  • Falta de engajamento da equipe (Lack of staff engagement): feedback incentive programme, internal communication feedback.
  • Processos judiciais (Legal processes): consultant review policy compliance civil responsibility insurance.
  • Falha na comunicação interna (Failures on communication): regular effective communication.
  • Desastres naturais (Natural desasters): evacuation drills (seguros).
  • Perda de produtividade (productivity loss): efficient time management .
• Legend:
  • Impacto: Alto (High), Médio (Medium), Baixo (Low)
  • Probabilidade: Alta (High), Média (Medium), Baixa (Low)
  • Nível do Risco: Alto (High), Médio (Medium), Baixo (Low)
  • Prazo: Semanas (Weeks)