Calculus: Understanding Limits and Continuity

Calculus

Limits and Continuity

• Limits describe a function's behavior, $f(x)$, as $x$ approaches a specific value, $c$.
• The notation for a limit is: $\lim_{x \to c} f(x) = L$ which means "$f(x)$ approaches $L$ as $x$ approaches $c$."
• When evaluating limits graphically and numerically, observe the trend of $f(x)$ as $x$ gets closer to $c$.
• If $f(x)$ is well-behaved at $x = c$, then $\lim_{x \to c} f(x) = f(c)$.
• The limit of a constant is that constant: $\lim_{x \to c} k = k$.
• The limit of $x$ as $x$ approaches $c$ is $c$: $\lim_{x \to c} x = c$.
• The limit of $x^n$ as $x$ approaches $c$ is $c^n$: $\lim_{x \to c} x^n = c^n$.
• The limit of the $n$th root of $x$ as $x$ approaches $c$ is the $n$th root of $c$: $\lim_{x \to c} \sqrt[n]{x} = \sqrt[n]{c}$.
• Factoring can simplify limits, for example: $\lim_{x \to 3} \frac{x^2 + x - 12}{x - 3} = \lim_{x \to 3} x + 4 = 7$
• Rationalizing the numerator can help in evaluating limits: $\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x} = \lim_{x \to 0} \frac{1}{\sqrt{x+1} + 1} = \frac{1}{2}$
• $\lim_{x \to c^+} f(x)$ represents the right-handed limit.
• $\lim_{x \to c^-} f(x)$ represents the left-handed limit.
• The limit $\lim_{x \to c} f(x) = L$ exists if and only if $\lim_{x \to c^+} f(x) = L$ and $\lim_{x \to c^-} f(x) = L$.
• $\lim_{x \to c} f(x) = \infty$ means $f(x)$ 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, trigonometric functions, exponential functions, and logarithmic functions are continuous on their domains.

Differentiation

• The slope of a tangent line is: $m = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$
• The derivative of a function is: $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$
• The derivative of a constant is zero: $\frac{d}{dx} [c] = 0$
• The power rule for derivatives: $\frac{d}{dx} [x^n] = nx^{n-1}$
• Constant multiple rule: $\frac{d}{dx} [cf(x)] = cf'(x)$
• Sum rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$
• Difference rule: $\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$
• 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: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$

Applications of Differentiation

• $f(c)$ is the minimum of $f$ on $I$ if $f(c) \le f(x)$ for all $x$ in $I$.
• $f(c)$ is the 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 $[a, b]$ has both a minimum and a maximum on the interval.
• The Mean Value Theorem states that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a $c$ 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 $I$.
• Newton's method formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Integration

• Indefinite integral: $\int f(x) dx = F(x) + C$ where $F'(x) = f(x)$
• Area under a curve: Area $= \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$
• Definite integral: $\int_a^b f(x) dx = \lim_{\Delta x \to 0} \sum_{i=1}^{n} f(x_i) \Delta x$
• Fundamental Theorem of Calculus part 1: If $f$ is continuous on $[a, b]$, then $\frac{d}{dx} [\int_a^x f(t) dt] = f(x)$
• Fundamental Theorem of Calculus part 2: 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 $u = g(x)$
• Integration of $\frac{1}{x}$: $\int \frac{1}{x} dx = ln |x| + C$

Design for Testability (DFT)

Motivation

• Detect manufacturing defects, reduce test costs, improve yield and reliability.
• Testing involves applying test vectors to inputs, observing outputs, and comparing with expected values.
• Internal nodes that are hard to control and observe make circuits difficult to test. Complex circuits require many test vectors.

Observability and Controllability

• Observability: Ease of observing an internal node from the primary outputs.
• Controllability: Ease of setting an internal node to 0 or 1 via primary inputs.
• High controllability and observability makes testing easier.

Fault Models

• Used to model physical defects like shorts, opens, and transistor issues.
• Fault models simplify testing.
• Stuck-At Fault: A popular model assuming any node can be stuck at 0 or 1.
  • Stuck-at-0: Node is always 0.
  • Stuck-at-1: Node is always 1.
  • It is technology independent and represents many common failures.
  • Doesn't cover all failures (e.g., shorts).

Test Generation

• Generates test vectors automatically (ATPG) or manually.
• ATPG goals include high fault coverage and minimum test vectors.
• ATPG Challenges: NP-complete, large circuits require long test generation time.

Design for Testability (DFT) Techniques

Ad-hoc DFT

• Simple techniques such as partitioning, adding test points, avoiding asynchronous or redundant logic.
  • It is easy to implement and has low overhead.
  • It is not systematic with limited fault coverage.

Scan-Based DFT

• Makes all flip-flops controllable and observable.
• It replaces flip-flops with scan flip-flops which have normal and scan modes.
  • In scan mode, flip-flops form a scan chain (shift register). Scan in test patterns and scan out test results.
  • It is systematic and provides high fault coverage.
  • High overhead, performance degradation
• A Scan Flip-Flop contains inputs (Data In DI, Scan In SI, Clock CLK, Mode), outputs (Data Out DO), multiplexer (MUX) and Master-Slave Flip-Flop. The multiplexer selects between DI (normal mode) and SI (scan mode) and feeds it to the Master-Slave Flip-Flop

Built-In Self-Test (BIST)

• Test the circuit using on-chip test circuitry.
• It contains on-chip test pattern generator (TPG) and on-chip output response analyzer (ORA).
  • Enables at-speed testing and reduces test cost.
  • High overhead, may not achieve high fault coverage.

Summary

• DFT improves controllability and observability.
• Scan-based DFT and BIST are popular.
• DFT adds overhead, but reduces test cost and improves yield.

Matrix Calculus Rules

Multiplication with 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 $k$.
• Example: $2 \cdot \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$

Addition

• If $A$ and $B$ are $m \times n$ matrices, then $A + B$ is an $m \times n$ matrix where elements are the sum of corresponding elements in $A$ and $B$.
• Example: $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$

Subtraction

• If $A$ and $B$ are $m \times n$ matrices, then $A - B$ is an $m \times n$ matrix where elements are the difference of corresponding elements in $A$ and $B$.
• Example: $\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} - \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}$

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.
• $(AB){ij} = \sum{k=1}^{n} A_{ik}B_{kj}$
• Example: $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$

Transposition

• If $A$ is an $m \times n$ matrix, $A^T$ is an $n \times m$ matrix where rows in $A$ are columns in $A^T$ and vice versa.
• Example: $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$

Inverse Matrix

• If $A$ is an $n \times n$ matrix and if there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$, where $I$ is an identity matrix, then $A^{-1}$ is the inverse matrix of $A$.

Risco Matrix (Risk Matrix)

Table Summary

• This risk matrix outlines potential risks, their categories, impacts, probabilities, risk levels, recommended actions, responsible parties, and deadlines.

Risk Categories

• The risk categories include: Acquisition, Operational, Security, Personnel, Legal, Quality, Financial, Technology, Communication, Imagem (Image), Meio Ambiente (Environment) and Segurança do Trabalho (Occupational Safety).

Specific Risks:

• Equipment delivery delays (Acquisition, High Risk): Requires close monitoring, supplier communication, and contingency plans.
• Power failure (Operational, Medium Risk): Mitigation via generator installation and electrical inspections.
• The above and similar risks are detailed in the Risco Matrix Table

Level of Risk

• High Risks require the most attention, with medium level risks requiring some action and lower level risks requiring monitoring

Timeframe

• Deadlines (deadlines) range from 1 to 4 weeks (Sem - Semanas).