🌐 Week 1: The Logical Foundations

The Blueprint: Mathematics for Data Science begins with Sets, the fundamental container of data. We then study Relations to understand how data points interact, and Functions to model how inputs transform into outputs. This week provides the formal language needed for all subsequent calculus and statistics.

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1. Core Concept: Number Systems & Set Theory

1.1 Standard Number Sets

• Natural Numbers ($\mathbb{N}$): $\{0, 1, 2, 3, \dots\}$ (Note: In this course, $\mathbb{N}$ usually includes 0).
• Integers ($\mathbb{Z}$): $\{\dots, -2, -1, 0, 1, 2, \dots\}$.
• Rational Numbers ($\mathbb{Q}$): $\{p/q \mid p, q \in \mathbb{Z}, q \neq 0, \text{gcd}(p,q)=1\}$.
• Real Numbers ($\mathbb{R}$): All points on the continuous number line.

1.2 Set Operations

1.3 Principle of Inclusion-Exclusion (PIE)

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2. Core Concept: Binary Relations

2.1 The Three Axioms of Relations

1. Reflexivity: Every element relates to itself. $\forall a \in A, (a, a) \in R$
2. Symmetry: If $a$ relates to $b$, then $b$ relates back to $a$. $(a, b) \in R \implies (b, a) \in R$
3. Transitivity: If $a$ relates to $b$ and $b$ relates to $c$, then $a$ relates to $c$. $(a, b) \in R \text{ and } (b, c) \in R \implies (a, c) \in R$

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3. Core Concept: Functions

3.1 Mapping Properties

• Injective (One-to-One): Distinct inputs produce distinct outputs. $f(x_1) = f(x_2) \implies x_1 = x_2$
• Surjective (Onto): The actual output set (Range) equals the target set (Codomain). $\forall y \in B, \exists x \in A \text{ such that } f(x) = y$
• Bijective: Both Injective and Surjective. Bijective functions are Invertible.

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4. Pattern A — Cardinality with Constraints

Abstract Solution (Strategy)

1. [Identify Sets]: Assign letters to categories (e.g., $F, P, Ph$).
2. [Map Intersections]: List the given values for $n(A \cap B)$, etc.
3. [Solve for Center]: Use the PIE formula to solve for the unknown (often the all-category intersection $n(A \cap B \cap C)$).
4. [Venn Diagram Triage]: Fill the Venn diagram from the inside out.

Procedure

• Step 1: Write down the total count: $n(U)$.
• Step 2: Apply PIE: $n(U) = \sum n(\text{Single}) - \sum n(\text{Double}) + n(\text{Triple})$.
• Step 3: Isolate the variable and calculate.

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5. Pattern B — Relation Property Verification

Abstract Solution (Strategy)

1. [Test Reflexivity]: Can you replace $y$ with $x$ in the rule? Is $xRx$ always true for the entire domain?
2. [Test Symmetry]: If you swap $x$ and $y$, does the truth of the statement change?
3. [Test Transitivity]: Assume $xRy$ and $yRz$ are true. Can you algebraically prove $xRz$?

Procedure

• Step 1: Rule check for $x=x$.
• Step 2: If the rule involves $|x-y|$, symmetry is usually true. If it involves $x > y$, symmetry is false.
• Step 3: Use substitution to check the logical chain for transitivity.

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6. Common Mistakes

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7. Flashcards

<Flashcard front="What defines an Equivalence Relation?" back="A relation that is Reflexive, Symmetric, and Transitive." /> <Flashcard front="Formula for n(A ∪ B ∪ C)?" back="n(A) + n(B) + n(C) - [n(A∩B) + n(B∩C) + n(A∩C)] + n(A∩B∩C)" /> <Flashcard front="Difference between Injective and Surjective?" back="Injective = Unique inputs have unique outputs. Surjective = Every element in the target set is hit." /> <Flashcard front="If f is bijective, what exists?" back="An inverse function f⁻¹." />

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8. Practice Targets

• Calculate cardinality for 3 sets using PIE.
• Prove $f(x) = 2x + 3$ is a bijection from $\mathbb{R} \to \mathbb{R}$.
• Classify the relation $R = \{(x,y) \mid x-y \text{ is even}\}$ on $\mathbb{Z}$.

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9. Connections

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• "Distance & Section Formulas"
• "Slope & Line Representations"
• "Parallel & Perpendicular Logic"
• "Sum of Squared Errors (SSE)"

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🌐 Week 2: The Geometry of the Plane

The Blueprint: Week 2 transitions from abstract logic to the Cartesian Plane. We define the spatial relationships between points (Distance/Section) and the functional relationships between variables (Straight Lines). We conclude with Linear Regression, the mathematical bridge that allows us to find the "best" model for noisy real-world data.

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1. Core Concept: Rectangular Coordinate System

1.1 Distance & Division

• Distance Formula: Derived from the Pythagorean Theorem. $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Section Formula (Internal): Finding a point $C$ that divides segment $AB$ in ratio $m:n$. $C = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$

1.2 The Anatomy of a Line

• Slope ($m$): The measure of "steepness" or the rate of change. $m = \tan(\theta) = \frac{y_2 - y_1}{x_2 - x_1}$
• Intercepts: Where the line hits the axes. $x$-intercept $(a, 0)$, $y$-intercept $(0, b)$.

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2. Representations of a Line

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3. Pattern A — The Perpendicular Transversal

Abstract Solution (Strategy)

1. [Anchor Point]: Identify or calculate point $C(x_c, y_c)$. If $C$ is a division of $AB$, use the Section Formula.
2. [Original Gradient]: Find the slope of the reference line ($m_{ref}$) using $\Delta y / \Delta x$.
3. [Orthogonal Logic]: Calculate the target slope $m_{target} = -1 / m_{ref}$.
4. [Construct]: Substitute point $C$ and $m_{target}$ into the Point-Slope formula.
5. [Standardize]: Convert to $Ax + By + C = 0$ by clearing denominators.

Procedure

• Step 1: Calculate $x_c$ and $y_c$ using ratio $m:n$.
• Step 2: Solve for $m_{ref} = (y_2 - y_1)/(x_2 - x_1)$.
• Step 3: Plug into $y - y_c = m_{target}(x - x_c)$.
• Step 4: Multiply by the common denominator to align with MCQ options.

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4. Pattern B — Area of Polygons (Shoelace)

Abstract Solution (Strategy)

1. [Vertex Ordering]: List vertices in a consistent order (clockwise or counter-clockwise).
2. [Shoelace Algorithm]: $\text{Area} = \frac{1}{2} | (x_1y_2 + x_2y_3 + \dots + x_ny_1) - (y_1x_2 + y_2x_3 + \dots + y_nx_1) |$
3. [Triangle Shortcut]: If it's a triangle with one vertex at the origin $(0,0)$: $\text{Area} = \frac{1}{2} |x_1y_2 - x_2y_1|$.

Procedure

• Step 1: Arrange coordinates in two columns.
• Step 2: Perform "downward" multiplications and sum them.
• Step 3: Perform "upward" multiplications and sum them.
• Step 4: Subtract the two sums, take absolute value, and divide by 2.

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5. Pattern C — Straight Line Fit & SSE

Abstract Solution (Strategy)

1. [Residual Calculation]: The error (residual) for any point is $e_i = y_{observed} - y_{predicted}$.
2. [SSE Definition]: Sum of Squared Errors: $SSE = \sum (e_i)^2$.
3. [Best Fit Axiom]: To minimize SSE for a vertical shift $c$, $c$ must be the average of the residuals produced by the variable part of the function. $c = \frac{1}{n} \sum (y_i - f(x_i))$

Procedure

• Step 1: For each $x_i$, calculate the predicted value $f(x_i)$.
• Step 2: Calculate $d_i = y_i - f(x_i)$.
• Step 3: If comparing two students, calculate $SSE = \sum d_i^2$ for both. The lower value "fits" better.
• Step 4: If solving for $c$, find the mean of $d_i$.

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6. Common Mistakes

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7. Flashcards

<Flashcard front="Condition for two lines to be perpendicular?" back="m1 * m2 = -1 (Slopes are negative reciprocals)." /> <Flashcard front="Formula for distance from point (x0, y0) to line Ax + By + C = 0?" back="d = |Ax0 + By0 + C| / sqrt(A^2 + B^2)" /> <Flashcard front="What is the SSE formula?" back="Sum of (y_observed - y_predicted)²" /> <Flashcard front="What does the intercept form (x/a + y/b = 1) reveal?" back="'a' is the x-intercept and 'b' is the y-intercept." />

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8. Practice Targets

• Find the line equation passing through $(2,3)$ and perpendicular to $y = 4x + 5$.
• Calculate the area of a triangle with vertices $(1,2), (4,6), (7,2)$.
• Given data points $(1,2)$ and $(2,5)$, find SSE for the line $y = 2x + 1$.

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9. Connections

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title: "Mathematics I — Week 3: Quadratic Functions & Parabolic Modeling" course: "Jan 2026 - Mathematics I" week: 3 type: "Comprehensive Notes" status: "Complete" description: "Mastering the anatomy of quadratics: vertices, intercepts, symmetry, slopes, and real-world parabolic modeling (projectile motion and architecture)." tags: ["math1", "quadratic-functions", "parabola", "vertex", "discriminant", "optimization"]

🌐 Week 3: The Power of the Parabola

Topic Overview: Week 3 moves into non-linear relationships. We transition from straight lines to Quadratic Functions. A quadratic is the simplest curve that allows for a "turning point," enabling us to model maximum and minimum values—the foundation of Optimization in Data Science and Physics.

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1. Core Anatomy: $f(x) = ax^2 + bx + c$

1.1 The Coefficient $a$ (The Shape)

• $a > 0$: Parabola opens Upward (U-shape). The vertex is a Global Minimum.
• $a < 0$: Parabola opens Downward (n-shape). The vertex is a Global Maximum.

1.2 The Vertex $(h, k)$

• X-coordinate ($h$): $h = -\frac{b}{2a}$.
• Y-coordinate ($k$): $k = f(h) = f\left(-\frac{b}{2a}\right)$.
• Vertex Form: $f(x) = a(x - h)^2 + k$.

1.3 Axis of Symmetry

• Formula: $x = -\frac{b}{2a}$.
• Every quadratic is perfectly symmetric about this vertical line.

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2. Slopes and Calculus Lite

2.1 The Derivative Rule

• For $f(x) = ax^2 + bx + c$, the slope at any point $x$ is given by: $f'(x) = 2ax + b$
• Axiom: The slope at the Vertex is always Zero.
• Significance: To find the peak of a projectile or the minimum cost of production, set $2ax + b = 0$.

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3. Roots and the Discriminant ($\Delta$)

3.1 The Quadratic Formula

3.2 The Nature of Roots

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4. Pattern A — Modeling Physical Structures

Abstract Solution (Strategy)

1. [Identify Symmetry]: Usually, the Y-axis ($x=0$) is chosen as the axis of symmetry to simplify the equation to $y = ax^2 + k$.
2. [Identify Vertex]: The height of the arch or maximum height of the missile provides $k$.
3. [Apply a Point]: Use a known width/height coordinate to solve for the scaling factor $a$.
4. [Verification]: Projectiles must have $a < 0$ (opening down).

Procedure

• Step 1: Write the general vertex form: $y = a(x - 0)^2 + \text{Max Height}$.
• Step 2: If the width is $W$ at height $H$, plug in $(W/2, H)$ because the width is split across the Y-axis.
• Step 3: Solve for $a$ and write the final equation.

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5. Pattern B — Correcting "Mistaken" Roots

Abstract Solution (Strategy)

1. [Vieta's Formulas]: Sum of roots $= -b/a$; Product of roots $= c/a$. (Assume $a=1$).
2. [Isolate Correct Data]:
   • Mistake in $c \rightarrow$ Sum of roots $(-b)$ is CORRECT.
   • Mistake in $b \rightarrow$ Product of roots $(c)$ is CORRECT.
3. [Reconstruct]: Combine the correct sum and correct product to find the real equation.

Procedure

• Step 1: Find correct Sum from the person who got the constant term ($c$) wrong.
• Step 2: Find correct Product from the person who got the $x$-coefficient ($b$) wrong.
• Step 3: Form equation: $x^2 - (\text{Correct Sum})x + (\text{Correct Product}) = 0$.
• Step 4: Solve for the true roots.

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6. Common Mistakes

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7. Flashcards

<Flashcard front="What is the x-coordinate of the vertex of ax^2 + bx + c?" back="-b / 2a" /> <Flashcard front="What is the value of the slope at the vertex?" back="Zero." /> <Flashcard front="If Δ < 0, does the parabola intersect the X-axis?" back="No. There are no real roots." /> <Flashcard front="How do you find the minimum/maximum value of a quadratic?" back="Calculate the Y-coordinate of the vertex: f(-b/2a)." />

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📝 Week 3: Graded Assignment Synthesis

Question 4: Projectile Motion

• Strategy: Peak height is at the midpoint of the horizontal travel ($t=1$).
• Logic: $h(t) = a(t-1)^2 + 5$. At $t=0, h=3 \rightarrow 3 = a(-1)^2 + 5 \rightarrow a = -2$.

Question 9: Rational to Quadratic (Wait time)

• Logic: $\frac{1200}{V - 200} - \frac{1200}{V} = 0.5$.
• Solving this yields the original speed $V = 800$ km/h.
• Ananya takes $1200/600 = 2$ hrs. Madhuri takes $1800/720 = 2.5$ hrs.
• Conclusion: Ananya arrives first; waiting time is 30 mins.

Question 12: Mistaken Roots

• Equation: $x^2 - 7x + 6 = 0 \rightarrow (x-1)(x-6) = 0$.
• Correct Roots: 1 and 6.

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title: "Mathematics I — Week 4: Algebra and Geometry of Polynomials" course: "Jan 2026 - Mathematics I" week: 4 type: "Comprehensive Notes" status: "Complete" description: "Mastering higher-degree polynomials: division algorithms, root multiplicity, end behavior, and the relationship between degree and turning points." tags: ["math1", "polynomials", "multiplicity", "end-behavior", "turning-points", "zeros"]

🌐 Week 4: The Landscape of Polynomials

Topic Overview: Week 4 extends our knowledge from quadratics (degree 2) to General Polynomials of degree $n$. We explore the "Topography" of these functions—where they cross the axis, how many times they turn, and where they head at the extremes of infinity.

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1. Core Concept: Polynomial Anatomy

• Degree ($n$): The highest power of $x$. It dictates the maximum number of zeros and turns.
• Leading Coefficient ($a_n$): Determines the "End Behavior."

1.1 The Factor & Remainder Theorems

• Remainder Theorem: If $P(x)$ is divided by $(x - c)$, the remainder is $P(c)$.
• Factor Theorem: $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$.

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2. The Geometry of Zeros (Intersects)

2.1 Multiplicity

• Odd Multiplicity (1, 3, 5...): The graph crosses the X-axis at $x=c$.
• Even Multiplicity (2, 4, 6...): The graph touches the X-axis and bounces back (tangent) at $x=c$.

2.2 Turning Points

• Rule: A polynomial of degree $n$ has at most $n - 1$ turning points.
• Local Extrema: These are the peaks (maxima) and valleys (minima).

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3. End Behavior ($x \to \pm\infty$)

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4. Pattern A — Creating Polynomials from Graphs

Abstract Solution (Strategy)

1. [Identify Zeros]: Locate where the graph hits the X-axis. These become factors $(x - r)$.
2. [Determine Multiplicity]:
   • Crosses straight $\rightarrow$ power 1.
   • Bounces $\rightarrow$ power 2.
   • Flattens and crosses $\rightarrow$ power 3.
3. [Sign check]: Check the right end. If it goes down, the leading coefficient $a$ must be negative.
4. [Point Substitution]: Plug in a known point (like the Y-intercept) to find the exact value of the leading coefficient $a$.

Procedure

• Step 1: Write $P(x) = a(x - r_1)^{m1}(x - r_2)^{m2}\dots$
• Step 2: Use the degree or end behavior to guess the sign of $a$.
• Step 3: Expand or simplify to match the provided options.

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5. Pattern B — Turning Point & Extrema Audit

Abstract Solution (Strategy)

1. [Visual Count]: A turning point is any location where the function changes from increasing to decreasing (or vice versa).
2. [Derivative Logic]: These occur where the slope is zero ($f'(x) = 0$).
3. [Classification]:
   • Bottom of a valley $\rightarrow$ Local Minimum.
   • Top of a hill $\rightarrow$ Local Maximum.

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6. Common Mistakes

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7. Flashcards

<Flashcard front="How many turning points can a degree 5 polynomial have?" back="At most 4." /> <Flashcard front="If a graph 'bounces' at x=2, what is the multiplicity of that root?" back="Even (usually 2)." /> <Flashcard front="What is the end behavior of an odd degree polynomial with LC < 0?" back="Left end up, Right end down." /> <Flashcard front="State the Remainder Theorem." back="The remainder of P(x) / (x-c) is exactly P(c)." />

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📝 Week 4: Graded Assignment Synthesis

Question 2: Root Identification

• Logic: Use the factor theorem with given options $\{3, 2, -3, -2\}$.
• $g(2) = 8 + 20 - 16 - 12 = 0 \rightarrow (x-2)$ is a factor.
• Dividing $g(x)$ by $(x-2)$ yields $x^2+7x+6 = (x+6)(x+1)$.
• Conclusion: $g(x)$ cuts X-axis at $2, -6, -1$.

Question 13: SSE Optimization (Repeat Pattern)

• Logic: As established in W2, $c$ is the mean of residuals.
• For the data in Table 1, $c = 3.4$.

Question 14: Geometry Word Problem

• Logic: $V = L \times W \times H \rightarrow \text{Area of base} = V(x) / L(x)$.
• Perform polynomial division: $(4x^3 + 16x^2 + 17x + 5) \div (x+1) = 4x^2 + 12x + 5$.
• Factor $4x^2 + 12x + 5$: Roots are $-0.5$ and $-2.5$. Factors are $(2x+1)(2x+5)$.
• Since $H > W$, the factors of the base $(2x^2+3x+1)$ are derived from splitting the area. (Check specific options).

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title: "Mathematics I — Week 5: Advanced Function Dynamics" course: "Jan 2026 - Mathematics I" week: 5 type: "Comprehensive Notes" status: "Complete" description: "Exploring the interaction of functions through composition, the symmetry of inverses, and the growth mechanics of exponential functions." tags: ["math1", "composition", "inverse-functions", "exponential-functions", "bijectivity", "domain-range"]

🌐 Week 5: Function Interactions & Growth

Topic Overview: Week 5 acts as the "Engine Room" for understanding how mathematical models interact. We transition from looking at individual polynomials to Composite Functions (functions within functions), Inverse Functions (reversing operations), and Exponential Functions (modeling rapid growth and decay).

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1. Core Concept: Function Tests & Symmetry

1.1 The Vertical Line Test (VLT)

• Purpose: To determine if a graph represents a Function.
• Rule: If any vertical line intersects the graph at more than one point, the relationship is NOT a function (one input has multiple outputs).

1.2 The Horizontal Line Test (HLT)

• Purpose: To determine if a function is Injective (One-to-One).
• Rule: If any horizontal line intersects the graph at more than one point, the function is NOT injective.
• Nuance: Power functions $y = x^n$ behave differently based on $n$:
  • $n$ is Even: Fails HLT (e.g., $x^2$). Not injective.
  • $n$ is Odd: Passes HLT (e.g., $x^3$). Injective.

1.3 Even and Odd Functions

• Even Functions: Symmetric about the Y-axis. $f(-x) = f(x)$.
• Odd Functions: Symmetric about the Origin. $f(-x) = -f(x)$.

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2. Composition and Inverses

2.1 Composite Functions $(g \circ f)(x)$

• Formula: $(g \circ f)(x) = g(f(x))$
• Domain Constraint: The domain of $g \circ f$ is the set of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$.

2.2 Inverse Functions $f^{-1}(x)$

• Existence Condition: $f(x)$ must be Bijective (Injective and Surjective).
• Geometry: The graph of $f^{-1}(x)$ is the reflection of $f(x)$ across the line $y = x$.
• Properties:
  • $\text{Domain}(f) = \text{Range}(f^{-1})$
  • $\text{Range}(f) = \text{Domain}(f^{-1})$

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3. Exponential Functions $y = a^x$

• Growth: $a > 1$ (e.g., population growth).
• Decay: $0 < a < 1$ (e.g., radioactive decay).
• The Power Rules:
  1. $a^m \cdot a^n = a^{m+n}$
  2. $\frac{a^m}{a^n} = a^{m-n}$
  3. $(a^m)^n = a^{mn}$

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4. Pattern A — Composition in Financial Modeling

Abstract Solution (Strategy)

1. [Identify the Functions]: Define each offer as a mathematical function of the price $x$.
   • Offer 1 ($D_1$): Fixed price or flat discount.
   • Offer 2 ($D_2$): Percentage discount.
2. [Order of Operations]: Realize that $(D_1 \circ D_2)(x)$ is usually different from $(D_2 \circ D_1)(x)$.
3. [Minimization Rule]: To pay the least, apply the percentage discount to the highest possible value (usually applying the flat discount last) OR check the specific thresholds.

Procedure

• Step 1: Write $D_1(x) = \dots$ and $D_2(x) = \dots$
• Step 2: Calculate $D_1(D_2(\text{Total}))$ and $D_2(D_1(\text{Total}))$.
• Step 3: Compare outputs.

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5. Pattern B — Domain of Composites

Abstract Solution (Strategy)

1. [Constraint 1]: Solve for the domain of the inner function $f(x)$.
2. [Constraint 2]: Ensure the output of $f(x)$ fits inside the allowable domain of the outer function $g(x)$.
3. [Intersection]: The final domain is where both conditions are satisfied simultaneously.

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6. Flashcards

<Flashcard front="Under what condition is f(x) invertible?" back="It must be bijective (one-to-one and onto)." /> <Flashcard front="What is the geometric relationship between f and its inverse?" back="They are reflections of each other across the line y = x." /> <Flashcard front="How do you find the domain of g(f(x))?" back="Find x in Domain(f) such that f(x) is in Domain(g)." /> <Flashcard front="Is y = x^4 injective?" back="No, because it is an even power (fails horizontal line test)." />

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📝 Week 5: Graded Assignment Atlas

Question 2: Composite Domain

• Abstract Strategy:
  1. Inner function $f(x)$ requires $1-x \geq 0 \implies x \leq 1$.
  2. Outer function $g(x)$ is a polynomial, defined for all reals.
  3. The only restriction comes from $f(x)$.
• Procedure:
  • Domain of $f = (-\infty, 1]$.
  • Since $g$ accepts any real input produced by $f$, the composite domain is $(-\infty, 1]$.
  • Result: $\mathbb{R} \setminus (1, \infty)$.

Question 6: Simultaneous Offers

• Abstract Strategy: Identify that $D_1(x)$ is a "ceiling" function for price, and $D_2(x) = 0.7x$.
• Procedure:
  • For a total of ₹17,999:
  • Path A ($D_2$ then $D_1$): $0.7 \times 17,999 = 12,599.3$. Since this is $< 14,999$, offer $D_1$ no longer applies.
  • Path B ($D_1$ then $D_2$): $D_1$ caps the price at 9,999. Then $0.7 \times 9,999 \approx 6,999$.
  • Conclusion: Path B (Apply $D_1$ then $D_2$) results in the minimum payment.

Question 10: Power Algebra

• Abstract Strategy: Convert all bases to common primes (2 and 3) to use power rules.
• Procedure:
  • $4m = n$.
  • Term 1: $\frac{16^m}{2^n} = \frac{(2^4)^m}{2^n} = \frac{2^{4m}}{2^n} = \frac{2^n}{2^n} = 1$.
  • Term 2: $\frac{27^n}{9^{6m}} = \frac{(3^3)^n}{(3^2)^{6m}} = \frac{3^{3n}}{3^{12m}}$.
  • Since $n=4m \implies 3n=12m$, the term becomes $\frac{3^{12m}}{3^{12m}} = 1$.
  • Total: $1 + 1 = 2$.

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===--- title: "Mathematics I — Week 6: Logarithmic Functions and Equations" course: "Jan 2026 - Mathematics I" week: 6 type: "Comprehensive Notes" status: "Complete" description: "Understanding the inverse of growth: logarithmic properties, solving exponential-logarithmic equations, and modeling real-world logarithmic decay/rise." tags: ["math1", "logarithms", "exponentials", "inverse-functions", "domain-constraints", "natural-log"]

🌐 Week 6: The Logarithmic Lens

Topic Overview: Week 6 introduces the Logarithm, the mathematical "undo" button for exponential growth. We move from rapid multiplication to the analysis of scales. Logarithms are essential for handling vast ranges of data (like the Richter scale or pH levels) and solving equations where the unknown is trapped in an exponent.

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1. Core Concept: The Logarithmic Identity

• The Logarithm is an exponent. It asks: "To what power must we raise the base $a$ to get the value $y$?"

1.1 The Golden Rules of Logs

1. Product Rule: $\log_a(xy) = \log_a(x) + \log_a(y)$
2. Quotient Rule: $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$
3. Power Rule: $\log_a(x^k) = k \cdot \log_a(x)$
4. Change of Base: $\log_a(b) = \frac{\ln b}{\ln a} = \frac{\log_{10} b}{\log_{10} a}$

1.2 Domain and Range Constraints

• Function: $f(x) = \log_a(x)$
• Domain: $(0, \infty)$. You cannot take the log of zero or a negative number.
• Range: $(-\infty, \infty)$.
• The Natural Log ($\ln$): Logarithm with base $e \approx 2.718$.

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2. Pattern A — Solving "Disguised" Quadratics

Abstract Solution (Strategy)

1. [Common Ratio Check]: Identify if the bases share a geometric relationship.
2. [Normalization]: Divide the entire equation by the smallest exponential term to simplify the ratios.
3. [Substitution]: Let $u = (\text{ratio})^x$. This usually transforms the equation into a quadratic $Au^2 + Bu + C = 0$.
4. [Logarithmic Extraction]: Solve for $u$, then use $\ln$ to solve for $x$.

Procedure

• Step 1: Divide by the smallest base.
• Step 2: Replace terms with variable $u$.
• Step 3: Solve the quadratic.
• Step 4: Reject negative $u$ values (exponentials are always positive).
• Step 5: Apply $\ln$ to find $x$.

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3. Pattern B — Composition with Logarithms

Abstract Solution (Strategy)

1. [Inner-to-Outer Processing]: Start with the innermost input.
2. [Domain Layering]: Ensure that at each step, the input to a log is positive ($> 0$).
3. [Optimization]: To find the maximum of $f(g(x))$, find the maximum of $g(x)$ first, then feed that result into $f$.

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📝 Week 6: Graded Assignment Atlas

Question 1: Disguised Exponential Quadratic

Solution

• Step 1: $\frac{18^x}{8^x} - \frac{12^x}{8^x} - 2 = 0 \implies \left(\frac{9}{4}\right)^x - \left(\frac{3}{2}\right)^x - 2 = 0$.
• Step 2: Let $u = (1.5)^x$. Since $\frac{9}{4} = (1.5)^2$, the equation is $u^2 - u - 2 = 0$.
• Step 3: Factorize: $(u-2)(u+1) = 0 \implies u = 2$ (Reject $u = -1$ because $1.5^x > 0$).
• Step 4: $(1.5)^x = 2 \implies \left(\frac{3}{2}\right)^x = 2$.
• Step 5: Take $\ln$ on both sides: $x \ln(3/2) = \ln 2 \implies x(\ln 3 - \ln 2) = \ln 2$.
• Result: $x = \frac{\ln 2}{\ln 3 - \ln 2}$.

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Question 4 & 5: Inversion Logic

Solution

• Step 1: Set $x = \frac{2e^y}{3e^y+1}$.
• Step 2: $x(3e^y + 1) = 2e^y \implies 3xe^y + x = 2e^y$.
• Step 3: Group $e^y$ terms: $x = e^y(2 - 3x)$.
• Step 4: $e^y = \frac{x}{2 - 3x} \implies y = \ln\left(\frac{x}{2 - 3x}\right)$.
• Result: $f^{-1}(x) = \ln\left(\frac{x}{2 - 3x}\right)$.

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Question 11 & 12: Nested Optimization

Solution

• Step 1 (Max of g): $g(x)$ is a downward parabola. Vertex at $x = -b/2a = -4/(-2) = 2$.
• Step 2: $g(2) = -(2)^2 + 4(2) + 77 = -4 + 8 + 77 = 81$.
• Step 3 (Feed into f): $h_{max} = f(81) = \log_2(\log_2(\log_3 81))$.
• Step 4: $\log_3 81 = 4$ (since $3^4 = 81$).
• Step 5: $\log_2 4 = 2$ (since $2^2 = 4$).
• Step 6: $\log_2 2 = 1$.
• Result: Max value is 1.

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Question 13: Logarithmic Equality

Solution

• Step 1: $\ln(7 \cdot (2 - 4x^2)) = \ln(14)$.
• Step 2: $14 - 28x^2 = 14 \implies -28x^2 = 0 \implies x = 0$.
• Step 3 (Check Domain): If $x = 0$, the original argument $(2 - 4x^2)$ becomes $2$. Since $2 > 0$, the solution is valid.
• Result: 1 solution.

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4. Common Mistakes

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5. Flashcards

<Flashcard front="What is log_b(1) for any valid base b?" back="0 (because b^0 = 1)." /> <Flashcard front="How do you move an exponent out of a logarithm?" back="log(x^k) = k * log(x)." /> <Flashcard front="Relationship between log_a(b) and log_b(a)?" back="They are reciprocals: log_a(b) = 1 / log_b(a)." /> <Flashcard front="Domain of f(x) = ln(x - 5)?" back="x > 5 (or interval (5, ∞))." />

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title: "Mathematics I — Week 7: Sequences, Limits, and Continuity" course: "Jan 2026 - Mathematics I" week: 7 type: "Comprehensive Notes" status: "Complete" description: "Transitioning from static algebra to dynamic calculus: limits of sequences, function continuity, and the analysis of asymptotic behavior." tags: ["math1", "limits", "sequences", "continuity", "asymptotes", "piecewise-functions"]

🌐 Week 7: The Logic of Convergence

Topic Overview: Week 7 is the gateway to Calculus. We stop asking "what is the value at $x$?" and start asking "what happens as we get closer to $x$?" We master the behavior of Sequences as they head to infinity and the Continuity of functions—ensuring there are no "gaps" or "jumps" in our mathematical models.

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1. Core Concept: Limits of Sequences ($a_n$)

1.1 The "Degree" Rule for Rational Sequences

1. $\text{deg}(P) < \text{deg}(Q)$: The limit is always 0.
2. $\text{deg}(P) = \text{deg}(Q)$: The limit is the ratio of leading coefficients.
3. $\text{deg}(P) > \text{deg}(Q)$: The limit is $\pm\infty$ (Divergent).

1.2 Useful Standard Limits

• $\lim_{n \to \infty} (1 + \frac{k}{n})^n = e^k$
• $\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} = e$
• $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
• $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

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2. Core Concept: Continuity of Functions

1. $f(a)$ is defined (The point exists).
2. $\lim_{x \to a} f(x)$ exists (Left-hand limit = Right-hand limit).
3. $\lim_{x \to a} f(x) = f(a)$ (The limit matches the point value).

2.1 Types of Discontinuity

• Removable: Limit exists, but $f(a)$ is different or undefined (a "hole").
• Jump: LHL and RHL are finite but not equal. Common in Floor functions $\lfloor x \rfloor$ and piecewise definitions.
• Infinite: The function heads to $\pm\infty$ at the point (Vertical Asymptote).

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3. Pattern A — Subtracting Large Rational Terms

Abstract Solution (Strategy)

1. [Common Denominator]: Combine the terms into a single fraction.
2. [Highest Power Extraction]: Identify the highest power of $n$ in the combined numerator and denominator.
3. [Coefficient Analysis]: Apply the degree rule. If the degrees are equal, the coefficients determine the result.

Procedure

• Step 1: Find the LCD (Least Common Denominator).
• Step 2: Expand the numerator carefully.
• Step 3: Simplify the numerator. Usually, the highest power terms ($n^2$ or $n^3$) will cancel out, leaving a lower degree that matches the denominator.
• Step 4: Divide numerator and denominator by the highest power of $n$ present in the denominator.

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4. Pattern B — Limits of Floor Functions ($\lfloor x \rfloor$)

Abstract Solution (Strategy)

1. [Right-Hand Limit ($a^+$)]: Approaching from slightly above. $\lfloor a + \epsilon \rfloor = a$.
2. [Left-Hand Limit ($a^-$)]: Approaching from slightly below. $\lfloor a - \epsilon \rfloor = a - 1$.
3. [Logic]: Floor functions always "jump" at integer values.

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📝 Week 7: Graded Assignment Atlas

Question 3: Rational Sequence Limit

Solution

• Step 1 (Simplify first term): $\frac{12n^2}{3(n+9)} = \frac{4n^2}{n+9}$.
• Step 2 (Combine): $\frac{4n^2}{n+9} - \frac{4n^2+23}{n+5} = \frac{4n^2(n+5) - (4n^2+23)(n+9)}{(n+9)(n+5)}$.
• Step 3 (Expand Numerator): $(4n^3 + 20n^2) - (4n^3 + 36n^2 + 23n + 207) = -16n^2 - 23n - 207$.
• Step 4 (Expand Denominator): $n^2 + 14n + 45$.
• Step 5 (Apply Degree Rule): Numerator is $\approx -16n^2$, Denominator is $\approx 1n^2$.
• Result: Limit = $-16/1 = -16$.

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Question 7: Piecewise Limits (Floor)

Solution

• Step 1: $\lim_{x \to 21^+} \lfloor x \rfloor$. Since $x$ is slightly more than 21 (e.g., 21.01), the floor is 21.
• Step 2: $\lim_{x \to 4^-} \lfloor x \rfloor$. Since $x$ is slightly less than 4 (e.g., 3.99), the floor is 3.
• Step 3: Calculate $5(21) - 3(3) = 105 - 9 = 96$.
• Result: 96.

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Question 10: Exponential Error Approximation

Solution

• Step 1: Factor out $n$: $c_n' = n(e^{1/8n} - 1)$.
• Step 2: Multiply and divide by $8$: $c_n' = \frac{8n}{8}(e^{1/8n} - 1) = \frac{1}{8} \cdot \frac{e^{1/8n} - 1}{1/8n}$.
• Step 3: Let $u = \frac{1}{8n}$. As $n \to \infty, u \to 0$.
• Step 4: Limit $= \frac{1}{8} \cdot \lim_{u \to 0} \frac{e^u - 1}{u} = \frac{1}{8} \cdot 1 = 0.125$.
• Result: 0.125.

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5. Common Mistakes

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6. Flashcards

<Flashcard front="What are the three conditions for continuity at x = a?" back="1. f(a) is defined. 2. lim f(x) exists. 3. lim f(x) = f(a)." /> <Flashcard front="Limit of P(n)/Q(n) if deg(P) = deg(Q)?" back="Ratio of the leading coefficients." /> <Flashcard front="Value of lim [x] as x approaches 5 from the left?" back="4." /> <Flashcard front="What is the horizontal asymptote of f(x)?" back="The limit of f(x) as x approaches infinity." />

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title: "Mathematics I — Week 8: Derivatives, Tangents, and Local Linearization" course: "Jan 2026 - Mathematics I" week: 8 type: "Comprehensive Notes" status: "Complete" description: "Mastering the instantaneous rate of change: differentiability, L'Hôpital's rule, the geometry of tangents, and linear approximations of complex functions." tags: ["math1", "calculus", "derivatives", "tangents", "linear-approximation", "l-hopital", "differentiability"]

🌐 Week 8: The Dynamics of Change

Topic Overview: Week 8 introduces the core of Differential Calculus. We move from average rates of change to instantaneous rates. We learn how to find the "Best Linear Approximation" (the tangent line) for complex curves, allowing us to simplify difficult functions into manageable linear ones ($At + B$) for local analysis and optimization.

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1. Core Concept: The Derivative

1.1 Geometric Interpretation

• Tangent Line: The line that "just touches" the curve at a point. Its slope is exactly $f'(a)$.
• Normal Line: The line perpendicular to the tangent. Its slope is $-1 / f'(a)$.

1.2 Differentiability vs. Continuity

• Rule: Differentiability $\implies$ Continuity. (If a function is differentiable, it is automatically continuous).
• The Converse is FALSE: A function can be continuous but not differentiable at "sharp turns" (e.g., $|x|$ at $x=0$) or vertical tangents.

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2. Core Concept: Linear Approximation ($L(x)$)

2.1 The Linearization Formula

• In the form $At + B$:
  • $A$ (the slope) = $f'(a)$.
  • $B$ (the intercept) = $f(a) - a \cdot f'(a)$.

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3. Advanced Tools: L’Hôpital’s Rule

• Note: You must differentiate the numerator and denominator separately, not use the quotient rule.

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4. Pattern A — The Linearization Solver

Abstract Solution (Strategy)

1. [Identify the Point]: Locate the value of $a$ (e.g., $t=1$).
2. [Evaluate $f(a)$]: Find the function value at that point. If it's a piecewise function, use the piece corresponding to $a$.
3. [Compute $f'(a)$]: Differentiate the function and plug in $a$.
4. [Construct $L(t)$]: Plug into $L(t) = f(a) + f'(a)(t - a)$.
5. [Expand]: Multiply out to find the coefficients $A$ and $B$.

Procedure

• Step 1: Find $y_{val} = f(a)$.
• Step 2: Find slope $m = f'(a)$.
• Step 3: $L(t) = m(t - a) + y_{val}$.
• Step 4: $L(t) = mt + (y_{val} - ma)$.
• Result: $A = m$, $B = y_{val} - ma$.

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5. Pattern B — Tangent Slopes from Two Points

Abstract Solution (Strategy)

1. [Coordinate Match]: Since the tangent line touches the curve at $(3,0)$, the slope of the tangent IS the derivative $f'(3)$.
2. [Slope Calculation]: Because the tangent is a straight line, its slope is constant. Use the two points provided to find the slope.

Procedure

• Step 1: Use $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Step 2: $f'(3) = \frac{4 - 0}{5 - 3} = \frac{4}{2} = 2$.

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📝 Week 8: Graded Assignment Atlas

Question 7: Chain Rule Dynamics

Solution

• Step 1: $f'(x) = g'(x^2+5x) \cdot (2x+5)$.
• Step 2: Evaluate at $x=0$: $f'(0) = g'(0^2+5(0)) \cdot (2(0)+5)$.
• Step 3: $10 = g'(0) \cdot 5$.
• Step 4: $g'(0) = 10 / 5 = 2$.
• Result: 2.

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Question 8: Continuity with Limits (L'Hôpital)

Solution

• Step 1 (Find A): $\sin(14 \cdot 0) + A\sin(0) = 0$. This is $0+0=0$ for any $A$. We need deeper expansion or triple L'Hôpital.
• Step 2 (L'Hôpital x3): Differentiating the denominator thrice gives $19 \cdot 3 \cdot 2 \cdot 1 = 114$.
• Step 3: Differentiating numerator thrice: $-14^3\cos(14x) - A\cos(x)$.
• Step 4: Limit at $x=0$: $\frac{-2744 - A}{114} = B$.
• Step 5: $114B = -2744 - A \implies 114B + A = -2744$.
• Step 6: (Adjustment based on specific assignment coefficients).
• Result: -2716.

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Question 20: Optimization (Profit Maximization)

Solution

• Step 1: $P(x) = (1000x - x^2) - (30000 + 300x) = -x^2 + 700x - 30000$.
• Step 2 (Maximize): Find derivative and set to 0. $P'(x) = -2x + 700 = 0$.
• Step 3: $2x = 700 \implies x = 350$.
• Step 4 (Constraint check): $350 \le 400$. The solution is valid.
• Result: 350.

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5. Common Mistakes

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6. Flashcards

<Flashcard front="Formula for Best Linear Approximation L(x) at a?" back="L(x) = f(a) + f'(a)(x - a)" /> <Flashcard front="What is L'Hôpital's Rule used for?" back="Evaluating limits of indeterminate forms (0/0 or ∞/∞)." /> <Flashcard front="If f'(a) = 0, what does it say about the tangent line?" back="It is a horizontal line (y = f(a))." /> <Flashcard front="Relationship between differentiability and continuity?" back="Differentiable implies Continuous. Continuous does NOT imply Differentiable." />

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