Mathematics I — Week 12: Full Course Synthesis & Exam Revision Map

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0. Entire course in one map

text
W1  Sets → Relations → Functions
        ↓
W2  Coordinates → Straight Lines
        ↓
W3  Quadratics
        ↓
W4  General Polynomials
        ↓
W5  Functions → Composition → Inverses → Exponentials
        ↓
W6  Logarithms
        ↓
W7  Sequences → Limits
        ↓
W8  Derivatives
        ↓
W9  Integrals
        ↓
W10 Graph Theory → BFS/DFS → DAGs
        ↓
W11 Weighted Graphs → Shortest/Longest Paths

1. Algebra + functions: Weeks 1–6
2. Calculus: Weeks 7–9
3. Discrete maths + graph algorithms: Weeks 10–11

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Part I — Weeks 1–6

Algebra, Functions & Their Graphs

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Week 1 — Sets, Relations & Functions

Core set notation

$$x\in A$$ $$A\subseteq B$$ $$\varnothing$$ $$|A|$$ $$\mathcal P(A)$$ $$|A|=n,$$ $$\boxed{ |\mathcal P(A)|=2^n }$$

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Set operations

$$A\cup B$$ $$A\cap B$$ $$A-B$$ $$A^c$$ $$\boxed{ (A\cup B)^c=A^c\cap B^c }$$ $$\boxed{ (A\cap B)^c=A^c\cup B^c }$$

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Cartesian product

$$\boxed{ A\times B = \{(a,b):a\in A,b\in B\} }$$ $$|A|=m,\qquad|B|=n,$$ $$\boxed{ |A\times B|=mn }$$

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Relation properties — must know cold

Reflexive

$$\boxed{ aRa }$$

Symmetric

$$\boxed{ aRb\Rightarrow bRa }$$

Antisymmetric

$$\boxed{ aRb\land bRa\Rightarrow a=b }$$

Transitive

$$\boxed{ aRb\land bRc\Rightarrow aRc }$$

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Equivalence relation

$$\boxed{ \text{reflexive + symmetric + transitive} }$$

Partial order

$$\boxed{ \text{reflexive + antisymmetric + transitive} }$$

Highest-value trap

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Functions

$$f:A\to B,$$

• $A$ = domain
• $B$ = codomain
• actual outputs = range

$$\boxed{ \text{Range}\subseteq\text{Codomain} }$$

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Injective / one-one

$$f(a)=f(b)\Rightarrow a=b$$

Surjective / onto

$$\text{Range}=\text{Codomain}$$

Bijective

$$\boxed{ \text{injective + surjective} }$$

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Week 2 — Coordinate Geometry & Straight Lines

Distance

$$\boxed{ d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }$$

Midpoint

$$\boxed{ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) }$$

Slope

$$\boxed{ m= \frac{y_2-y_1}{x_2-x_1} }$$

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Types of slopes

$$m>0$$ $$m<0$$ $$m=0$$ $$m\text{ undefined}$$

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Line equations

Slope-intercept

$$\boxed{ y=mx+c }$$

Point-slope

$$\boxed{ y-y_1=m(x-x_1) }$$

General form

$$\boxed{ Ax+By+C=0 }$$ $$\boxed{ m=-\frac AB }$$

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Parallel lines

$$\boxed{ m_1=m_2 }$$

Perpendicular lines

$$\boxed{ m_1m_2=-1 }$$

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Collinearity

$$\boxed{ x_1(y_2-y_3)+ x_2(y_3-y_1)+ x_3(y_1-y_2)=0 }$$

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Week 3 — Quadratics

$$\boxed{ f(x)=ax^2+bx+c }$$ $$a\neq0.$$

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Direction

$$a>0,$$ $$a<0,$$

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Axis of symmetry

$$\boxed{ x=-\frac b{2a} }$$

Vertex

$$\boxed{ \left( -\frac b{2a}, f\left(-\frac b{2a}\right) \right) }$$

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Quadratic formula

$$\boxed{ x= \frac{-b\pm\sqrt{b^2-4ac}}{2a} }$$ $$\boxed{ \Delta=b^2-4ac }$$

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Nature of roots

$$\Delta>0$$ $$\Delta=0$$ $$\Delta<0$$

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Root relationships

$$\alpha,\beta,$$ $$\boxed{ \alpha+\beta=-\frac ba }$$ $$\boxed{ \alpha\beta=\frac ca }$$

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Three useful quadratic forms

$$ax^2+bx+c$$ $$a(x-r_1)(x-r_2)$$ $$a(x-h)^2+k$$

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Week 4 — Polynomials

$$P(x)=a_nx^n+\cdots+a_0.$$ $$n.$$ $$a_n.$$

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Division algorithm

$$\boxed{ P(x)=D(x)Q(x)+R(x) }$$ $$\deg R<\deg D.$$

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Remainder theorem

$$x-a,$$ $$\boxed{ P(a) }$$

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Factor theorem

$$\boxed{ x-a\text{ factor} \iff P(a)=0 }$$ $$\boxed{ \text{root} \iff \text{factor} \iff x\text{-intercept} }$$

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Multiplicity

$$(x-r)^m,$$ $$\boxed{\text{crosses the axis}}$$ $$\boxed{\text{touches/bounces}}$$

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End behavior

• parity of degree
• sign of leading coefficient

even degree → same-direction ends
odd degree → opposite-direction ends

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Week 5 — Composition, Inverses & Exponentials

Vertical line test

Horizontal line test

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Composition

$$\boxed{ (f\circ g)(x)=f(g(x)) }$$ $$f\circ g \neq g\circ f$$

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Inverse

$$f(a)=b,$$ $$f^{-1}(b)=a.$$ $$\boxed{ f^{-1}(f(x))=x }$$ $$\boxed{ f(f^{-1}(x))=x }$$

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Inverse trap

$$\boxed{ f^{-1}(x)\neq\frac1{f(x)} }$$

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Finding inverse

1. Write: $$y=f(x)$$
2. Swap $x,y$.
3. Solve for $y$.
4. Check domain/range.

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Exponential function

$$\boxed{ f(x)=a^x }$$ $$a>0,\qquad a\neq1.$$ $$\mathbb R$$ $$(0,\infty)$$ $$a>1$$ $$0<a<1$$

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Week 6 — Logarithms

$$\boxed{ a^x=y \iff \log_a y=x }$$ $$a>0,\quad a\neq1,\quad y>0.$$

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Log laws

$$\boxed{ \log_a(xy) = \log_a x+\log_a y }$$ $$\boxed{ \log_a\left(\frac xy\right) = \log_a x-\log_a y }$$ $$\boxed{ \log_a(x^r)=r\log_a x }$$ $$\boxed{ \log_a x = \frac{\log_bx}{\log_ba} }$$

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Inverse identities

$$\boxed{ \log_a(a^x)=x }$$ $$\boxed{ a^{\log_a x}=x }$$

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Common log trap

$$\log(x+y)=\log x+\log y.$$

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Part II — Weeks 7–9

Calculus

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Week 7 — Sequences & Limits

Sequence

$$a:\mathbb N\to\mathbb R.$$

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Sequence limit

$$\boxed{ \lim_{n\to\infty}a_n=L }$$

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Basic limit

$$\boxed{ \lim_{n\to\infty}\frac1{n^p}=0 }$$ $$p>0.$$

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Rational degree rule

$$\frac{P(n)}{Q(n)},$$ $$\boxed{0}$$ $$\boxed{ \frac{\text{leading coefficients}} {} }$$

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Function limits

$$\boxed{ \lim_{x\to a}f(x)=L }$$ $$f(a)$$

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Left and right limits

$$\lim_{x\to a^-}f(x)$$ $$\lim_{x\to a^+}f(x).$$ $$\boxed{ \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) }$$

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Continuity

$$\boxed{ \lim_{x\to a}f(x)=f(a) }$$

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$0/0$ trap

$$\frac00$$

• factor
• cancel
• rationalize

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Tangent preview

$$\frac{f(x+h)-f(x)}h.$$ $$\boxed{ \lim_{h\to0} \frac{f(x+h)-f(x)}h }$$

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Week 8 — Derivatives

$$\boxed{ f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}h }$$

• tangent slope
• instantaneous rate of change
• local behavior

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Differentiability relationship

$$\boxed{ \text{differentiable} \Rightarrow \text{continuous} }$$ $$\text{continuous} \not\Rightarrow \text{differentiable}.$$ $$|x|$$

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Derivative rules

$$\boxed{ \frac d{dx}c=0 }$$ $$\boxed{ \frac d{dx}x^n=nx^{n-1} }$$ $$\boxed{ (fg)'=f'g+fg' }$$ $$\boxed{ \left(\frac fg\right)' = \frac{f'g-fg'}{g^2} }$$ $$\boxed{ (f\circ g)'(x) = f'(g(x))g'(x) }$$

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Common derivatives

$$\boxed{ \frac d{dx}e^x=e^x }$$ $$\boxed{ \frac d{dx}a^x=a^x\ln a }$$ $$\boxed{ \frac d{dx}\ln x=\frac1x }$$

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Tangent line

$$x=a,$$ $$\boxed{ y-f(a)=f'(a)(x-a) }$$

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Increasing/decreasing

$$f'(x)>0$$ $$f'(x)<0$$

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Critical points

$$\boxed{ f'(x)=0 }$$

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First derivative test

$$+\to-$$ $$-\to+$$

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L’Hôpital

$$\frac00$$ $$\frac{\infty}{\infty},$$ $$\boxed{ \lim\frac fg = \lim\frac{f'}{g'} }$$

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Week 9 — Integration

$$\boxed{ \int f(x)\,dx=F(x)+C }$$ $$F'(x)=f(x).$$

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Power rule

$$n\neq-1,$$ $$\boxed{ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C }$$ $$\boxed{ \int\frac1x\,dx = \ln|x|+C }$$

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Exponential integral

$$\boxed{ \int e^x\,dx=e^x+C }$$

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Fundamental Theorem of Calculus

$$F'=f,$$ $$\boxed{ \int_a^b f(x)\,dx = F(b)-F(a) }$$

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Signed area

$$+$$ $$-$$ $$|f(x)|.$$

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Area between curves

$$f(x)\geq g(x),$$ $$\boxed{ A= \int_a^b [f(x)-g(x)]\,dx }$$

top minus bottom.

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Substitution

$$\boxed{ \int F'(g(x))g'(x)\,dx = F(g(x))+C }$$

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Calculus in one picture

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LIMIT
"What happens nearby?"
   ↓
DERIVATIVE
"How fast is it changing?"
   ↓
INTEGRAL
"How much accumulated?"

$$\boxed{ \text{derivative} \leftrightarrow \text{integral} }$$

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Part III — Weeks 10–11

Graph Theory & Algorithms

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Week 10 — Graph Fundamentals

$$\boxed{ G=(V,E) }$$

• $V$ = vertices
• $E$ = edges

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Degree

$$\deg(v)$$ $$\boxed{ \sum_v\deg(v)=2|E| }$$

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Directed graphs

$$\deg^-(v)$$ $$\deg^+(v)$$ $$\boxed{ \sum_v\deg^-(v) = \sum_v\deg^+(v) = |E| }$$

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Complete graph

$$K_n,$$ $$\boxed{ |E|=\frac{n(n-1)}2 }$$

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Representations

$$A_{ij}=1$$

text
A: B, C
B: A
C: A

$$O(V+E)$$ $$O(V^2).$$

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BFS

• queue
• level-by-level exploration
• shortest paths in unweighted graphs

$$\boxed{ O(V+E) }$$

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DFS

• stack/recursion
• explores deeply
• backtracking

$$\boxed{ O(V+E) }$$

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Tree

• connected
• acyclic

$$\boxed{ |E|=n-1 }$$

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DAG

$$\boxed{ \text{Directed Acyclic Graph} }$$

• dependencies
• scheduling
• prerequisites

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Week 11 — Weighted Graph Algorithms

Path weight

$$\boxed{ \sum w(e) }$$

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Relaxation

$$\boxed{ d[v] \leftarrow \min(d[v],d[u]+w(u,v)) }$$

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Algorithm choice

Unweighted

$$\boxed{\text{BFS}}$$

Weighted, nonnegative

$$\boxed{\text{Dijkstra}}$$

Negative weights

$$\boxed{\text{Bellman–Ford}}$$

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Dijkstra

$$\boxed{ w(e)\geq0 }$$

repeatedly finalize nearest unprocessed vertex.

$$O((V+E)\log V).$$

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Bellman–Ford

$$|V|-1$$ $$\boxed{ O(VE) }$$

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Negative cycle

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Longest path in DAG

$$\boxed{ d[v] = \max_{u\to v} (d[u]+w(u,v)) }$$

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Transitive closure

Is $v$ reachable from $u$?

$$u\to v$$

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Adjacency matrix powers

$$\boxed{ (A^k)_{ij} }$$ $$k$$

walks, not necessarily simple paths.

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1. Highest-value formula sheet

Sets

$$|\mathcal P(A)|=2^{|A|}$$ $$|A\times B|=|A||B|$$

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Line

$$m=\frac{y_2-y_1}{x_2-x_1}$$ $$y-y_1=m(x-x_1)$$ $$m_1m_2=-1$$

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Quadratic

$$x_v=-\frac b{2a}$$ $$x= \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ $$\Delta=b^2-4ac$$

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Polynomial

$$P(a)=\text{remainder on division by }x-a$$ $$P(a)=0 \iff x-a\text{ factor}$$

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Composition/inverse

$$(f\circ g)(x)=f(g(x))$$ $$f^{-1}(f(x))=x$$

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Logs

$$a^x=y \iff \log_a y=x$$ $$\log(xy)=\log x+\log y$$ $$\log(x^r)=r\log x$$

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Limits

$$\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x)$$

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Derivative

$$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}h$$ $$(x^n)'=nx^{n-1}$$ $$(fg)'=f'g+fg'$$ $$\left(\frac fg\right)' = \frac{f'g-fg'}{g^2}$$

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Integral

$$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C$$ $$\int_a^b f = F(b)-F(a)$$

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Graphs

$$\sum_v\deg(v)=2|E|$$ $$\text{BFS/DFS}=O(V+E)$$ $$\text{tree: }|E|=|V|-1$$

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2. Most dangerous conceptual traps

Trap 1

$$aRb\Rightarrow bRa$$ $$aRb\land bRa\Rightarrow a=b.$$

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Trap 2

Is it a function?

Is it one-one?

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Trap 3

$$f^{-1}(x)$$ $$\frac1{f(x)}.$$

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Trap 4

$$\log(x+y) \neq \log x+\log y.$$

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Trap 5

$$\frac00$$

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Trap 6

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Trap 7

$$f'(x)=0$$

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Trap 8

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Trap 9

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Trap 10

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3. Dependency map

text
Functions
├── composition
│   └── chain rule
│
├── inverse
│   └── logarithms
│
└── graphs
    ├── limits
    ├── derivatives
    └── integrals

Relations
└── graph theory
    ├── adjacency
    ├── reachability
    └── transitive closure

Matrices
└── adjacency matrix powers

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4. Suggested exam-priority hierarchy

Tier A — must be reliable

• relation properties
• functions/injectivity/surjectivity
• straight-line equations and slope
• quadratic roots/vertex/discriminant
• polynomial roots/factor/remainder theorem
• composition/inverse
• logarithm rules
• limits
• derivative rules
• critical points
• basic integration
• BFS vs DFS
• Dijkstra vs Bellman–Ford

Tier B — important

• multiplicity/end behavior
• continuity
• tangent equations
• L’Hôpital
• area between curves
• DAG/topological ordering
• transitive closure
• adjacency matrix powers

Tier C — polish after fundamentals

• deeper proofs
• rarer graph identities
• unusual transformations
• more elaborate limit manipulations

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5. The course as mental models

Sets

Which objects belong?

Relation

Which pairs are connected?

Function

Each input gets exactly one output.

Slope

change in $y$ per change in $x$.

Quadratic

symmetric curved function with one turning point.

Polynomial

degree + roots + multiplicities + leading term determine much of the graph.

Exponential

repeated multiplicative growth.

Logarithm

what exponent produced this value?

Limit

what value is behavior approaching?

Derivative

instantaneous local change.

Integral

accumulated change.

Graph

objects connected by relationships.

BFS

explore outward in layers.

DFS

go deep, then backtrack.

Dijkstra

repeatedly lock in nearest safe vertex.

Bellman–Ford

repeatedly propagate improvements everywhere.

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6. Final mastery checklist

• Can I distinguish all four relation properties?
• Can I classify injective/surjective/bijective functions?
• Can I construct a line from two points?
• Can I analyze a quadratic completely?
• Can I infer polynomial graph behavior from factors?
• Can I use remainder/factor theorem quickly?
• Can I compose functions without reversing order?
• Can I find an inverse and respect its domain?
• Can I solve exponential/log equations?
• Can I evaluate basic limits and recognize $0/0$?
• Can I differentiate standard functions correctly?
• Can I identify extrema from $f'$?
• Can I integrate basic polynomials and compute definite integrals?
• Can I distinguish signed area from actual area?
• Can I trace BFS and DFS?
• Can I recognize trees and DAGs?
• Can I choose BFS vs Dijkstra vs Bellman–Ford?
• Can I interpret adjacency-matrix powers?

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Quick Recall & Flashcards

• Equivalence Relation vs Partial Order distinction :: Equivalence is Symmetric; Partial Order is Antisymmetric
• Vertex of parabola vs Critical point of derivative :: Vertex $x = -b/(2a)$ is where $f'(x) = 2ax + b = 0$
• Shortest path algorithm decision: Unweighted graph :: Breadth-First Search (BFS) in $O(|V| + |E|)$
• Shortest path algorithm decision: Nonnegative weights :: Dijkstra's Algorithm in $O((|V| + |E|)\log |V|)$
• Shortest path algorithm decision: General weights (negative edges allowed) :: Bellman–Ford Algorithm in $O(|V| \cdot |E|)$
• Fundamental theorem linking derivatives and integrals :: Fundamental Theorem of Calculus (FTC): accumulation vs rate of change

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Practice & Self-Check