Maths revision week 1 to 12

Course Summary

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Course Summary

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For Qualifier exam click here.

Week 1 to 4 (Qualifier)

Week 1: Relations and Venn Diagrams

In the first week, we focused on understanding different types of relations: reflexive, symmetric, transitive, and equivalence relations. We also explored the use of Venn diagrams to solve problems involving sets.

Important Concepts:

Reflexive Relation: A relation $R$ on a set $A$ is reflexive if $\forall a \in A, (a, a) \in R$ .
Symmetric Relation: A relation $R$ on a set $A$ is symmetric if $\forall a, b \in A, (a, b) \in R ⟹ (b, a) \in R$ .
Transitive Relation: A relation $R$ on a set $A$ is transitive if $\forall a, b, c \in A, (a, b) \in R \land (b, c) \in R ⟹ (a, c) \in R$ .
Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

Venn Diagram Formula:

$| A \cup B \cup C | = | A | + | B | + | C | - | A \cap B |$

$- | A \cap C | - | B \cap C | + | A \cap B \cap C |$

Example:

Suppose we have three groups of people: those who like football, hockey, and cricket. Let:

$| A | = 30$ (football)
$| B | = 25$ (hockey)
$| C | = 20$ (cricket)
$| A \cap B | = 10$
$| A \cap C | = 5$
$| B \cap C | = 8$
$| A \cap B \cap C | = 3$

Using the formula, the number of people who like exactly one sport is:

| A \cup B \cup C | = 30 + 25 + 20 - 10 - 5 - 8 + 3 = 55

Week 2: Lines and Distance

This week, we covered various geometric concepts related to lines and distances.

Important Concepts:

Midpoint Formula: The midpoint of a line segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is: $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Distance Formula: The distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is: $\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
Equation of a Line: The general form is $A x + B y + C = 0$ .
Distance Between Two Parallel Lines: The distance between $A x + B y + C_{1} = 0$ and $A x + B y + C_{2} = 0$ is: $\frac{| C_{2} - C_{1} |}{\sqrt{A^{2} + B^{2}}}$
Sum of Squared Errors: A measure used in regression analysis to determine the accuracy of a model.

Week 3: Quadratic Functions

We delved into quadratic functions, their properties, and their graphs.

Important Concepts:

Quadratic Function: A function of the form $f (x) = a x^{2} + b x + c$ .
Vertex: The highest or lowest point on the graph of a parabola, given by: $(- \frac{b}{2 a}, f (- \frac{b}{2 a}))$
Discriminant: Determines the nature of the roots of a quadratic equation $a x^{2} + b x + c = 0$ , given by: $Δ = b^{2} - 4 a c$

Week 4: Polynomials

This week, we explored polynomials and their properties.

Important Concepts:

Zeros of a Polynomial: The values of $x$ for which $P (x) = 0$ .
Factors of a Polynomial: Expressions that can be multiplied to get the polynomial.
Intercepts: Points where the graph intersects the axes.
Multiplicity of Roots: The number of times a particular root appears.
End Behavior: The behavior of the graph as $x$ approaches $\pm \infty$ .
Turning Points: Points where the graph changes direction.

Mid-Term Focus (Weeks 5 to 8)

For the mid-term quiz, more weightage will be given to topics from weeks 5 to 8, but questions from weeks 1 to 4 will also be included.

Week 5: Functions

We studied different types of functions and their compositions.

Important Concepts:

One-to-One Function: A function where each element of the range is mapped to by exactly one element of the domain.
Onto Function: A function where every element of the range is mapped to by at least one element of the domain.
Composition of Functions: Combining two functions $f$ and $g$ to form $f (g (x))$ .

Week 6: Logarithmic Functions

This week, we focused on logarithmic functions and their properties.

Important Concepts:

Logarithmic Function: The inverse of an exponential function, given by $y = \log_{b} (x)$ .
Properties of Logarithms: $\log_{b} (x y) = \log_{b} (x) + \log_{b} (y)$ $\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)$ $\log_{b} (x^{y}) = y \log_{b} (x)$
Concept of Limits: Understanding the behavior of functions as they approach specific points.
L'Hôpital's Rule: A method to evaluate limits of indeterminate forms: $lim_{x \to c} \frac{f (x)}{g (x)} = lim_{x \to c} \frac{f^{'} (x)}{g^{'} (x)}$

Week 8: Continuity and Differentiability

In Week 8, we focused on the concepts of continuity and differentiability, which are fundamental in calculus.

Continuity:

A function

f (x)

is continuous at a point

x = a

if the following three conditions are met:

$f (a)$ is defined.
$lim_{x \to a} f (x)$ exists.
$lim_{x \to a} f (x) = f (a)$ .

This means there are no breaks, jumps, or holes in the graph of the function at

x = a

Differentiability: A function $f (x)$ is differentiable at a point $x = a$ if the derivative $f^{'} (a)$ exists. This implies that the function is smooth and has no sharp corners or cusps at $x = a$ .

Equation of Tangent and Linear Approximation

Equation of Tangent: The equation of the tangent line to the curve $y = f (x)$ at the point $(a, f (a))$ is given by: $y - f (a) = f^{'} (a) (x - a)$ This line touches the curve at exactly one point and has the same slope as the curve at that point.

Linear Approximation: Linear approximation is used to approximate the value of a function near a given point. The linear approximation of $f (x)$ near $x = a$ is: $f (x) \approx f (a) + f^{'} (a) (x - a)$ This is essentially the equation of the tangent line and provides a good approximation for values of $x$ close to $a$ .

End-Term Focus: Graph Theory

Question will be from week 1 to 11

For the end-term exam, more weightage will be given to graph theory. Here are the key topics:

Week 9: Integration

Integration: Integration is the process of finding the integral of a function, which represents the area under the curve of the function. The definite integral of $f (x)$ from $a$ to $b$ is given by: $\int_{a}^{b} f (x) d x$

Riemann Sum: The Riemann sum is a method for approximating the definite integral of a function. It involves dividing the interval $[a, b]$ into smaller subintervals and summing up the areas of rectangles formed by the function values at specific points within the subintervals.

Graph Theory

Breadth-First Search (BFS) and Depth-First Search (DFS): BFS and DFS are fundamental algorithms for traversing graphs. BFS explores all neighbors of a vertex before moving on to the next level, while DFS explores as far as possible along each branch before backtracking.

Directed Acyclic Graph (DAG): A DAG is a directed graph with no cycles. It is important for representing tasks with dependencies.

Topological Sorting: Topological sorting is used to order the vertices of a DAG such that for every directed edge $u v$ , vertex $u$ comes before $v$ . This is crucial for scheduling tasks.

Longest Path in DAG: Finding the longest path in a DAG is important for determining the maximum time required to complete a sequence of tasks.

Dijkstra's Algorithm: Dijkstra's algorithm finds the shortest path from a source vertex to all other vertices in a graph with non-negative weights. It does not work with negative weights.

Bellman-Ford and Floyd-Warshall Algorithms: These algorithms handle graphs with negative weights but no negative cycles. Bellman-Ford works by iterating over all edges, while Floyd-Warshall considers all pairs of vertices.

Minimum Cost Spanning Tree (MCST): An MCST connects all vertices in a graph with the minimum total edge weight. Kruskal's and Prim's algorithms are used to find the MCST.

Kruskal's Algorithm: Kruskal's algorithm sorts all edges and adds them one by one to the spanning tree, ensuring no cycles are formed.

Prim's Algorithm: Prim's algorithm starts with a single vertex and grows the spanning tree by adding the smallest edge that connects a vertex in the tree to a vertex outside the tree.

Both algorithms yield the same MCST, and it's useful to cross-verify the results obtained from one algorithm with the other.