IITM-Foundation-Course-Mathematics-1

01 Construction of Subsets and set operations

A well-defined collection of distinct objects called elements or members. 1️⃣ Set Comprehension: Writing Sets in Comprehensive Form Set comprehension is a way to define a set by describing the properties its members must satisfy instead of listing elements. Notation: $$ A = \{x \mid \text{property of } x\} $$Reads as: “Set $ A $ is the set of all $ x $ such that $ x $ satisfies the property.”

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02 Examples of Set Operations and Counting Problems

A well-defined collection of distinct objects called elements or members. Here is a detailed guide covering set notations and solving counting problems using Venn diagrams with 10 examples, including diagrams and emojis to enhance clarity. 1️⃣ Understanding Set Notations Basic Set Notations and Symbols: Set: Denoted by curly braces ${}$. Example: $A = {1,2,3}$ Element of a Set: $x \in A$ means $x$ is in set $A$. Not an Element: $x \notin A$ means $x$ is not in set $A$. Subset: $A \subseteq B$ means every element of $A$ is in $B$. Proper Subset: $A \subset B$ means $A$ is subset but not equal to $B$. Empty Set: $\emptyset$, set with no elements. Universal Set: $U$, all elements under consideration. Union: $A \cup B = {x | x \in A \text{ or } x \in B}$ Intersection: $A \cap B = {x | x \in A \text{ and } x \in B}$ Set Difference: $A - B = {x | x \in A \text{ and } x \notin B}$ Complement: $A^c = U - A$ Visual Diagram of Set Operations:

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03 Functions

A well-defined collection of distinct objects called elements or members. 1️⃣ Define Function, Domain, Co-domain, and Range Function: A relation $f$ from set $A$ (domain) to set $B$ (co-domain) that assigns each element in $A$ exactly one element in $B$. Denoted as $f: A \to B$. Domain ($A$): The set of input values over which the function is defined. Example: $A = {1,2,3}$. Co-domain ($B$): The set where outputs of the function lie. Example: $B = {a,b,c,d}$. Range: The subset of the co-domain actually mapped by the function. For example, if $f(1) = a, f(2) = b, f(3) = a$, then range $= {a,b}$. Diagram: Function Mapping

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04 Mathematics for Data Science 1

IIT Madras has launched the BS in Data Science and Applications. In this program, the course contents are delivered online and can be studied by anyone from anywhere, while the monthly quizzes and final semester exams will have to be attended in-person at designated centres.

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05 Natural Numbers and their operations

Of course! Let’s dive deep into the world of Natural Numbers and their operations. We’ll start from the very basics and build up to the properties that govern them. Identifying Natural Numbers and Integers Natural numbers are positive whole numbers, used for counting: 1, 2, 3, 4, 5, ….12 Integers include all positive and negative whole numbers and zero: …, -3, -2, -1, 0, 1, 2, 3, ….3

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06 Rational Numbers

Here is a detailed, emoji-enhanced guide to rational numbers, ordering, GCD, and density, with diagrams and markdown image code for added visuals. 🧮 Define Rational Numbers A rational number is any number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.123

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07 Real and Complex Numbers

Real Numbers (R) include all rational numbers plus all irrational numbers (numbers that cannot be expressed as fractions). This makes the real numbers a complete set covering every possible point on the number line. 1️⃣ How Rational Numbers Extend to Real Numbers Rational Numbers ($\mathbb{Q}$) are numbers expressed as fractions $\frac{p}{q}$, where $p, q$ are integers, $q \neq 0$. Examples: $\frac{1}{2}, 0.75, -3$.12 Real Numbers ($\mathbb{R}$) include all rational numbers plus all irrational numbers (numbers that cannot be expressed as fractions). This makes the real numbers a complete set covering every possible point on the number line.3 Diagram:

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08 Relations

A well-defined collection of distinct objects called elements or members. 1️⃣ Compute the Cartesian Product of Two Non-Empty Sets Definition: The Cartesian Product $ A \times B $ of sets $ A $ and $ B $ is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$. $$ A \times B = \{ (a, b) \mid a \in A, b \in B \} $$ Example: $ A = {1, 2}, B = {x, y} $ Then: $$ A \times B = \{(1,x), (1,y), (2,x), (2,y)\} $$ Diagram: Cartesian Product

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09 Set Examples

A well-defined collection of distinct objects called elements or members. 1️⃣ Describe the Membership of Sets Membership refers to whether an element belongs to a set or not. Notation: If $x$ is an element of set $A$, we write: $$ x \in A $$pronounced as “$x$ belongs to $A$.” - If $x$ is not an element of $A$, write:

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10 Set Theory

A well-defined collection of distinct objects called elements or members. 1️⃣ Define Set Concepts with Notations Set ( $ A $ ): A well-defined collection of distinct objects called elements or members. Example: $ A = {1, 2, 3, 4} $ 📦 Think of it as a box containing elements. Cardinality ( $ |A| $ ): The number of elements in set $ A $. Example: If $ A = {1, 2, 3} $, then $ |A| = 3 $. 🔢 Counts how many items are in the set. Subset ( $ B \subseteq A $ ): Set $ B $ is a subset of $ A $ if every element of $ B $ is also in $ A $. Example: $ B = {1, 2} \subseteq A = {1, 2, 3} $. 🤝 Everything in $ B $ is in $ A $. Proper Subset ( $ B \subset A $ ): $ B $ is a subset of $ A $, but $ B \neq A $. Example: $ B = {1, 2} \subset A = {1, 2, 3} $. 🚫 $ B $ does not contain all elements of $ A $. Power Set ( $ \mathcal{P}(A) $ ): The set of all subsets of $ A $, including the empty set and $ A $ itself. Example: For $ A = {1, 2} $, $ \mathcal{P}(A) = {\emptyset, {1}, {2}, {1,2}} $. 📚 All possible collections you can make from the set. Empty Set ( $ \emptyset $ ): The set with no elements. Example: $ \emptyset = {} $. ❌ A box with nothing inside. Diagram illustrating sets and subsets:

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11 Relations examples

A well-defined collection of distinct objects called elements or members. Learning Outcomes: List the number of elements in the cartesian product of two finite sets. Interpret relations as a subset of the cartesian product. Represent data tables as relations.

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12 Functions Examples

A well-defined collection of distinct objects called elements or members. https://youtu.be/6YrUx0bAnuQ Learning Outcomes: Find the domain and range of a given function. Understand what is the maximum value, minimum value of a function. Understand local maximum and local minimum. Identify whether one function grows faster than another.

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13 Prime Numbers

A well-defined collection of distinct objects called elements or members. https://youtu.be/98NeCye2i3k Learning Outcomes: Argue why the set of prime numbers is infinite. Identify if a given number is prime or not.

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14 Why is a number irrational?

A well-defined collection of distinct objects called elements or members. https://youtu.be/xUZeskT6WiQ Learning Outcomes: Understand why root 2 is an irrational number. Explain why not every number can be expressed in p/q form.

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15 Set versus Collections

A well-defined collection of distinct objects called elements or members. https://youtu.be/kM9nVJks1a8 Learning Outcomes: Express natural numbers as sets. Understand Russell’s paradox (why not every collection can be called a set). Define class.

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16 Degrees of infinity

A well-defined collection of distinct objects called elements or members. https://youtu.be/yjJ_pualJr4 Learning Outcomes: Compare the cardinalities of two infinite sets. Know what countable sets are. Explain why the cardinalities of N,Z, and Q are the same. Argue that the real numbers are not countable.

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17 Week 1 | Practice Assignemnt

Question 1: Venn Diagram Interpretation (from file image_5c141c.png) The Question: Given below is a Venn diagram for sets of students who take Maths, Physics, and Statistics. Which of the option(s) is (are) correct? [Venn diagram shows three overlapping circles: Maths (regions A, D, G, F), Physics (regions B, E, G, F), and Statistics (regions C, D, G, E)] D is the set of students who take both Maths and Statistics. $D \cup E \cup F \cup G$ is the set of all students who take at least two subjects. E is a subset of the set of the students who have not taken Maths. $Maths \setminus D$ is the set of all students who have taken only Maths. $Physics \setminus (D \cup G \cup E)$ is the set of all students who have taken only Physics. Core Concepts: Set Operations in Venn Diagrams

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18 Week 1 | Graded Assignment

Question 1: Irrational Numbers The Question: Which of the following are irrational numbers? 2.99999999 $(\sqrt{8} + \sqrt{2})(\sqrt{12} - \sqrt{3})$ $\frac{\sqrt{8} + \sqrt{2}}{\sqrt{8} - \sqrt{2}}$ $\frac{\sqrt{6} + \sqrt{3}}{\sqrt{6} - \sqrt{3}}$ Core Concepts: Rational vs. Irrational Numbers Rational Number: Expressible as fraction $p/q$, where $p,q$ are integers, $q \ne 0$. Decimals terminate or repeat. Irrational Number: Cannot be expressed as a simple fraction. Decimals non-terminating and non-repeating. Detailed Solution:

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19 Rectangular Coordinate system

A well-defined collection of distinct objects called elements or members. https://youtu.be/RfG2NLXqGE8 Learning Outcomes Know the basics of the rectangular coordinate system and the quadrants. Recognize, show, or plot a point on the coordinate plane. 1️⃣ Basics of the Rectangular Coordinate System and Quadrants The rectangular coordinate system (also called the Cartesian coordinate system) uses two number lines that intersect at right angles at a point called the origin (0,0).

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20 Distance formula

A well-defined collection of distinct objects called elements or members. https://youtu.be/aDhyAkXiDOY 1️⃣ Distance of a Point from the Origin To compute the distance of a point $P(x, y)$ from the origin $(0, 0)$ in the rectangular coordinate system, use the distance formula: $$ OP = \sqrt{x^2 + y^2} $$This formula comes straight from the Pythagorean Theorem, since the movement from $(0, 0)$ to $(x, y)$ forms the two right-angle legs.

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21 Distance of a line from a given point

A well-defined collection of distinct objects called elements or members. https://youtu.be/tYSZ4L0X3kY Learning Outcomes ● Explain the concept of intercepts of a line on the axes. ● Calculate the distance of a point from a line using the general linear expression. ● Determine the distance between two parallel lines.

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22 Equation of a perpendicular line passing through a point

A well-defined collection of distinct objects called elements or members. https://youtu.be/CjeHgCXhi4k Learning Outcomes ● Find the general equation of a line passing through a point and is perpendicular to a given line. Exercise Questions 1) The equation of a line passing through the point $(3,4)$ and perpendicular to the line $3x + 4y - 8 = 0$ is Concepts: The slope of the given line $3x + 4y - 8 = 0$: Rearranged to $y = -\frac{3}{4}x + 2$, slope $ m_1 = -\frac{3}{4} $. The perpendicular slope $ m_2 $ satisfies $ m_1 \cdot m_2 = -1 $: $$ m_2 = \frac{4}{3} $$ Using point-slope form for a line through $(3,4)$ with this slope: $$ y - 4 = \frac{4}{3}(x - 3) $$$$ 3(y - 4) = 4(x - 3) \implies 3y - 12 = 4x - 12 \implies 3y - 4x = 0 \implies 4x - 3y = 0 $$None of the options match this directly, so multiply both sides to create equivalent forms:

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23 Equation of parallel and perpendicular lines in general form

A well-defined collection of distinct objects called elements or members. https://youtu.be/gJuJtTYmbSs Learning Outcomes ● Determine the general conditions for the slope of a line parallel or perpendicular to a given line. ● Find the general equation of a line parallel or perpendicular to a given line. Exercise Questions 1) Which of the following statements are true? Concepts: Two lines $Ax + By + C = 0$ and $A’x + B’y + C’ = 0$ are parallel if $\frac{A}{B} = \frac{A’}{B’}$. They are perpendicular if the product of their slopes ($m_1 \cdot m_2$) is $-1$, i.e., $-A/B \cdot -A’/B’ = -1 \implies \frac{A}{B} \cdot \frac{A’}{B’} = -1$. Examine Each Option: a) $2x + 3y - 8 = 0$ and $3x - y - 2 = 0$ are parallel? Slopes: First: $m_1 = -2/3$ Second: $m_2 = -3/(-1) = 3$ Not equal, so not parallel. b) $3x + 5y - 10 = 0$ and $6x + 10y - 26 = 0$ are parallel? Second equation is just 2 × first equation, so slopes are both $-3/5$. True: They are parallel. c) $6x + 8y - 20 = 0$ and $4x - 3y = 0$ are perpendicular? Slopes: First: $m_1 = -6/8 = -3/4$ Second: $m_2 = -4/(-3) = 4/3$ Product: $(-3/4) \times (4/3) = -1$ True: They are perpendicular. d) $2x - 3y + 8 = 0$ and $3x + 2y - 18 = 0$ are not perpendicular to each other? Slopes: First: $m_1 = -2/(-3) = 2/3$ Second: $m_2 = -3/2$ Product: $(2/3) \times (-3/2) = -1$ But option says “are not perpendicular,” which is False (these are perpendicular). e) $4x + 5y + 10 = 0$ and $10x - 8y - 16 = 0$ are perpendicular? Slopes: First: $m_1 = -4/5$ Second: $m_2 = -10/(-8) = 5/4$ Product: $(-4/5) \times (5/4) = -1$ True: They are perpendicular. Correct Statements:

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24 General equation of line

A well-defined collection of distinct objects called elements or members. https://youtu.be/_3Iidm8NnbM Learning Outcomes ● Understand the general equation of a line. ● Find the different forms of the equation of a line including slope-point form, slope-intercept form, two-point form and intercept form from the general linear equation.

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25 Practice_Assignment_2

A well-defined collection of distinct objects called elements or members. Week 02 - Tutorial 01 https://youtu.be/ggpFtRiDUFA Week 02 - Tutorial 02 https://youtu.be/Qr7Kou01yj8

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26 Representation of a Line-1

A well-defined collection of distinct objects called elements or members. https://youtu.be/fKUK8xeuWNo Learning Outcomes Obtain equations of horizontal and vertical lines on the coordinate plane. Find the equation of a line in point-slope form and two-points form. 1️⃣ Equations of Horizontal and Vertical Lines Horizontal Lines: These lines run parallel to the $x$-axis. All points on a horizontal line have the same $y$-coordinate. Equation:

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27 Representation of a Line-2

A well-defined collection of distinct objects called elements or members. https://youtu.be/zM0q4y-y4so Learning Outcomes Find the equation of a line in slope-intercept form and intercept form. Exercise Questions Hello! On this Wednesday evening here in India, let’s work through these questions about the different forms of linear equations.

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28 Straight line fit

A well-defined collection of distinct objects called elements or members. https://youtu.be/xdZHsFuyBZM Learning Outcomes ● Associate the study on lines with real-world problems. ● Predict the suitable line among the given set of lines for a given set of points. ● Find the best fit line among the given set of lines using the Sum Squared Error Method for a given data.

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29 Section formula

A well-defined collection of distinct objects called elements or members. https://youtu.be/B5yv8zPrkek 1. Finding the Ratio in Which a Point Divides a Line Segment Suppose you have points $ A(x_1, y_1) $ and $ B(x_2, y_2) $, and a third point $ P(x, y) $ somewhere between them (or on the extension), and you want to know the ratio $ m:n $ in which $ P $ divides $ AB $.

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30 Area of triangle

A well-defined collection of distinct objects called elements or members. https://youtu.be/x62fodF7ezk 1️⃣ Compute the Area of a Trapezium A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides called the bases. Let the lengths of the parallel sides be $ a $ and $ b $. Let the height (distance between the parallel sides) be $ h $. The area $ A $ of the trapezium is given by:

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31 Parallel and perpendicular lines

A well-defined collection of distinct objects called elements or members. https://youtu.be/CXhBGVfmtBg 1️⃣ Why the Slope of a Line Does Not Uniquely Determine the Line The slope measures the steepness and direction of a line but does not uniquely determine a line by itself. Reason: Many different lines can share the same slope but differ in their y-intercept (where they cross the y-axis). Changing the y-intercept shifts the line up or down without changing the slope. Example: Lines $y = 2x + 3$ and $y = 2x - 1$ both have slope 2 but are distinct because their y-intercepts differ. 2️⃣ Characterize Parallel and Perpendicular Lines Using Slope Parallel Lines: Have the same slope but different y-intercepts. They never intersect. $$ \text{If } m_1, m_2 \text{ are slopes, then } m_1 = m_2 \implies \text{lines are parallel} $$ Perpendicular Lines: Intersect at right angles (90°). Slopes are negative reciprocals of each other. $$ m_1 \times m_2 = -1 $$For example, if one line has slope $\frac{3}{4}$, a line perpendicular to it has slope $-\frac{4}{3}$.

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32 Slope of a line

A well-defined collection of distinct objects called elements or members. https://youtu.be/V-b3BL8DAvU 1️⃣ Understand the Concept of Slope of a Line The slope of a line measures its steepness and direction in the coordinate plane. It is the ratio of the vertical change (“rise”) to the horizontal change (“run”) between two points on the line.

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33 Graded Assignment 2

Graded Assignment - 2 solution

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34 Quadratic functions

A well-defined collection of distinct objects called elements or members. https://youtu.be/U4gr4zMosMk Learning Outcomes Compare quadratic functions and linear functions. Define a quadratic function and represent it using a parabola. Identify different terms in a quadratic function and comprehend the importance of their coefficients. Represent the ‘axis of symmetry’ and ‘vertex’ used to represent parabola. Exercise Questions 🤯

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35 Examples of Quadratic = 🧐

A well-defined collection of distinct objects called elements or members. https://youtu.be/dvJKbgIPG8Q Learning Outcomes Determine the minimum and maximum value of a quadratic function. Explain the concept of range and domain of a quadratic function. Demonstrate the ability to apply these concepts in real life scenarios. Exercise Questions 🤯

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36 Examples of Quadratic functions

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37 Quadratic

A well-defined collection of distinct objects called elements or members.

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38 Quadratic

A well-defined collection of distinct objects called elements or members.

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39 Quadratic formula

A well-defined collection of distinct objects called elements or members. https://youtu.be/1shGS3xfmwc Learning Outcomes The student can solve quadratic equations with irrational roots. The student will be able to solve the quadratic equation using the quadratic formula. The student can choose the appropriate method to solve a quadratic equation. The student can appreciate the use of discriminant in finding the nature of roots for quadratic equations. The student can understand the concept of the ‘axis of symmetry’. Exercise Questions 🤯

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40 Slope | Line & Parabola

A well-defined collection of distinct objects called elements or members. https://youtu.be/kaByg1VQxqk Exercise Questions 🤯 Hello! On this Wednesday evening here in India, I’d be happy to guide you through these questions. They are great examples of how the concepts of parabolas and their slopes (derivatives) are applied in different scenarios.

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41 Slope of quadratic function

A well-defined collection of distinct objects called elements or members. https://youtu.be/1_Y2cZMrcfY Learning Outcomes Define the slope of a quadratic function. Compute the slope of any given parabola. Differentiate between slopes of linear equations and slopes of quadratic equations. Exercise Questions 🤯

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42 Solution of quadratic equation using Factorization

A well-defined collection of distinct objects called elements or members. https://youtu.be/ZRXWkzFSZzU Learning Outcomes The student will be able to solve the quadratic equation in an algebraic manner using factorization. Students can identify the intercept form of the quadratic equation and convert the intercept form into the normal form. Exercise Questions 🤯

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43 Solution of quadratic equation using graph

A well-defined collection of distinct objects called elements or members. https://youtu.be/RzIBZbGov4w Learning Outcomes Define the standard form of quadratic equations and relate this with the quadratic function. Compute roots of equations and zeros of a function. Solve quadratic equations using the graphing technique. Exercise Questions

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44 Solution of quadratic equation using Square method

A well-defined collection of distinct objects called elements or members. https://youtu.be/qvUX-jgtogg Exercise Questions 🤯 Hello! On this Wednesday evening here in India, I’d be happy to explain these questions. They all revolve around a very powerful technique for working with quadratic equations called “Completing the Square”.

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45 Summary Lecture (Quadratic Functions)

A well-defined collection of distinct objects called elements or members. https://youtu.be/JkbFTVFw1yQ

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46 Graded Assignment 3

A well-defined collection of distinct objects called elements or members.

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47 Polynomials

A well-defined collection of distinct objects called elements or members. https://youtu.be/2PTcxG8e6os Learning Outcomes Define a polynomial. Distinguish between a layman’s perspective and a mathematician’s perspective about a polynomial. Understand the origin of the word ‘polynomial’. Identify if a given function is a polynomial or not. Identify types of polynomials based on the number of variables.

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48 Polynomials

A well-defined collection of distinct objects called elements or members. https://youtu.be/s9zDRWmgsb4 Learning Outcomes Define the degree of a polynomial. Understand the degree of zero polynomial. Classify polynomials based on degrees of polynomials.

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49 Polynomials

A well-defined collection of distinct objects called elements or members.

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50 Polynomials

A well-defined collection of distinct objects called elements or members.

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51 Polynomials

A well-defined collection of distinct objects called elements or members.

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52 Polynomials

A well-defined collection of distinct objects called elements or members.

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53 Polynomials

A well-defined collection of distinct objects called elements or members.

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54 Polynomials

A well-defined collection of distinct objects called elements or members.

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55 Polynomials

A well-defined collection of distinct objects called elements or members.

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56 Polynomials

A well-defined collection of distinct objects called elements or members.

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57 Polynomials

A well-defined collection of distinct objects called elements or members.

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58 Polynomials

A well-defined collection of distinct objects called elements or members.

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59 Polynomials

A well-defined collection of distinct objects called elements or members.

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60 Polynomials

A well-defined collection of distinct objects called elements or members.

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61 Graded Assignment 4

A well-defined collection of distinct objects called elements or members.

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62 One-to-One Function | Definition & Tests

A well-defined collection of distinct objects called elements or members. https://youtu.be/CiZrP2-cpGM Learning Outcomes: The student will be able to (a) Perform vertical line test and horizontal line test to find whether a given relation is function or not. (b) Understand reversibility of a function. (c) Properly define one-to-one function.

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63 Examples

A well-defined collection of distinct objects called elements or members. https://youtu.be/em8Lw7g776M Learning Outcomes: The student will be able to (a) Understand one-to-one functions using examples. (b) Describe increasing and decreasing functions. (c) Identify the class of functions that are one-to-one.

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64 Exponential

A well-defined collection of distinct objects called elements or members. https://youtu.be/0l47NHJy-f4 Learning Outcomes: The student will be able to (a) Know the laws of exponents. (b) Define exponential function in standard form. (c) Understand the conditions for base and exponents.

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65 Exponential

A well-defined collection of distinct objects called elements or members. https://youtu.be/dMvBjIeJe80 Learning Outcomes: The student will be able to (a) Understand the behaviour of exponential functions via graphs. (b) Find the domain, range, intercepts, asymptotes, increasing or decreasing, end behaviour of exponential functions.

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66 Exponential

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67 Exponential

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68 Exponential

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69 Exponential

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70 Exponential

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71 Exponential

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72 Graded_assignment_5

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73 Logarithmic Functions

A well-defined collection of distinct objects called elements or members. https://youtu.be/G5A7imv2Otc Learning Outcomes: To learn inverse of an exponential function and understand its properties Learn ‘7-rule’ Understand the domain and range of exponential functions and logarithmic functions Solve problems on domains and ranges of exponential functions and logarithmic functions To plot graph of inverse function of exponential functions

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74 Logarithmic Functions

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75 Logarithmic Functions

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76 Logarithmic Functions

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77 Logarithmic Functions

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78 Logarithmic Functions

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79 Logarithmic Functions

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80 Logarithmic Functions

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81 Graded_assignment_6

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82 Limits

A well-defined collection of distinct objects called elements or members. https://youtu.be/0WHixZdnxTQ Learning Outcomes: To learn inverse of an exponential function and understand its properties Learn ‘7-rule’ Understand the domain and range of exponential functions and logarithmic functions Solve problems on domains and ranges of exponential functions and logarithmic functions To plot graph of inverse function of exponential functions

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83 Limits

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84 Limits

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85 Limits

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86 Limits

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87 Limits

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88 Graded_Assignment_7

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89 Limits and continuity

A well-defined collection of distinct objects called elements or members. https://youtu.be/do-iDBxQCpw

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90 Limits and continuity

A well-defined collection of distinct objects called elements or members. https://youtu.be/do-iDBxQCpw

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91 Limits and continuity

A well-defined collection of distinct objects called elements or members. https://youtu.be/do-iDBxQCpw

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92 Limits and continuity

A well-defined collection of distinct objects called elements or members. https://youtu.be/do-iDBxQCpw

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93 Limits and continuity

A well-defined collection of distinct objects called elements or members. https://youtu.be/do-iDBxQCpw

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94 Graded_Assignment_8

A well-defined collection of distinct objects called elements or members. https://youtu.be/do-iDBxQCpw

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95 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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96 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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97 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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98 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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99 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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100 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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101 Critical points | local maxima and minima

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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102 Graded_Assignment_9

A well-defined collection of distinct objects called elements or members. https://youtu.be/wUPkaBwF1-Y

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103 Sets and Relations

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104 Introduction to Graphs

A well-defined collection of distinct objects called elements or members. https://youtu.be/wtQ6RtTwCVk Learning Outcomes: To explain the relation as a subset of the Cartesian product Formal introduction to graphs: vertices, edges, directed and undirected graph, path, walk Concept of path and reachability in directed graphs

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105 Introduction to Graphs

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106 Introduction to Graphs

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107 Introduction to Graphs

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108 Introduction to Graphs

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109 Introduction to Graphs

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110 Introduction to Graphs

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111 Introduction to Graphs

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112 Introduction to Graphs

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113 Introduction to Graphs

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114 Introduction to Graphs

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115 Graded_Assignment_10

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116 Longest Paths in DAGs

A well-defined collection of distinct objects called elements or members. https://youtu.be/934a17SB5NY Learning Outcomes: The student will be able to: find the order of the dependencies using topological sorting. Calculate the longest path to a node using a topological sorting algorithm.

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