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SEASON 1 EP 3 -- DIFFERENTIABILITY

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 S1 EPISODE 3 DIFFERENTIABILITY INTRODUCTION:- Have you ever wondered what makes a function differentiable? A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean? It means that a function is differentiable everywhere its derivative is defined. So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable. Differentiability in Calculus refers to the property of a function that allows it to have a derivative at a given point. In other words, a function is differentiable at a point if its derivative exists at that point. The concept of differentiability is central to calculus and has significant implications in both theoretical and applied mathematics. Derivative Definition:- The derivative of a function f ( x ) at a point  x = a x = a  is defined as the limit: f ′ ( a ) = lim ⁡ h → 0 f ( a + h ) − f ( a ) h f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} ​ If...
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SEASON 1 EP 2 -- CONTINUITY

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  S1  EPISODE 2 CONTINUITY INTRODUCTION:-  The concept of limits is a fundamental building block of calculus and mathematical analysis. It is used to define continuity, derivatives, and integrals.  A function is said to be continuous if it can be drawn without picking up the pencil. Otherwise, a function is said to be discontinuous. Similarly, Calculus in  maths  , a function f(x) is continuous at x = c, if there is no break in the graph of the given function at the point. (c, f(c)) DEFINITION:- A function f ( x ) is said to be continuous at a point  x = c x = c  if the following three conditions are satisfied: The function is defined at c c : f ( c )  exists. f(c) \text{ exists.} The limit of the function exists as x x  approaches c c : lim ⁡ x → c f ( x )  exists. \lim_{x \to c} f(x) \text{ exists.} The limit equals the function's value: lim ⁡ x → c f ( x ) = f ( c ) . \lim_{x \to c} f(x) = f(c). Types of Continuity:- a) Continuit...
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SEASON 1 EP 1 -- LIMITS

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  S1 EPISODE 1  LIMITS INTRODUCTION:- The concept of limits is a fundamental building block of calculus and mathematical analysis. It is used to define continuity, derivatives, and integrals. In Mathematics, a limit   is defined as a value that a function approaches the output for the given input values. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns about the  behaviour  of the function at a particular point. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For  definite   integrals   , the upper limit and lower limits are defined properly. Whereas in indefinite the integrals are expressed without limits, and it will have an arbitrary constant while integrating the function. DEFINITION:- A limit describes the value that a function approaches as the input (or variable) ap...
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CEIL, FLOOR AND COMPOSITE FUNCTIONS

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  CEIL , FLOOR AND COMPOSITE FUNCTION CEIL AND FLOOR FUNCTION:- INTRODUCTION:- In Mathematics and Computer Programming, two important functions are used quite often. One is the floor function ,  and the other is the ceiling function  For example, the floor and ceiling of a decimal 3.31 are 3 and 4 respectively.  So with the help of these two functions, we get the nearest integer in a number line  of a given decimal. CEILING FUNCTION:- The ceiling function, denoted as ⌈ x ⌉, is a mathematical function that rounds a real number  x x  up to the smallest integer greater than or equal to x x . In simpler terms, it gives the "smallest" integer that is not less than x x . Formal Definition: For any real number x x , the ceiling function ⌈ x ⌉ \lceil x \rceil  is defined as: ⌈ x ⌉ = the smallest integer  n  such that  n ≥ x \lceil x \rceil = \text{the smallest integer } n \text{ such that } n \geq x ⌈ x ⌉ = the smal...
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