Integration

Differential Integration is a key concept for Math Olympiad contenders. Explore the seamless integration of differentiation and integration, unraveling the intricate interplay between these two fundamental operations. Master the techniques of differential integration and elevate your problem-solving prowess to Olympiad-level heights. Join us as we demystify this intriguing mathematical fusion, empowering you to tackle challenging problems with confidence and precision

Differential Integration

Area Under the Curve is an essential concept for Math Olympiad. Delve into the world of functions, graphs, and integrals as we explore how to calculate and interpret the Area Under the Curve. Sharpen your problem-solving skills and gain a competitive edge as we break down this fundamental topic crucial for success in math competitions.

Area Under Curve

AUC Lecture-1

AUC Lecture-3

AUC Lecture-2

1. A sequence of real numbers is defined as follows :

$x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{9}{x_{n}}\right) \forall n \geq 0$

with $\mathrm{x}_{0} \in \mathrm{R}^{+}, \mathrm{y}_{\mathrm{n}}=\log _{3}\left(\frac{\mathrm{x}_{\mathrm{n}}-3}{\mathrm{x}_{\mathrm{n}}+3}\right)$ and $\mathrm{z}_{\mathrm{n}}=\log _{2}\left(\frac{\mathrm{x}_{\mathrm{n}}+3}{\mathrm{x}_{\mathrm{n}}-3}\right)$

which of the following statements are false ?
(A) If $x_{0}=6, y_{8}=-128$
(B) If $x_{0}=5, z_{9}=512$
(C) If $\mathrm{x}_{0}=5, z_{9}=256$
(D) If $\mathrm{x}_{0}=6, \mathrm{y}_{9}=-256$

Lorem ipsum dolor sit amet consectetur, adipisicing elit. Veniam temporibus mollitia eaque provident cum expedita ipsa aliquam dolores, nostrum consectetur repudiandae quos dignissimos?

5. The fraction loss of KE of the system (rod + particle) during collision is

(A) $\frac{1}{2}$     (B) $\frac{3}{4}$

(C) $\frac{2}{3}$     (D) None of these

$x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{9}{x_{n}}\right) \forall n \geq 0$with $\mathrm{x}_{0} \in \mathrm{R}^{+},\mathrm{y}_{\mathrm{n}}=\log _{3}\left(\frac{\mathrm{x}_{\mathrm{n}}-3}{\mathrm{x}_{\mathrm{n}}+3}\right)$ and $\mathrm{z}_{\mathrm{n}}=\log _{2}\left(\frac{\mathrm{x}_{\mathrm{n}}+3}{\mathrm{x}_{\mathrm{n}}-3}\right)$

which of the following statements are false ?
(A) If $x_{0}=6, y_{8}=-128$
(B) If $x_{0}=5, z_{9}=512$
(C) If $\mathrm{x}_{0}=5, z_{9}=256$
(D) If $\mathrm{x}_{0}=6, \mathrm{y}_{9}=-256$

⚡JEE adv⚡

$x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{9}{x_{n}}\right) \forall n \geq 0$with $\mathrm{x}{0} \in \mathrm{R}^{+},\mathrm{y}{\mathrm{n}}=\log {3}\left(\frac{\mathrm{x}{\mathrm{n}}-3}{\mathrm{x}{\mathrm{n}}+3}\right)$ and $\mathrm{z}{\mathrm{n}}=\log {2}\left(\frac{\mathrm{x}{\mathrm{n}}+3}{\mathrm{x}_{\mathrm{n}}-3}\right)$

1. The fraction loss of KE of the system (rod + particle) during collision is

(A) $\frac{1}{2}$     (B) $\frac{3}{4}$

(C) $\frac{2}{3}$     (D) None of these

asdfasdf

which of the following statements are false ?
(A) If $x_{0}=6, y_{8}=-128$
(B) If $x_{0}=5, z_{9}=512$
(C) If $\mathrm{x}{0}=5, z{9}=256$
(D) If $\mathrm{x}{0}=6, \mathrm{y}{9}=-256$

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1. he fraction loss of KE of the system (rod + particle) during collision is

(A) $\frac{1}{2}$     (B) $\frac{3}{4}$

(C) $\frac{2}{3}$     (D) None of these

asefwe

IIT-JEE Mains

The value of integral
$\int_0^4 \min \{|x-1|,|x-2|,|x-3|\} d x=\frac{a}{b}$ where $a$ and $b$ are co-prime, then the value of $(a+b)$ is

Options And Answer(s)
1 - 3
(2) 5
(3) 7
(4) 9

Solution

Required value

\[
=\frac{1}{2}(1)(1)+2\left(\frac{1}{2} \times 1 \times \frac{1}{2}\right)+\frac{1}{2} \times 1 \times 1=\frac{3}{2}
\]

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