Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Differentiation Ch-5 Class 12 | Continuity @ Differentiability

Differentiation Chapter - 5, Class 12
Continuity @ Differentiability

Method of explaining continuity and differentiable, method of finding the derivative by First Principal. Differentiation of polynomial functions, logarithmic functions , exponential functions and trigonometric functions, product rule, quotient rule in differentiation  


DIFFERENTIABILITY

f(x) is said to be differentiable if left hand differentiation is equal to the right hand differentiation.
\[\lim_{x\to a^{-}}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a^{+}}\frac{f(x)-f(a)}{x-a}\]

DERIVATIVE BY FIRST PRINCIPAL

\[f^{'}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\; (OR) \\ f^{'}(x)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\]

PRODUCT RULE OF DIFFERENTIATION

\[\left ( uv \right )^{'}=u^{'}v+uv^{'}\]

QUOTIENT RULE OF DIFFERENTIATION

\[\left ( \frac{u}{v} \right )^{'}=\frac{u'v-uv'}{v^{2}} \]

DIFFERENTIATION OF TRIGONOMETRIC AND OTHER FUNCTIONS

Derivative of polynomial functions
\[\frac{d}{dx}\left ( Constant \right )=0\]
\[\frac{d}{dx}x^{n}=nx^{n-1}\]
\[\frac{d}{dx}\left ( x^{5} \right )=5x^{4} \]
\[\frac{d}{dx}\left ( ax+b \right )=a\]
\[\frac{d}{dx}\left ( ax^{2}+bx \right )=2ax+b \]
\[\frac{d}{dx}\left ( ax^{5}+bx^{4}+cx^{3}+20 \right )=5ax^{4}+4bx^{3}+3cx^{2}\]
Derivative of logarithmic functions
\[\frac{d}{dx}logx=\frac{1}{x}\]
\[\frac{d}{dx}\left ( \frac{1}{x} \right )=-\frac{1}{x^{2}}\]
\[\frac{d}{dx}\left ( \frac{1}{x^{2}} \right )=-\frac{2}{x^{3}}\]
\[\frac{d}{dx}\left ( \frac{1}{x^{3}} \right )=-\frac{3}{x^{4}}\]
Derivative of exponential functions
\[\frac{d}{dx}e^{x}=e^{x}\]
\[\frac{d}{dx}e^{2x}=2e^{2x}\]
\[\frac{d}{dx}e^{5x}=5e^{5x}\]
\[\frac{d}{dx}e^{ax}=ae^{ax}\]
\[\frac{d}{dx}a^{x}=a^{x}\frac{d}{dx}x\: loga=a^{x}\: loga\]
\[\frac{d}{dx}2^{x}=2^{x}\frac{d}{dx}x\: log\: 2=2^{x}\: log\: 2\]
Derivative of trigonometric functions
\[\frac{d}{dx}sinx=cosx\]
\[\frac{d}{dx}cosx=-sinx\]
\[\frac{d}{dx}tanx=sec^{2}x\]
\[\frac{d}{dx}cotx=-cosec^{2}x\]
\[\frac{d}{dx}secx=secx\; tanx\]
\[\frac{d}{dx}cosecx=-cosecx\; cotx\]

DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS 

\[\frac{d}{dx}sin^{-1}x=\frac{1}{\sqrt{1-x^{2}}}\]
\[\frac{d}{dx}cos^{-1}x=\frac{-1}{\sqrt{1-x^{2}}}\]
\[\frac{d}{dx}sec^{-1}x=\frac{1}{x\sqrt{x^{2}-1}}\]
\[\frac{d}{dx}cosec^{-1}x=\frac{-1}{x\sqrt{x^{2}-1}}\]
\[\frac{d}{dx}tan^{-1}x=\frac{1}{1+x^{2}}\]
\[\frac{d}{dx}cot^{-1}x=\frac{-1}{1+x^{2}}\]

Method of writing the derivatives

First order derivatives can be written as \[\frac{dy}{dx}=y'=y_{1}\]Second order derivatives can be written as \[\frac{d^{2}y}{dx^{2}}=y''=y_{2}\]Third order derivatives can be written as\[\frac{d^{3}y}{dx^{3}}=y'''=y_{3}\]

LOGARITHMIC FUNCTIONS

Any function of type  y =   is called logarithmic function
If base b is replaced by 10 then it is called common logarithm.
If base b is replaced by 'e' then it is called natural logarithm . 
Natural logarithm is denoted by lnx. 
In place of lnx we simple write logx \[log\: mn=log\;m+log\;n\]\[log\frac{m}{n}=logm-logn\]\[log\: m^{n}=n\: log\: m\]

ROLL'S THEOREM

If f(x) is a continuous function in [a, b] and is differentiable in (a, b) and f(a) = f(b) then there exist some c ∊ (a, b) such that f ' (c) = 0

Algorithm for proving Roll's Theorem

a) Explain the continuity of the function f(x) in close interval [a, b]

b) Explain the differentiable of the function f(x) in open interval (a, b). If function is differentiable then find f '(x).

c) Check whether f(a) = f(b)

d) If all the above three conditions are satisfied then there exist some c ∊ (a, b) such that f ' (c) = 0.
Use this equation to find the value of c.

If  c ∊ (a, b) then Roll's Theorem verified.

NOTE:  f '(x) is called the slope of tangent and when f '(x) = 0 then slope is become parallel to the x-axis.
In the figure below  red lines shows the tangents are parallel to the x-axis at all the points where f '(x) = 0

MEAN VALUE THEOREM

If f(x) is a continuous function in [a,b] and is differentiable in (a,b) then there exist some c ∊ (a, b)  such that    \[f'(c)=\frac{f(b)-f(a)}{b-a}\]

Algorithm for proving Mean Value Theorem

a) Explain the continuity of the function f(x) in close interval [a, b]

b) Explain the differentiable of the function f(x) in open interval (a, b). If function is differentiable then find f '(x).

c) When both the above conditions are satisfied the use  formula given below to find the value of c. \[f'(c)=\frac{f(b)-f(a)}{b-a}\]
If  c ∊ (a, b) then Mean Value Theorem is verified. 



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