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Notes - Mathematics Olympiads -Application of Derivatives

                                                                                         Application of Derivatives    
  • Continuity and Differentiability of a function: Let a function \[y=f(x)\] is said to be continuous at
\[x=a\] then \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,\,\,f(x)=f(a)\] Generally, a function is said to be continuous at \[x=a\] when the graph of that function can be drawn/sketched without lefting the pencil.    
  • Differentiation: The process of finding out the differentiability/derivatives of the function \[y=f(x)\]in the interval (a, b) is said to be differentiation.
 
  • Derivatives of f(x): Let \[y=f(x)\] is continuous in interval [a, b]. Let a point \[c\in (a,b)\]
Then function \[y=f(x)\] is differentiable at \[x=c\] i.e. \[\underset{x\to c}{\mathop{\lim }}\,\frac{f(x+c)-(c)}{c}=f'(c)\]   Solved Example
  • Find derivative of \[y=\sin x.\] by 1st principle:
    • Let \[y=f(x)=\sin x\]    ...(1) Let \[\delta \]x be the more...
    Catgeory: 12th Class

    Notes - Mathematics Olympiads -Maxima Minima Function

                                                                                                  Maxima and Minima of the Function  
    • Local maximum: A function \[y=f(x)\]is said to have local maximum value at a point\[x=a.\]
    If \[f(x)\le f(a)\]\[\forall x\in (a-h,a+h)\]and \[h>0\] i.e.\[f(a)\] is the greatest value of all values of \[f(x)\]in the interval\[(a-h,a+h)\] The point\[x=a\]is said to be the point of local maximum of the function     ·                    
    • Local minimum: A function\[y=f(x)\]is said to have a local minimum value at a point \[x=a.\] If
    \[f(x)\ge f(a)\]\[\forall x\in (a-h,a+h)\]for \[h>0\] i.e.\[f(a)\] is the smallest value of all the value of \[f(x)\] in the interval \[(a-h,a+h)\] The Point\[x=a\] is said to be the point of local minimum of the function\[f(x)\].   Note: The points more...
    Catgeory: 12th Class

    Notes - Mathematics Olympiads -Integral Calculus

      Integral Calculus   Integration is the inverse process of differentiation i.e. the process of finding out the integral of the integrand function is called integration. e.g. If \[\frac{d}{dx}\{F(x)\}=f(x),\]then\[\int{f(x).dx=F(x)+C}\] Here function f(x) is said to be integrand and value of\[\int{f(x).dx=F(x)}\] is said to be integral value the function, \[c=\] Integration constant  
    • Some Basic Properties of Indefinite Integration
     
    Derivatives Integrals:
    1. \[\frac{d}{dx}({{x}^{n}})=n.{{x}^{n-1}}\] \[\int{{{x}^{n}}.dx=\frac{{{x}^{n}}+1}{n+1}+c}\] where \[c=\]integration constant.
    2. \[\frac{d}{dx}({{e}^{x}})={{e}^{x}}\] \[\int{{{e}^{x}}.dx={{e}^{x}}+c}\]
    3. \[\frac{d}{dx}\{\log (x)\}=\frac{1}{x}\] \[\int{\frac{1}{x}.dx=\log x+c}\]
    4. \[\frac{d}{dx}({{a}^{x}})={{a}^{x}}.\log a\] \[\int{{{a}^{x}}.dx=\frac{{{a}^{x}}}{\log a}+c}\]
    5. more...
    Catgeory: 12th Class

    Notes - Mathematics Olympiads -Differential Equations

                                                                                     Differential Equations  
    • Definition: An equation involving independent variable, dependent variable and its derivatives is said to be differential equation.
    e.g. An equation of the form of \[\therefore \,\,\,\,\,y=f(x,y,p)\]is said to be differential equation, where\[p=\frac{dy}{dx}\]e.g. (1) \[y={{\left( \frac{dy}{dx} \right)}^{3}}+5\]              (2) \[{{x}^{2}}\frac{{{d}^{2}}y}{d{{x}^{2}}}+y\frac{dy}{dx}=5y+c\] etc.  
    • Order and Degree of differential Equation:
    The order of highest order derivative occurred in differential equation is said to be the order of the differential equation whereas power/exponent of the highest order derivative term in the different equation whereas power/exponent of the highest order derivative term in the differential equation is said to be the degree of the differential equation. e.g. (1) \[\frac{{{d}^{3}}y}{d{{x}^{3}}}+2{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}+6{{\left( \frac{dy}{dx} \right)}^{6}}+7y=0\] Here order of differential equation be 3 Degree of differential equation = 1 (2) \[{{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right]}^{\frac{3}{2}}}=\frac{{{d}^{3}}y}{d{{x}^{3}}}\] Order of this differential equation more...
    Catgeory: 12th Class

    Notes - Mathematics Olympiads -Vectior Algebra

                                                                                                   Vector Algebra   ·                 
    • As we know that the some quantities have only magnitude and some have magnitude as well as direction, e.g. distance, force, work, current etc.
    On this basis, all physical quantities are divided into two groups: (i)  Scalar quantities. (ii) Vector quantities.  
    • Scalar quantity: A quantity which has only magnitude and does not have direction is said to be scalar quantity. For example: distance, speed, work-done etc.
     
    • Vector quantity: A quantity which has magnitude as well as direction and also obeys the addition law of triangle is said to be vector quantity. For example: Displacement, Force, Velocity etc.
     
    • Representation of vectors: A vector is symbolically represented by putting an arrow sign (\[\to \]) on more...
    Catgeory: 12th Class

    Notes - Mathematics Olympiads - Three D Geometry

                                                                                   Three Dimensional Geometry ·                
    • Direction cosines: Let P(a, b, c) be any point. We join P to origin O. Let the line OP makes an angle \[\alpha ,\beta ,\gamma \] with positive direction of x-axis, y-axis and z-axis respectively. Then \[\cos \alpha ,\cos \beta \] and \[\cos \gamma \] are called the direction cosine of the directed line OP. If the angle is measured in clockwise direction then the direction angles are replaced by their supplements i.e. \[\pi -\alpha ,\]\[\pi -\beta ,\]and\[\pi -\gamma \] respectively. It is generally denoted by \[\ell \], m and n respectively i.e. \[\ell =\cos \alpha ,\]\[m=\cos \beta \]and \[n-\cos \gamma \]
    \[\Rightarrow \] \[{{\ell }^{2}}+{{m}^{2}}+{{n}^{2}}=1\]  
    • Direction Ratio: The three number a, b, c proportional to the direction cosines \[\ell \]m, n of a more...
    Catgeory: 12th Class

    Notes - Mathematics Olympiads - Probability

                                                                                     Probability    
    • Conditional Probability: The probability of event A is called the conditional probability of A given that event B has already occurred. It is written as \[P(A|B)\]or \[P\left( \frac{A}{B} \right)\].Mathematically it is given by the formula \[P(A|B)=\frac{P(A\bigcap B)}{P(B)}\]
    Event A is independent of B if the conditional probability of A given B is the same as the unconditional probability as A,i.e, they are independent if \[P(A|B)=P(A)\] 'Gender gap' in politics is a well-known example of conditional probability and independence in the real life. Suppose candidate A receive 55% of the entire vote and the only 45% of the female vote. Let \[P(R)=\] Probability that a random person voted for A But \[P\left( \frac{W}{R} \right)=\] Probability of a random women voted for A \[\therefore P(R)=0.55\]and \[P\left( \frac{W}{R} \right)=0.45\] Then \[P\left( \frac{W}{R} \right)\] is said to be the more...
    Catgeory: 12th Class

    Test Tube Babies

    The technique of in-vitro fertilization and in-vitro development followed by the embryo-transfer in the uterus of the normal female to start the development and finally leading to normal birth, is called test tube baby. History : First attempt to produce a test tube baby was made by a Italian scientist, Dr. Petrucci (1959 A.D.). Although, this human embryo survived for only 29 days, but his experiment opened a new filed of biological science. The first test tube baby was born to Lesley and Gilbert Brown on July 25, 1978, in Oldham, England. Mrs. Brown had obstructed Fallopian tubes. Dr.Patiricke Steptoe and Dr. Robert Edward both from England experimented on Mrs. Brown successfully. the world's first test tube baby (a baby girl) was named as Louise Joy Brown. Later, test tube babies were also born in Australia, United States and some other countries. India's first test tube baby was born on more...
    Catgeory: 12th Class

    Population Trends in the World and India

    Population Trends in the World The distribution of human population is not uniform throughout the world. Only about one third of the total land area is inhabited. Of the inhabited areas, some are thickly populated, others sparsely. This depends upon the availability of the requirements of life. About 56% of the total world population resides in Asia alone. Bangladesh is the most thickly populated country, and Australia, the most thinly populated. Annual Birth, Death and Growth Rates for Human Population in 1973
    S. No Region more...
    Catgeory: 12th Class

    Population trends in Madhya Pradesh and Chhattisgarh

    Population : India’s population according to the provisional totals of Census of India 2001 at the 00.00 hours of Ist March 2001 is 1,027,015,247, out of which the population of Madhya Pradesh is 60,385,118, state thus contributing 5.87% share to India’s total population. Madhya Pradesh is the 7th largest state population-wise whereas it is 2nd in terms of its geographical spread and contributes 9.38% to the country’s total area of 3,287,263 sq.km.   Population 1901 to 2001 in Madhya Pradesh
    Year Population more...
    Catgeory: 12th Class

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