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हार्मोनल परिवर्तन के कारण कम उम्र में लड़कियों में हो रहा शारीरिक परिवर्तन

हार्मोनल परिवर्तन के कारण कम उम्र में लड़कियों में हो रहा शारीरिक परिवर्तन प्रदुषण और जंक फूड घटा रहा मुश्किल भरे दिन की उम्र लड़कियों में समय से पहले हार्मोनल चेंज के मामले बढ़ रहे हैं। शारीरिक बदलाव के लिए औसत उम्र 13 से 14 साल मानी जाती है, अब 8 से 11-12 साल में वजन बढ़ने जैसी समस्या आ रही है। कम उम्र में मुश्किल भरे दिनों का सामना करना पड़ रहा है। विशेषज्ञों की मानें तो समस्या प्रदुषण और जंक फूड के अत्यधिक सेवन से बढ़ रही है। शरीर में हार्मोनल परिवर्तन, किसी प्रकार की सिस्ट और ट्यूमर जैसे कारण सामने आ रहे हैं। इनके लिए मुख्य कारण निम्न हैं- लड़कियों में हार्मोनल परिवर्तन से पीरियड जल्दी आते हैं। आनुवंशिक समस्या इसके लिए जिम्मेदार हो सकती है। तनाव से भी हार्मोनल परिवर्तन होते हैं। आयरन तथा विटामिन-डी जैसे पोषण तत्वों की कमी भी हार्मोनल परिवर्तन के लिए जिम्मेदार हैं। प्रदुषण के सम्पर्क में रहने पर भी ये समस्या होती है। शरीर में कही भी सिस्ट या ट्यूमर होने पर भी ये सम्भव है। अगर 8 साल से 12 साल की उम्र के बीच किसी बालिका के शरीर में तेजी से परिवर्तन हो तो मुश्...
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Calculus | Mathematics | BSc

Calculus

Differential Calculus, Integral Calculus and Differential Equation





Authors: Dr. Vimal Saraswat, Dr. Anil Kumar Menaria, Dr. Chandrapal Singh Chouhan

ISBN : 978-93-94954-67-0

Price: Rs. 395.00 

Publisher: Himanshu Publications, Hiran Magri Udaipur; Himanshu Publications Prakash House, Ansari Road, New Delhi

E-mail : info@sacademy.co.in

Phone: +91 9664392614

To buy this book click on Calculus by Saraswat

This book includes the following topics 

Pedal Equations and Derivative of the Length of an Arc

  • Polar co-ordinates
  • Relation between cartesian and polar co-ordinates
  • Angle between radius vector and tangent
  • Angle of intersection of two polar curves
  • Polar tangent, subtangent, normal, subnormal and their lengths
  • Perpendicular from pole to tangent and its length
  • Pedal equation
  • Differential coefficient of length of the arc
    • When the equation of curve is in cartesian form (x, y); When the equation of curve is in polar form (r, ፀ)
  • Other formula

Mean Value Theorem

  • Introduction
  • Rolle’s Theorem
  • Geometrical interpretation of Rolle’s theorem
  • Algebraic interpretation of Rolle’s theorem
  • Another form of Rolle’s theorem
  • Lagrange’s mean value theorem
  • Another form of Lagrange’s mean value theorem
  • Geometrical interpretation of Lagrange’s mean value theorem
  • Some important deduction from Lagrange’s mean value theorem
  • Cauchy’s mean value theorem or first mean value theorem
  • Another useful form of Cauchy’s mean value theorem
  • General mean value theorem
  • Second mean value theorem
  • Generalised mean value theorem or Taylor’s theorem with Lagrange’s form of remainder
  • Taylor’s theorem with Cauchy’s form of remainder
  • Taylor’s theorem with Schlömilch and Roche form of remainder
  • Maclaurin’s theorem with Schlömilch and Roche form of remainder
  • Power series
  • Taylor series
  • Maclaurin’s series
  • Power series expansion of some useful basic functions

Asymptotes

  • Introduction
  • Asymptote
  • Condition for asymptote
  • Asymptotes of general algebraic form of the curve
  • Number of asymptotes of a curve
  • Asymptotes parallel to the direction axes
  • Alternative method for finding asymptotes
  • Intersection of curve and its asymptotes
  • Position of curve with respect to asymptote

Curvatures

  • Introduction)
  • Some important definitions
  • A formula for radius of curvature)
  • Cartesian formula for radius of curvature
  • Parametric formula for radius of curvature
  • Polar formula for radius of curvature
  • Pedal formula for radius of curvature
  • Tangential polar formula for radius of curvature
  • Miscellaneous formula for radius of curvature
  • Coordinate of centre of curvature
  • Length of chord of curvatures

Curve Tracing

  • Introduction
  • Concavity and convexity
  • Test of concavity or convexity with respect to coordinate axes
  • Point of inflexion
  • Point of undulation
  • Singular points
  • Multiple points
  • Types of double points
  • Types of cusp
  • Species of cusp
  • Necessary condition for the existence of double point
  • Curve tracing
  • Working method for tracing of cartesian curve
  • Working method for tracing of polar curves
  • Working method for tracing of parametric curve

Beta and Gamma Function

  • Beta function
  • Properties of beta function
  • Gamma function
  • Recurrence formula of gamma function
  • Relation between beta and gamma functions
  • Gamma formula
  • Duplication formula
  • Properties of gamma function

Quadrature

  • Introduction
  • Area bounded by cartesian curve and direction axes
  • Common area of two cartesian curve
  • Area and mass by double integration
  • Area bounded by polar curves and radius vectors
  • Area of closed curve

Rectification

  • Introduction
  • Length of arc of different form of curves
  • Intrinsic equation of a curve
  • To find the intrinsic equation of curve
  • Length of arc of an evolute

Differential Equations of First Order & First Degree

  • Differential equation
  • Ordinary and partial differential equations
  • Order and degree of a differential equation
  • Solution of differential equation
  • Equation of the first order and first degree
  • Equations in which variables are separable
  • Homogeneous differential equation
  • Equations reducible to a homogeneous equation
  • Linear differential equation
  • Bernoulli’s differential equation reducible to the linear form
  • Exact differential equation
  • Method of finding out the general solution of an exact differential equation
  • Equation reducible to an exact differential equation
  • Methods of finding out the integrating factors
    • By inspection; Rules of finding the integrating factors

Differential Equations of First Order & Higher Degree

  • Introduction
  • Equation solvable for p
  • Equation solvable for y
  • Lagrange’s equation
  • Equation solvable for x
  • Clairaut’s equation
  • Singular solutions
    • Geometrical meaning of singular solutions; Method of finding singular solutions in general case
  • Extraneous loci and its type
  • Procedure of finding the singular solutions and extraneous loci

Geometrical Meaning of Differential Equation

  • Introduction
  • Geometrical meaning of differential equation
  • Family of curves
  • Equation of family of curve
  • Trajectories
    • Orthogonal trajectory; Self orthogonal family of curves
  • Equation of orthogonal trajectory of a family of curves
  • Applications of orthogonal trajectories

Linear Differential Equations with Constant Coefficient

  • Introduction
  • General form of linear differential equation
  • Differential operator
  • Inverse operator
  • Homogeneous and non-homogeneous linear differential equation
  • Complementary function and particular integral
  • Method of finding complementary function
    • When all roots of auxiliary equations are different; When roots of auxiliary equation are same; When the roots of auxiliary equation are complex; When the roots of auxiliary equation are complex and same; When the roots of auxiliary equation are surd roots; When the roots of auxiliary equation are surd and equal
  • General method for finding particular integral
  • Short method for finding particular integral
  • Short method for finding P.I., when Q = eax
  • Short method for finding P.I., when Q = sin ax or cos ax
  • Short method for P.I., when Q = xn
  • Short method of P.I., when Q = eax.v, where v is any function of x
  • Short method of P.I., when Q = x.v, where v is any function of x

Homogeneous Linear Differential Equation

  • Introduction
  • Method for solving homogeneous linear differential equation
  • Methods for finding the complementary function
  • Alternative method of finding homogeneous linear differential equations
  • Differential equations reducible to homogeneous linear form
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