Differential Calculus
Calculus Contents D
     Applications of the derivative
      Tangent, normal, subtangent and subnormal
         Property of the parabola
         Property of power functions
         Property of the exponential function
      Angle between two curves
         Angle between two curves examples
      Differential of a function
         Use of differential to approximate the value of a function
         Rules for differentials
         Differentials of some basic functions
     Higher order derivatives and higher order differentials
      Higher order derivatives
         Higher order derivatives examples
         Higher derivative formula for the product - Leibniz formula
         Higher derivatives of composite functions
         Higher derivatives of composite functions examples
         Higher derivatives of implicit functions
         Higher derivatives of implicit functions examples
         Higher derivatives of parametric functions
         Higher derivatives of parametric functions examples
      Higher order differentials
         Higher order differentials examples
      Applications of differentiation - the graph of a function and its derivative
      Definition of increasing and decreasing
         Increasing/decreasing test
      Rolle's theorem
      The mean value theorem
         Generalization of the mean value theorem
         Cauchy's mean value theorem or generalized mean value theorem
      L'Hospital's rule - limits of indeterminate forms
         Applications of L'Hospital's rule - evaluation of limits of indeterminate forms, examples
    a)  Applications of differentiation - the graph of a function and its derivatives
      Generalization of the mean value theorem, concavity of the graph of a function
      Concavity of the graph of a function
         Concave up and concave down definition
         Points of inflection
         Points of inflection and concavity of the sine function
         Points of inflection and concavity of the cubic polynomial
      Approximate solution to an equation, Newton's method (or the Newton-Raphson method)
         Use of Newton's method example
    b)  Applications of differentiation - the graph of a function and its derivatives
      Taylor's theorem (Taylor's formula) - The extended mean value theorem
         The proof of Thaylor's theorem
         Maclaurin's formula or Maclaurin's theorem
      Representing polynomial using Maclaurin's and Taylor's formula
         Representing polynomial using Maclaurin's and Taylor's formula examples
      The approximation of the exponential function by polynomial using Taylor's or Maclaurin's formula
         Properties of the power series expansion of the exponential function
      The approximation of the sine function by polynomial using Taylor's or Maclaurin's formula
         Properties of the power series expansion of the sine function
      Extreme points, local (or relative) maximum and local minimum
         The first derivative test
         The second derivative test and concavity
      Finding and classifying critical (or stationary) points
         Finding extreme points
         Points of inflection
         Finding points of inflection
         Finding and classifying critical (or stationary) points examples
 
 
 
 
 
 
 
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