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ANALYSIS III

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Learning Outcomes

  • express the fundamental properties of n-dimensional eucledian space.
  • explain the n-dimensional euclidean space.
  • calculate the norm of a vector.
  • express the properties of norm.
  • calculate the distance between two points in n-dimensional eucledian space.
  • express the Cauchy-Schwarz inequality.
  • exemplify open and closed subsets of n-dimensional euclidean space.
  • express the fundamental concepts of functions of several variables.
  • define the graph of a function.
  • determine the level set of a function.
  • sketch the graph of a real valued function of two variables.
  • explain the limit and continuity of a function of several variables.
  • define limit at a point of a function.
  • characterize the limit by sequences.
  • express the properties of limit.
  • define continuity at a point of a function.
  • explain the differences between limit and continuity.
  • express the properties of continuous functions.
  • define uniformly continuous function.
  • use derivative to determine the properties of a function.
  • express the differentiability of a multivariable function at a point.
  • express the properties of the derivative.
  • determine the partial derivatives of a multivariable function.
  • express the chain rule.
  • calculate the directional derivative of a multivariable function.
  • find the gradient of a multivariable function.
  • find the equation for the tangent plane through a point of the graph of a multivariable function.
  • express the higher order derivatives.
  • express the Taylor's theorem for multivariable functions.
  • investigate the maxima and minima for multivariable functions.
  • determine the critical points of a multivariable function using partial derivatives.
  • identify a critical point of a multivariable function using the second partial derivative test.
  • solve optimization problems using the concept of critical points.
  • explain the method of Lagrange multipliers.
  • express the implicit and inverse function theorems.
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