۲۹ اسفند ۱۳۹۸
دسته:
کد : 241

» عنوان : Calculus – Engineering Mathematics Study Materials

زبان : انگلیسی

تعداد صفحات : ۳۷

توضیح : فرمول های پرکاربرد ریاضی مهندسی

عناوین :

  • Functions of Single Variable Limits
  • L’ Hospital’s Rule
  • Continuity and Discontinuity
  • Differentiability
  • Mean Value Theorems
  • Functions of Two Variables
  • Computing the Derivative
  • Partial Derivatives
  • Maxima and Minima of Functions  of Two Variables
  • Taylor and Maclaurin Series
  • Some Standard Integrations
  • Definite Integral
  • Multiple Integrals
  • Change of Order of Integration
  • Applications of the Definite Integral
  • Volume: Slicing and Disks
  • Vector
  • Product of Two Vectors
  • Vector Calculus
  • Directional Derivative
  • Divergence
  • Curl
  • Line Integral
  • Surfaces
  • Volume Integral
  • Stoke’s Theorem
  • Green Theorem
  • Gauss’s Divergence Theorem

 

به عنوان مثال :

MAXIMA AND MINIMA OF  FUNCTIONS OF TWO VARIABLES

A function f(x, y) is said to have a maximum or  minimum at x = a, y = b, according as f(a + h, b + k) is  less or greater than f(a, b) for all positive or negative  small values of h and k.

In other words, if Δ = f(a + h, b + k) – f(a, b) is of the  same sign for all small values of h, k and if this sign is  negative, then f(a, b) is a maximum. If this sign is  positive, then f(a, b) is a minimum.  A maximum or minimum value of a function is called  its extreme value.

Above figure shows a graph of the function f(x) and OA = a, i.e., f(x) has a maximum value for x = a because f(a) has a value more than the values of f(x) for every value of x between B and B’. f(x) is said to be a maximum at x = a, even though value of f(x) at x = a should be greater than all other values of f(x) in some small neighbourhood. Thus a maximum value of f(x) is not necessarily the greatest value of f(x). In fact, a curve might have several maxima (and minima).

Finding Absolute Maxima and Minima  Values  Working Rules:

If f is a differentiable function in [a b] except at finitely  many points, then to find absolute maximum and  absolute minimum values, adopt following procedure: (i) Evaluate f(x) at the points, where f’(x) = 0.  (ii) Evaluate f(a) and f(b).

Then maximum of these values is the absolute maxima and minimum of these values is called  absolute minima.

MULTIPLE INTEGRALS

Calculus Let a single-valued and bounded function f(x,y) of two  independent variables x,y be defined in a closed region  R of the xy-plane. Divide the region R into sub-regions  by drawing lines parallel to co-ordinate axes. Number  the rectangles which lie entirely inside the region R  from 1 to n.

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