» عنوان : Calculus – Engineering Mathematics Study Materials
زبان : انگلیسی
تعداد صفحات : ۳۷
توضیح : فرمول های پرکاربرد ریاضی مهندسی
- Functions of Single Variable Limits
- L’ Hospital’s Rule
- Continuity and Discontinuity
- 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
- Product of Two Vectors
- Vector Calculus
- Directional Derivative
- Line Integral
- 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.
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.