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df dy In each of the following, find and (j) f(x,y) = eaz+by

df dy In each of the following, find and (j) f(x,y) = eaz+by

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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df dy In each of the following, find and (j) f(x,y) = eaz+by
Transcribed Image Text:df dy In each of the following, find and (j) f(x,y) = eaz+by
Definition 3.1 The set of all ordered n-tuples of real numbers is called the n-dimensional Euclidean space and is denoted by R", that is, R" = {(r1, "2, ….. ,"n) :2; € R for all i}. An ordered n-tuple (x1, X2,.… . . ,¤n) in R" is called a point in R". Definition 3.2 A function in n variables is a set of ordered pairs of the form (P, w) in which no two distinct ordered pairs have the same first element, where P E R" and w E R. The set of all admissible values of P is called domain of the function and the set of all resulting values of w is called the range of the function. Definition 3.4 Let f be a function of æ and y. The partial derivative of f with respect to x, denoted by fz, fe etc.) such that its function value at any point (x, y) in the domain of ƒ is given by f(r + Ax, y) – f(x, y) f.(x, y) = lim Ar+0 Ar if this limit exists. The partial derivative of f with respect to y, denoted by fy» af , etc.) such that its function value at any point (x, y) in the domain of ƒ is given by f„(x, y) = lim Ay+0 f(x, y + Ay) – f(x, y) Ay if this limit exists. The process of finding a partial derivative is called partial differentiation.
Transcribed Image Text:Definition 3.1 The set of all ordered n-tuples of real numbers is called the n-dimensional Euclidean space and is denoted by R", that is, R" = {(r1, "2, ….. ,"n) :2; € R for all i}. An ordered n-tuple (x1, X2,.… . . ,¤n) in R" is called a point in R". Definition 3.2 A function in n variables is a set of ordered pairs of the form (P, w) in which no two distinct ordered pairs have the same first element, where P E R" and w E R. The set of all admissible values of P is called domain of the function and the set of all resulting values of w is called the range of the function. Definition 3.4 Let f be a function of æ and y. The partial derivative of f with respect to x, denoted by fz, fe etc.) such that its function value at any point (x, y) in the domain of ƒ is given by f(r + Ax, y) – f(x, y) f.(x, y) = lim Ar+0 Ar if this limit exists. The partial derivative of f with respect to y, denoted by fy» af , etc.) such that its function value at any point (x, y) in the domain of ƒ is given by f„(x, y) = lim Ay+0 f(x, y + Ay) – f(x, y) Ay if this limit exists. The process of finding a partial derivative is called partial differentiation.
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