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Function of Several Variables Basics Part 1
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OBJECTIVE
Find a function value for a function of several variables.
Sketch the graph of two variable
The function of two or more variables
The expression z = f(x, y) is a function of two variables if a unique value of z is obtained from each ordered pair of real numbers (x, y). The variables x and y are independent variables, and z is the dependent variable. The set of all ordered pairs of real numbers (x, y) such that f(x, y) exists is the domain of ƒ; the set of all values of f(x, y) is the range. Similar definitions could be given for functions of three, four, or more independent variables.
Quantities in nature usually are functions of, more than one variable. Examples:
If I put a deposit into an interest-bearing account and let it sit, the amount I have at the end of 3 years depends on P (how much my initial deposit is), r (the annual interest rate), and n (the number of compoundings per year) and we write A(p, r, n)
The concepts of continuity and differentiability of functions of one real variable can be extended to functions of two or more variables x, y, u, v, . . . (or x1, x2, x3, . . . ). Suppose that a quantity z, called the dependent variable, depends on two independent variables x and y. The dependence can often be expressed by an explicit formula such as z = x^2 +y^2, z = x^3 +y^2.
Find a function value for a function of several variables.
Sketch the graph of two variable
The function of two or more variables
The expression z = f(x, y) is a function of two variables if a unique value of z is obtained from each ordered pair of real numbers (x, y). The variables x and y are independent variables, and z is the dependent variable. The set of all ordered pairs of real numbers (x, y) such that f(x, y) exists is the domain of ƒ; the set of all values of f(x, y) is the range. Similar definitions could be given for functions of three, four, or more independent variables.
Quantities in nature usually are functions of, more than one variable. Examples:
If I put a deposit into an interest-bearing account and let it sit, the amount I have at the end of 3 years depends on P (how much my initial deposit is), r (the annual interest rate), and n (the number of compoundings per year) and we write A(p, r, n)
The concepts of continuity and differentiability of functions of one real variable can be extended to functions of two or more variables x, y, u, v, . . . (or x1, x2, x3, . . . ). Suppose that a quantity z, called the dependent variable, depends on two independent variables x and y. The dependence can often be expressed by an explicit formula such as z = x^2 +y^2, z = x^3 +y^2.