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7.3 Functions of Several Variables Tools to learn for functions of several variables Evaluating functions Finding domain and range Contour Maps (functions of 2 variables) Using functions of several variables to model real-life situations (how to ask and answer questions) Definition of function of 2 variables A function of two variables is a rule which defines a unique output, z = f (x, y), for every input pair of real numbers, (x, y), in a specified set of points D called the domain. For example: f (x, y) = x3 − 3y 2 is a function which gives an output for every pair of input values. f (0, 0) = 03 − 3(0)2 = 0, f (1, 2) = 13 − 3(2)2 = 1 − 3(4) = −11 f (−4, 3) = (−4)3 − 3(3)2 = 64 − 3(9) = 64 − 27 = 37. The domain of a function The domain of a function is very important and can be either 1 Specified by the problem, i.e. specific restrictions are given f (x, y) = x2 + y 2 such that − 1 < x < 1, −1 < y < 1 2 Assumed to be all points for which the function is valid p f (x, y) = 4 − x2 − y 2 More about domains.... what are invalid points? Recall from functions of one variable You cannot take the square root of a negative number, so if √ f (x, y) = 2x + 3y the assumed domain requires that 2x + 3y ≥ 0. 1 You cannot divide by zero, so if f (x, y, z) = the x − y 2 + 3z assumed domain requires that x − y 2 + 3z 6= 0. You cannot take the logarithm of 0 or a negative number, so if f (x, y) = ln(x2 − 3y) the assumed domain requires that x2 − 3y > 0. The range of a function The range of the function is the set of all possible values for the output. So for example if f (x, y, z) = x2 + y 2 + z 2 then we know that z = f (x, y, z) ≥ 0. Another example f (x, y) = 1 x+y Find the domain and range. The domain is all (x, y) such that x + y 6= 0 or x 6= −y and the range is all z except z = 0. Contour Maps and Level Curves Functions of two variables are often referred to as surfaces z = f (x, y). A contour map is a collection of (x, y) traces all drawn on the same picture. z = x2 + y2 4 Applications Example: The monthly payments M for an installment loan of P dollars taken out over t years at an annual interest rate r is given by Pr 12 M = f (P, r, t) = 1− h 1 1+(r/12) i12t . Find the monthly payment for a home mortgage of $350, 000 take out for 30 years at an annual interest rate of 5%. Plugging in 350000(.05) 12 M = f (350000, .05, 30) = 1− h 1 1+((.05)/12) i12(30) = $1869