Homework3Solns

Homework3Solns - SHORT CALCULUS Math 16C Sec 2 Spring 2008...

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Unformatted text preview: SHORT CALCULUS Math 16C Sec 2 Spring 2008 Homework #3 Solutions Peter Malkin Section 7.3 Question 18 The domain of the function ln( x + y ) is the set of points ( x, y ) such that x + y > 0. More succinctly, the domain is the set D = { ( x, y ) : x + y > } . Question 24 The domain of the function g ( x, y ) = 1 x- y is the set D = { ( x, y ) : x 6 = y } . Question 30 The contour map consists of curves e 1- x 2 + y 2 = C 1- x 2 + y 2 = ln ( C ), which are hyperbolas. The contour map of this surface matches (d). Section 7.4 Question 28 f ( x, y ) = ln( xy ) = 1 2 ln( xy ) = 1 2 (ln x + ln y ). So, f x ( x, y ) = 1 2 x and f y ( x, y ) = 1 2 y . Thus, f x (- 1 ,- 1) = 1 2(- 1) =- 1 2 and f y (- 1 ,- 1) = 1 2(- 1) =- 1 2 . Question 42 f ( x, y ) = ln( x 2 + y 2 + 1). f x ( x, y ) = 2 x ( x 2 + y 2 + 1) and f y ( x, y ) = 2 y ( x 2 + y 2 + 1) . So, f x ( x, y ) = 0 when x = 0, and f y ( x, y ) = 0 when y = 0. Thus, f x ( x, y ) = 0 and f y ( x, y ) = 0 at the point (0 , 0). Question 48 z = f ( x, y ) = x y . f x ( x, y ) = 1 y and f y ( x, y ) =- x y 2 . The slope of the surface in the x-direction at the point (3 , 1 , 3) is f x (3 , 1) = 1 1 = 1, and in the y-direction at the same point (3 , 1 , 3) is f y (3 , 1) =- 3 1 2 =- 3....
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Homework3Solns - SHORT CALCULUS Math 16C Sec 2 Spring 2008...

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