HW09-solutions - nguyen(avn365 HW09 staron(52970 This...

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nguyen (avn365) – HW09 – staron – (52970) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Determine f x - f y when f ( x,y ) = 2 x 2 + xy - 3 y 2 + 3 x - 2 y. 1. f x - f y = 3 x + 7 y + 1 2. f x - f y = 5 x - 5 y + 5 3. f x - f y = 5 x - 5 y + 1 4. f x - f y = 5 x + 7 y + 1 5. f x - f y = 3 x - 5 y + 5 6. f x - f y = 3 x + 7 y + 5 correct Explanation: After differentiation we see that f x = 4 x + y + 3 , f y = x - 6 y - 2 . Consequently, f x - f y = 3 x + 7 y + 5 . 002 10.0points Determine whether the partial derivatives f x , f y of f are positive, negative or zero at the point P on the graph of f shown in P x z y From the graph it thus follows that at P f x < 0 , f y = 0 . keywords: surface, partial derivative, first or- der partial derivative, graphical interpreta- tion 003 10.0points Determine f x when f ( x,y ) = (2 y - x 2 )( y 2 + x ) . 1. f x < 0 , f y > 0 2. f x = 0 , f y > 0 3. f x < 0 , f y < 0 4. f x = 0 , f y < 0 5. f x > 0 , f y > 0 6. f x > 0 , f y = 0 7. f x = 0 , f y = 0 8. f x < 0 , f y = 0 correct Explanation: The value of f x at P is the slope of the tangent line to graph of f at P in the x - direction, while f y is the slope of the tangent line in the y -direction. Thus the sign of f x indicates whether f is increasing or decreasing in the x -direction, or whether the tangent line in that direction at P is horizontal. Similarly, the value of f y at P is the slope of the tangent line at P in the y -direction, and so the sign of f y indicates whether f is increasing or decreasing in the y -direction, or whether the tangent line in that direction at P is horizontal.

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