課程名稱︰微積分甲下 課程性質︰工學院必修 課程教師︰官彥良 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰108.03.21 考試時限（分鐘）：30 試題 : 1. Let f(x,y) = (x^3+y^3)/(x^2+y^2) if (x,y) ≠ (0,0) 0 if (x,y) ＝ (0,0) (a) Is f(x,y) continuous at (0,0)? Justify your answer/ (b) Use the definition of f_x and f_y to find the gradient vector ▽f(0,0). (c) Is f_x(x,y) continuous at (0,0)? Justify your answer. 2. The base radius and height of a right circular cone are measured as 20 cm and 30 cm, respectively, with a possible error in measurement of as much as 0.1 cm in each. Use differentials to estimate the maximum error in the calcul- ated volume of the cone. 3. Let f(x,y) = (sin(x^3)-sin(y^3))/(x^2+y^2) if (x,y) ≠ (0,0) 0 if (x,y) ＝ (0,0) (a) Use the definition of f_x and f_y to find the gradient vector ▽f(0,0) (b) Use the definition of directional derivative to calculate D_uf(0,0), where u = (i-j)/√2. (c) Is f(x,y) differentiable at (0,0)? --

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