课程名称︰微积分甲下
课程性质︰工学院必修
课程教师︰官彦良
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰108.03.21
考试时限（分钟）：30
试题 :
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)?