作者wheata (仙人指路为马)
看板NTU-Exam
标题[试题] 99下 王金龙 微积分甲下 第一次小考
时间Thu Jun 9 10:08:13 2011
课程名称︰微积分甲下
课程性质︰必修
课程教师︰王金龙
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2011/3/10
考试时限（分钟）：40
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
A. Let f(x,y) be a real-valued function. Show that if f (x,y) and f (x,y)
x y
are both continuous at (0,0), then f(x,y) is differentiable at (0,0).
xy
┌ ────── if (x,y) ≠ (0,0)
B. Consider the function f(x,y) = │ (x^2+y^2)^2
│
└ 0 if (x,y) = (0,0)
(a) Show that f is continuous at (0,0).
(b) Show that f (0,0) and f (0,0) are both exist and evaluate their values.
x y
(c) Show that f is not differentiable at (0,0).
C. Consider the function
-1 -1
┌ (x^2)tan (y/x) - (y^2)tan (x/y) if x≠0 and y≠0
f(x,y) = │
└ 0 otherwise
(a) Evaluate f (x,y) and f (x,y) for x≠0 and y≠0.
x y
(b) Evaluate f (0,0) and f (0,0), and show that f is differentiable at (0,0).
x y
What is the tangent plane of the surface z = f(x,y) at (0,0,0)?
(c) Evaluate f (0,y) for y≠0 and f (x,0) for x≠0.
x y
Show that f (0,0)≠f (0,0).
xy yx
2
D. Let f(x,y), u(x,y) and v(x,y) be C functions. Suppose that f + f = 0
xx yy
and u = v , u = - v , show that the function
x y y x
ψ(x,y) = f(u(x,y),v(x,y)) also satisfies ψ + ψ = 0.
xx yy
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