1.确定功能是否
f（x; y）=
8
<

pxy + 1 1
y
x = 0和y = 0和xy 1;
0（x; y）=（0; 0）：

2.确定功能是否
f（x; y）=
8
<

2×2
ÿ
x4 + y2
[x; y）=（0; 0）
0（x; y）=（0; 0）：

3.找到函数的绝对最小值和绝对最大值
f（x; y）=

+ y2
</ s> </ s> </ s> </ s> </ s>
4; y 0克。
4.找到以下函数的绝对最小值和绝对最大值
f（x; y）= x2
y（4 x y）在以x轴，y轴和

5.球体每个点的温度x2
+ y2
+ z2
= 11由

。找到最高和最低

+ y2
+ z2
= 11并且

6.使用牛顿-拉夫森方法的一个迭代进行初始猜测（1; 1）

2倍
+ 4年2
9 = 0
18岁14×2
+ 45 = 0

7.找到函数的所有临界点w = x3
+ y3
12x 3y + 20。还可以阻止-

8.找到函数f（x; y）=的所有相对和绝对最大值和最小值
2倍
22
+ 2x + 4y + 2在闭合的第一个象限x 0，y 0
9.令w = x2
e2y
cos（3z）。在点（1; ln 2; 0）上找到dw = dt的值

10.找出f（x; y）= xy + 2x ln（x2
y）在开放的第一个象限中
x> 0和y>0。还要确定此临界点的性质（相对

11.找到并分类所有对于f（x; y）= 3×2的临界点
y + y3
3×2
3年2
+2。
12.确定功能是否

Chapter One
1. Determine whether the function
f(x; y) =
8
<
:
pxy + 1 1
xy
x = 0 and y = 0 and xy  1;
0 (x; y) = (0; 0):
is continuous at (0; 0) or not. Prove your result.
2. Determine whether the function
f(x; y) =
8
<
:
2×2
y
x4 + y2
(x; y) = (0; 0)
0 (x; y) = (0; 0):
is continuous at (0; 0) or not. Prove your result.
3. Find the absolute minimum and absolute maximum value of the function
f(x; y) =
px2 + y2 2x + 2 on the closed half disk R = f(x; y) : x2
+ y2

4; y  0g.
4. Find the absolute minimum and absolute maximum value of the function for
f(x; y) = x2
y(4 x y) in the closed region bounded by x axis, y axis and
the line x + y = 6.

5. The temperature at each point of the sphere x2
+ y2
+ z2
= 11 is given by
the function T(x; y; z) = 20 + 2x + 2y + z2
. Find the highest and lowest
temperature on the intersection of the given sphere x2
+y2
+z2
= 11 and the
plane x + y + z = 3.
6. Using ONE iteration of Newton-Raphson’s method with initial guess (1; 1)
to solve the system of linear equations
(
x2
+ 4y2
9 = 0
18y 14×2
+ 45 = 0
:

7. Find all critical points of the function w = x3
+y3
12x 3y+20. Also deter-
mine the nature of the critical points (relative maximum, relative minimum
8. Find all relative and absolute maxima and minima of the function f(x; y) =
x2
y2
+ 2x + 4y + 2 on the closed rst quadrant x  0, y  0.
9. Let w = x2
e2y
cos (3z). Find the value of dw=dt at the point (1; ln 2; 0) on the
curve x = cos t, y = ln(t + 2), z = t.
10. Find the critical point of f(x; y) = xy+2x ln(x2
y) in the open rst quadrant
x > 0 and y > 0. Also determine the nature of this critical point (relative
maximum, relative minimum or saddle point).
11. Find and classify all the critical points for f(x; y) = 3×2
y+y3
3×2
3y2
+2.
12. Determine whether the function

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