MA127-2018春 期中

1

单项选择题说明：选择一个正确答案。

1-1

If both $f_x(x,y)$ and $f_y(x,y)$ exist at $(x_0,y_0)$, then $f(x,y)$ is continuous at $(x_0,y_0)$

1-2

Let $f(x,y)=\{\begin{array}{ll}\frac{xy}{x^2+y^2},& (x,y)\ne (0,0)\\ 0,& (x,y)=(0,0)\end{array}$

1-3

For the $f(x,y)$ as in $(1)$ .both $f_x(0,0)$ and $f_y(0,0)$ exist

1-4

Nonzero vectors $u$ and $v$ are parallel if and only if $u\times v=0$

1-5

The surface $y^2-x^2=z$ is a hyperbolic paraboloid

2

解答题说明：请写出详细解答。

2-1

Suppose that the function $f(x,y)$ is differentiable, and $f(0,0)=1,f_x(0,0)=2,f_y(0,0)=3$ . Then $f(x,y)\approx$ $\underset{―}{}$ when both $x$ and $y$ are small(using the standard linear approximation at $(0,0)$

2-2

Find the distance from the point $(1,1,5)$ to the line

$$L:x=1+t,y=3-t,z=2t$$

2-3

Find the length of the curve

$$r(t)=(\sqrt{2}t)i+(\sqrt{2}t)j+(1-t^2)k$$

from $(0,0,1)$ to $(\sqrt{2},\sqrt{2},0)$

2-4

Find the normal vector and the curvature for the helix

$$r(t)=(a\cos t)i+(a\sin t)j+(bt)k,a,b⩾0,a^2+b^2\ne 0$$

2-5

Find $\underset{(x,y)\to (0,0)}{lim}\frac{x^{2}y}{x^2+y^2}$, if it exists；otherwise give the reason why the limit does not exist

2-6

Find $\frac{\partial w}{\partial v}$ when $u=-1,v=2$, if $w=xy+\ln z,x=\frac{v^2}{u},y=u+v,z=\cos u$

2-7

Find the critical points of the function $f(x,y)=x^4+y^4+4xy$, and use the second derivative test to classify each point as one where a saddle, local maximum or local minimum occurs

2-8

Find the point on the surface $z^2=xy+4$ closest to the origin

2-9

Use Taylor＇s formula for $f(x,y)=x{e}^y$ at the origin to find quadratic and cubic approximations of $f$ near the origin