MA127-2018春 期中考试

一

Determine whether the following statements are ture or false, No justification is necessary

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

At the point $(0,0)$, $f(x,y)$ is continuous

1-3

For the $f(x,y)$ as in problem 1-2, both $f_x(0,0)$ and $f_y(0,0)$ exist

1-4

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

1-5

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

1-6

If $f(x,y)$ and it's partial derivatives $f_x,f_y,f_{xy}$ and $f_{yx}$ are defined throughout an open region containing a point $(a,b)$, then $f_{xy}(a,b)=f_{yx}(a,b)$

二

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 $(0,0)$)

三

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

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

四

Find the length of the curve

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

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

五

Find the normal vector and the curvature for the helix

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

六

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

七

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

八

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

九

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

十

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