2023秋高数下期末试题(回忆版)

一

Determine whether the following limits exist:

(A) $\underset{(x,y)\to (1,2)}{lim}\frac{\ln (1-x+xy)}{x-1}$

(B) $\underset{x\to (0,0)}{lim}(x+y{)}^{x+y+xy}$

二

Let $f$ be the function defined by $f(x,y)=\{\begin{array}{ll}\frac{2x+1}{x^2+y^2}\sin (x^2+y^2)& \text{ if }(x,y)\ne (0,0)\\ 1& \text{ if }(x,y)=(0,0)\end{array}$

Determine if the function $f$ is contineuous at the origin

三

Let $n$ be the normal unit vector pointing inside the surface $3x^2+y^2+z^2=3$ .Compute the directional derivative of the function

$$f(x,y,z)=\frac{\sqrt{x^2+y^2+z^2}}{(y+z+1{)}^2}$$

at the point $(1,0,0)$ in the direction $n$

四

Find the equations of the tangent plane and the normal line for the surface

$$xy+z+2^{xy}=4$$

at the point $(1,1,1)$

五

Use the Lagrange multipliers to find the minimal and maximal values of the function

$$f^{\prime }(x,y,z)=x^{\frac{5}{2}}+y^{\frac{5}{2}}+z^{\frac{5}{2}}$$

on the sphere $x^2+y^2+z^2=1$

六

Compute the integral $I={\iint }_D{x}^2{y}^{2}dxdy$, where $D$ is the plane domain bounded by the curves $x=\frac{1}{2}$ and $y^2=2x$

七

Determine the area of the region bounded by the curves $r=\sin \theta$ and $r=\cos \theta$

八

Find the volume of the solid bounded by the surfaces $S_1$ and $S_2$,

$$\begin{array}{rl}& S_1:=\{(x,y,z):x^2+y^2+4z^2=9,z\ge 0\},\\ & S_2:=\{(x,y,z):z=\sqrt{x^2+y^2}\}\end{array}$$

九

Determine the work done by the vector field

$$F=e^{y+2z}(i+xj+2xk)$$

along the curve of the intersection of the surfaces $x^2+2y^2+3z^2=3$ and $x+y+z=0$ joining the points $A(1,-1,0)$ and $B(-1,1,0)$

十

Calculate the circulation of the vector field

$$F=y{z}^{2}i+2x{z}^{2}j+xyzk$$

along the curve of intersection of the sphere $x^2+y^2+z^2=1$ and the cone $z=\sqrt{x^2+y^2}$ traversed in the counterclockwise direction around the $z$-axis when viewed from above