2020春高数下期末试题(回忆版)

一

Determine whether the following statements are true or false? No justification is necessary

1-1

Equation $r=2\sin (\theta )(0\le \theta \le \pi )$ in polar form is a circle of radius 1 centered at $(0,1)$

1-2

If $f(x,y)=\sin x+\sin y$, then for any direction $u$, the directional derivative of $f(x,y)$ satisfies $-\sqrt{2}\le \mathrm{Du}f(x,y)\le \sqrt{2}$

1-3

If $u\ne 0$, and if $u\times v=u\times w$ and $u\cdot v=u\cdot w$, then $v=w$

二

Multiple Choice Qquestions : (only one correct answer for each of the following questions.)

2-1

Let $R:(x-1{)}^2+y^2\le 1$, then the integral $\iint Rf(x,y)dA$ is not equal to

(A) ${\int }_0^2{\int }_{-\sqrt{2x-x^2}}^{\sqrt{2x-x^2}}f(x,y)dydx$

(B) ${\int }_{-1}^1{\int }_{1-\sqrt{1-y^2}}^{1+\sqrt{1-y^2}}f(x,y)dxdy$

(C) ${\int }_0^{2\pi }{\int }_0^{1}f(1+r\cos \theta ,r\sin \theta )\cdot rdrd\theta$

(D) ${\int }_0^{2\pi }{\int }_0^{2\cos \theta }f(r\cos \theta ,r\sin \theta )\cdot rdrd\theta$

2-2

Which formula satisfies the conditions that function $f(x,y)$ has both partial dervivaties at $(0,0)$ when $f(0,0)=0$ ?

(A) $\frac{xy}{x^2+y^2}$

(B) $\frac{x^2-y^2}{x^2+y^2}$

(C) $\sqrt{x^2+y^2}\sin \frac{1}{x^2+y^2}$

(D) $\frac{x^4+y^2}{x^2+y^2}$

2-3

If $f(x,y)=3x+4y-a{x}^2-2a{y}^2-2bxy$ has only local maxima, then

(A) $2a^2>b^2$, and $a<0$

(B) $2a^2>b^2$, and $a>0$

(C) $2a^2<b^2$, and $a<0$

(D) $2a^2<b^2$, and $a>0$

三 Please fill in the blank for the questions below

3-1

If a plane is tangent to the surface $x^2-2y^2+z^2=2$, and parallel to $x-y+2z=0$, then the equation of the plane is $\underset{―}{}$

3-2

Let $f(x,y,z)={\left(\frac{x}{y}\right)}^{\frac{1}{2}}$, then af $(1,1,1)=$ $\underset{―}{}$

3-3

The equation of the plane through the line $x=-1+2t,y=3+t,z=-t$ and parallel to the line $x=-2t,y=t,z=1-t$ is $\underset{―}{}$

3-4

The circulation of the field $F=\mathrm{\nabla }\left(x{y}^2{z}^3\right)$ around the ellipse

$$C:r(t)=(\cos t)i+(4\sin t)j,0\le t\le 2\pi \text{, }$$

is $\underset{―}{}$

四

Find the area of region that lies inside the circle $r=3\sin \theta$ and outside the cardioid $r=1+\sin \theta$

五

Find the points on the curve

$$r(t)=(12\sin t)i-(12\cos t)j+5tk$$

at a distance $26\pi$ units along the curve from the point $(0,-12,0)$

六

Find the interval of convergence of the power series $\sum _{n=0}^{\mathrm{\infty }}\frac{2^n}{\ln (n+2)}{x}^n$

七

Find the real numbers $a,b(b\ne 0)$, which satisfy

$$\underset{x\to 0}{lim}\frac{\cos (\sin x)-\sqrt{1-x^2}}{x^a}=b$$

八

Find the absolute maximum and minimum values of $f(x,y)=e^{-x^2-y^2}(x^2+2y^2)$ on the close disk $x^2+y^2\le 4$

九

Evaluate the integal $\iiint z\sqrt{x^2+y^2+z^2}dV$, where $D$ is the solid bounded above by $z=1$ and below by $z=\sqrt{x^2+y^2}$

十

Calculate the line integral ${\int }_L\sin 2xdx+2(x^2-1)ydy$, bee here $L$ is the curve $y=\sin x$, from $(0,0)$ to $(\pi ,0)$

十一

Use the Stokes' Theorem to calculate the circulation of the field $F$ around the curve $C$ in the indicated direction, here $F=yi+xzj+x^{2}k$, and $C$ is the boundary of the triangle cut from the plane $x+y+z=1$ by the first octant, counterclockwise when viewed from above

十二

Use the Divergence Theorem to find the outward flux of $F$ across the boundary of the region $D$, here $F=x^{2}i+y^{2}j+z^{2}k$; and $D$ is the region cut from the solid cylinder $x^2+y^2\le 4$ by the planes $z=0$, and $z=1$