24-25-2-高等数学A（下）-期中

Q1

a)

Determine the convergence domain of the series $\sum_{n=0}^\infty\frac{x^n}{2^nn!}$ . [10 marks]

b)

Find the sum of the series $\sum_{n=0}^\infty\frac1{2^nn!}$ . [10 marks]

Q2

a)

Find the sine series $\sum_{n=1}^\infty b_n\sin nx$ of the function $f(x)=\frac\pi4-\frac x3$ defined on the interval $(0,\pi)$ . [10 marks]

b)

Calculate the sum function $S(x)$ of the above sine series $\sum_{n=1}^\infty b_n\sin nx$ on $[-\pi,\pi]$ . [5 marks]

Q3

Suppose

g(x,y)=\begin{cases} \frac{x^3+xy^2}{x^2-xy+y^2},&(x,y)\ne(0,0),\\ 0,&(x,y)=(0,0). \end{cases}

a)

Prove that the function $g(x,y)$ is continuous at the point $(0,0)$ . [10 marks]

b)

Determine if the function $g(x,y)$ is differentiable at point $(0,0)$ . [10 marks]

Q4

Suppose that the second derivative of the function $f(x)$ is continuous and $z(x,y)=\frac1xf(xy)+yf(x^2+y^2)$ .

a)

Find all partial derivatives of the function $z(x,y)$ . [10 marks]

b)

Find $\frac{\partial^2z}{\partial x\partial y}$ . [10 marks]

Q5

Suppose that the plane $\pi$ is tangent to the surface $S:x^2+2y^2+3z^2=6$ and parallel to the plane $x+4y-3z=0$ . Find the equation(s) of the plane $\pi$ . [10 marks]

Q6

Assume that area of $\Delta ABC$ is constant $M$ and the sides of $\Delta ABC$ have lengths $a,b$ and $c$ , respectively. Construct three perpendicular lines from a point $P$ inside $\Delta ABC$ to its sides. Please determine the location of point $P$ that maximizes the product of these three perpendiculars. [15 marks]