PastExamLabPastExamLab

成功大學 87 年度 微積分

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第 1 題10 分
Evaluate the left-hand limit of the function 1[z2−1]\frac{1}{[z^2-1]} at 1, i.e. lim⁡z→1−1[z2−1]\lim\limits_{z \to 1^-} \frac{1}{[z^2-1]}, where [z2−1][z^2-1] is the Gauss function of z2−1z^2-1.
第 2 題10 分
Let P(x)=anxn+a1xn−1+⋯+an−1x+anP(x) = a_n x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n be an nn-th degree polynomial such that its all the coefficients a0,a1,…,ana_0, a_1, \ldots, a_n are reals and satisfy the equation an1+an−12+⋯+a0n+1=0\frac{a_n}{1} + \frac{a_{n-1}}{2} + \cdots + \frac{a_0}{n+1} = 0. Show that the equation P(x)=0P(x) = 0 has at least one real root between 0 and 1.
第 3 題10 分
Find the tangent equation for the function f(x)=x4−x3f(x) = \sqrt{x^4 - x^3}.
第 4 題20 分
Determine whether the following two improper integrals are convergent? If an improper integral is convergent, find its limit. Otherwise, explain why the improper integral is divergent?
(a)10 分
∫1+∞dxx(x−1)\int_1^{+\infty} \frac{dx}{x(x-1)}
(b)10 分
∫−10dx(x−3)x+1\int_{-1}^0 \frac{dx}{(x-3)\sqrt{x+1}}
第 5 題10 分
Find the radius of convergence and the interval of convergence of the power series ∑n=2+∞(−1)nx2n−12n\sum\limits_{n=2}^{+\infty} (-1)^n \frac{x^{2n-1}}{2n}.
第 6 題10 分
Find the Maclaurin series of the function sin⁡−1x\sin^{-1} x.
第 7 題10 分
Find the maximum and minimum values of f(x,y,z)=x−2y+zf(x, y, z) = x - 2y + z, with the constraints x2+y2+z2=1x^2 + y^2 + z^2 = 1 and x+y+z=0x + y + z = 0.
第 8 題10 分
Find the volume of the solid enclosed by the cone z=rz = r and the cylinder r=3sin⁡θr = 3 \sin \theta in the first-octant.
第 9 題10 分
Let SS be the positively directed curve of the ellipse x29+y24=1\frac{x^2}{9} + \frac{y^2}{4} = 1. Evaluate the line integral ∫S(x4−3y)dx+(4x+2y2)dy\int_S (x^4 - 3y)dx + (4x + 2y^2)dy.
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