PastExamLabPastExamLab

成功大學 85 年度 微積分

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第 1 題10 分
A function ff is defined by f(x)=lim⁡n→∞x2n−1x2n+1f(x) = \lim\limits_{n \to \infty} \frac{x^{2n} - 1}{x^{2n} + 1}. Where is ff continuous?
第 2 題16 分
(a)8 分
Suppose ff is an even function and is differentiable at 00. Prove that f′(0)=0f'(0) = 0.
(b)8 分
Suppose ff is an odd function and differentiable everywhere. Prove that for every positive number aa, there exists a number cc in (−a,a)(-a, a) such that f′(c)=f(a)af'(c) = \frac{f(a)}{a}.
第 3 題10 分
Let f(x)={−x3−x2+x+1for x≤0e−xfor x>0f(x) = \begin{cases} -x^3 - x^2 + x + 1 & \text{for } x \leq 0 \\ e^{-x} & \text{for } x > 0 \end{cases}. Find the local maximum and minimum values of ff.
第 4 題16 分
(a)8 分
Evaluate the improper integral ∫0∞xe−xdx\int_0^{\infty} xe^{-x} dx.
(b)8 分
Show that there exists a positive number cc such that ∫0cxe−xdx=12∫0∞xe−xdx\int_0^c xe^{-x} dx = \frac{1}{2} \int_0^{\infty} xe^{-x} dx.
第 5 題10 分
If f(x)=sin⁡(x3)f(x) = \sin(x^3), find f(99)(0)f^{(99)}(0) and f(100)(0)f^{(100)}(0).
第 6 題10 分
Find the points on the sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1 where the tangent plane is parallel to the plane 2x+y−3z=32x + y - 3z = 3.
第 7 題12 分
Let (a,b,c)(a, b, c) be a fixed point in the first octant. Find the plane through this point that cuts off from the first octant the tetrahedron of minimum volume, and determine the resulting volume. Hint: Let the plane be xA+yB+zC=1\frac{x}{A} + \frac{y}{B} + \frac{z}{C} = 1.
第 8 題16 分
(a)8 分
Use polar coordinate to evaluate ∫02∫x4−x211+x2+y2dydx\int_0^{\sqrt{2}} \int_x^{\sqrt{4-x^2}} \frac{1}{1 + x^2 + y^2} dy dx.
(b)8 分
Evaluate the line integral ∫Cx3ydx−xdy\int_C x^3 y dx - x dy where CC is the circle x2+y2=1x^2 + y^2 = 1 with counterclockwise orientation.
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