PastExamLabPastExamLab

成功大學 101 年度 微積分

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第 1 題12 分
Define f(x)={e−1x2if x≠00if x=0f(x) = \begin{cases} e^{-\frac{1}{x^2}} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}
(a))6 分
Find f′(0)f'(0)
(b))6 分
Is f′(x)f'(x) continuous at x=0x=0 ?
第 2 題10 分
Evaluate ∫01x2⋅ln⁡(x3+1)dx\int_0^1 x^2 \cdot \ln(x^3 + 1)dx
第 3 題12 分
Show that sin⁡−1x1−x2=∑n=0∞22n(n!)2(2n+1)!x2n+1\frac{\sin^{-1}x}{\sqrt{1-x^2}} = \sum\limits_{n=0}^{\infty} \frac{2^{2n}(n!)^2}{(2n+1)!} x^{2n+1}, ∀x:∣x∣<1\forall x: |x| < 1
第 4 題12 分
The cardioid r=2(1+cos⁡θ)r = 2(1 + \cos\theta) is rotated about the polar axis y=0y=0. Find the area of the surface generated.
第 5 題10 分
Evaluate ∫08∫x21y4+1dydx\int_0^8 \int_{\sqrt{x}}^2 \frac{1}{y^4+1} dydx
第 6 題10 分
Convert ∫02∫02x−x2(x2+y2)dydx\int_0^2 \int_0^{\sqrt{2x-x^2}} (x^2 + y^2)dydx to polar coordinates and evaluate.
第 7 題10 分
Find ∂w∂x\frac{\partial w}{\partial x} at the point (x,y,z)=(2,−1,1)(x,y,z) = (2,-1,1) if w=x2+y2+z2w = x^2 + y^2 + z^2 and z3−xy+yz+y3=1z^3 - xy + yz + y^3 = 1
第 8 題12 分
Two sides of a triangle are 10cm10^{cm} and 15cm15^{cm}, and are increasing at 3cm/sec3^{cm}/_{sec} and 4cm/sec4^{cm}/_{sec}, respectively, which the included angle is π3\frac{\pi}{3} and decreasing at 0.5rad/sec0.5^{rad}/_{sec}. Is the third side increasing or decreasing ? at what rate ?
第 9 題12 分
Suppose the utility of purchases of x,y,zx, y, z units of three different kinds of product is given by u=5x12y12z12u = 5x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}, where the price per unit of the products is 2,2, 5, and 1,respectively.Ifaconsumerhas1, respectively. If a consumer has 90 to spend, how many units of each product should be purchased to achieve maximum utility ?
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