PastExamLabPastExamLab

成功大學 109 年度 微積分B

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第 1 題10 分
Find the following limits.
(a)5 分
lim⁡x→2x+4x−7\lim\limits_{x \to 2} \frac{x + 4}{x - 7}
(b)5 分
lim⁡x→0(1−7x)1x\lim\limits_{x \to 0} (1 - 7x)^{\frac{1}{x}}
第 2 題10 分
Evaluate the following integrals
(a)5 分
∫0π4xsec⁡2x dx\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx
(b)5 分
∫45x−3x2−6x+13 dx\int_4^5 \frac{x - 3}{\sqrt{x^2 - 6x + 13}} \, dx
第 3 題10 分
Evaluate ∂u∂x∣(0,1)\frac{\partial u}{\partial x}\big|_{(0,1)} for
u(x,y)=(ex−y6)∫−22x4−t2 dtu(x,y) = \left(e^x - \frac{y}{6}\right) \int_{-2}^{2x} \sqrt{4 - t^2} \, dt
Note: Answer must in numerical expression and natural constants like π\pi, ee, ... etc.
第 4 題10 分
Evaluate ∂u∂y∣(2,0)\frac{\partial u}{\partial y}\big|_{(2,0)} for
u(x,y)=h(x2+y2,3x−4y)u(x,y) = h(x^2 + y^2, 3x - 4y)
where
h(s,t)=t4s−t2+tan⁡sln⁡sh(s,t) = \frac{t}{4s} - t^2 + \frac{\tan s}{\ln s}
第 5 題10 分
Find the volume of the solid generated by rotating the curve
y=sin⁡(x2)y = \sin(x^2)
over 0≤x≤10 \leq x \leq 1 around the yy-axis.
第 6 題10 分
Evaluate the infinite sum
∑k=0∞1(k+2)k!\sum\limits_{k=0}^{\infty} \frac{1}{(k+2)k!}
by some manipulations of the Taylor series of f(x)=xexf(x) = xe^x.
第 7 題10 分
A dog is running along a semi-circular track with radius 1 km in counterclockwise direction with speed 0.1 km per minute (See Figure). Let hh be the distance between the dog and point AA. Find the rate of change of hh at point DD, half way between BB and CC.
第 7 題圖表
第 8 題10 分
Find (f−1)′(5)(f^{-1})'(5) for
f(x)=x5+2x3+2xf(x) = x^5 + 2x^3 + 2x
第 9 題10 分
Evaluate
∫01∫1−x29−x2ex2+y2 dy dx+∫13∫09−x2ex2+y2 dy dx\int_0^1 \int_{\sqrt{1-x^2}}^{\sqrt{9-x^2}} e^{x^2+y^2} \, dy \, dx + \int_1^3 \int_0^{\sqrt{9-x^2}} e^{x^2+y^2} \, dy \, dx
第 10 題10 分
Use Lagrange multiplier to find the extreme value of
f(x,y,z)=exyzf(x,y,z) = e^{xyz}
subject to the constraint
x3−y3+z3=24x^3 - y^3 + z^3 = 24
Also, indicate the value(s) you obtain is (are) maximum or minimum.
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