PastExamLabPastExamLab

成功大學 106 年度 微積分A

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第 1 題10 分
Given a curve CC in R2\mathbb{R}^2 defined by
ln⁡(1−x3+y3)−4=0.\ln(1 - x^3 + y^3) - 4 = 0.
Find the point on CC at which the tangent line is vertical.
第 2 題10 分
If the function
f(x)={sin⁡(4x)+a−2b3xx≠02a+bx=0f(x) = \begin{cases} \frac{\sin(4x)+a-2b}{3x} & x \neq 0 \\ 2a + b & x = 0 \end{cases}
is continuous at x=0x = 0, then (a,b)=?(a, b) = ?
第 3 題10 分
Write down the first three terms (three lowest order terms) of the Taylor series of tan⁡−1(2x)1−x\frac{\tan^{-1}(2x)}{1-x} at 00. (Hint: tan⁡−1u=∫?du\tan^{-1} u = \int ? du)
第 4 題10 分
Evaluate the following integral:
∫01∫x1xcos⁡(y5+2) dy dx.\int_0^1 \int_{\sqrt{x}}^1 x \cos(y^5 + 2) \, dy \, dx.
第 5 題10 分
From the equation
ex2+y2sin⁡(2x)=4y,e^{x^2} + y^2 \sin(2x) = 4y,
Solve dydx\frac{dy}{dx} in terms of xx and yy
第 6 題10 分
Find all values of aa so that the series
∑n=1∞sin⁡(1n3a−1+3)\sum\limits_{n=1}^{\infty} \sin \left( \frac{1}{n^{3a-1} + 3} \right)
is divergent.
第 7 題10 分
Compute the following improper integral
∫0∞e−4x2dx.\int_0^{\infty} e^{-4x^2} dx.
第 8 題10 分
Compute the line integral
∮CF⃗⋅dr⃗,\oint_C \vec{F} \cdot d\vec{r},
where
F⃗(x,y)=(4y+6ye2x,6x+3e2x)\vec{F}(x,y) = (4y + 6ye^{2x}, 6x + 3e^{2x})
and CC is the closed loop formed by traveling from (−2,0)(-2, 0) to (4,0)(4, 0) to (3,3)(3, 3) to (−1,3)(-1, 3) and back to (−2,0)(-2, 0) by straight lines.
第 9 題10 分
Given the function F:R3→RF : \mathbb{R}^3 \to \mathbb{R} by
F(x,y,z)=ex+y2+cos⁡z.F(x, y, z) = e^{x+y^2+\cos z}.
At (0,0,0)(0, 0, 0), find the direction along which the function decreases most rapidly and find the corresponding rate of change.
第 10 題10 分
A rectangular box is formed by cutting four equal corners from a square of side 3 and then folding up (see the figure below). Find the maximum possible volume of the box.
第 10 題圖表
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