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臺灣綜合大學系統 111 年度 微積分A

本頁整理這份微積分考卷的題目、解答與詳解步驟，可直接對照每題內容與答案重點。

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這份試題整理

依本站已整理題目自動彙整題數、分數與題型，方便先判斷這份考卷的出題結構。

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題數
10
題
總分
100
分
已整理詳解
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微積分A 其他年度考古題

本科目共 9 個年度已整理題目，可直接切換查閱。

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第 1 題10 分
For what values of aa and bb, a,b≠0a, b \neq 0, is the following equation true?
lim⁡x→0(eax−1x2−ax+bx)=12\lim\limits_{x \to 0} \left(\frac{e^{ax} - 1}{x^2} - ax + \frac{b}{x}\right) = \frac{1}{2}
第 2 題10 分
(a)5 分
Let f(x)=sec⁡2xf(x) = \sec^2 x on (0,π2)(0, \frac{\pi}{2}) and f−1f^{-1} be the inverse function of ff.
Find (f−1)′(4)=(f^{-1})'(4) = ____________.
(b)5 分
Let f=f(x,y)f = f(x, y) be a differentiable function of xx and yy, and let x=rsx = rs, y=r+sy = r + s and h(r,s)=f(x,y)=f(rs,r+s)h(r, s) = f(x, y) = f(rs, r + s). Assume ∂f∂x(1,2)=2\frac{\partial f}{\partial x}(1, 2) = 2, ∂f∂y(1,2)=1\frac{\partial f}{\partial y}(1, 2) = 1. Find ∂h∂s(1,1)=\frac{\partial h}{\partial s}(1, 1) = ____________.
第 3 題10 分
If f(x)=∫0x2(1−t2)et2dtf(x) = \int_0^{x^2} (1 - t^2)e^{t^2} dt, on what interval(s) is ff increasing?
第 4 題10 分
Let a>0a > 0 be a constant. Evaluate
∫0ax2(x2+a2)3/2dx.\int_0^a \frac{x^2}{(x^2 + a^2)^{3/2}} dx.
第 5 題10 分
Find the area of the region that lies inside the curve r=4sin⁡θr = 4\sin\theta and outside the curve r=2r = 2.
第 6 題10 分
For what real values of pp does the series ∑n=2∞1npln⁡n\sum\limits_{n=2}^\infty \frac{1}{n^p \ln n} converge?
第 7 題10 分
Let f(x)f(x) be the function defined by the power series
∑n=0∞(x+2)2n(n+3)!.\sum\limits_{n=0}^\infty \frac{(x + 2)^{2n}}{(n + 3)!}.

Try to express f(x)f(x) as an elementary function.
第 8 題10 分
Find the global maximum and global minimum of f(x,y,z)=x2+y2+z2f(x, y, z) = x^2 + y^2 + z^2 on the surface x2+2y2+3z2=1x^2 + 2y^2 + 3z^2 = 1 by using the method of Lagrange multipliers.
第 9 題10 分
Evaluate ∬RxyexydA\iint_R \frac{x}{y} e^{xy} dA, where RR is the region in the first quadrant bounded by lines y=xy = x, y=3xy = 3x, and the hyperbolas xy=1xy = 1, xy=3xy = 3.
第 10 題10 分
Evaluate the line integral ∫C(1−y3)dx+(x3+e−y2)dy\int_C (1 - y^3)dx + (x^3 + e^{-y^2})dy, where CC is the arc of the circle x2+y2=4x^2 + y^2 = 4 traversed counter-clockwise from (2,0)(2, 0) to (−2,0)(-2, 0).