PastExamLabPastExamLab

成功大學 84 年度 微積分

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第 1 題21 分
Suppose f:R→R:f(x)=∫0xt23+t3dt+2f : \mathbb{R} \to \mathbb{R} : f(x) = \int_0^x t^2 \sqrt{3 + t^3} dt + 2.
(i)4 分
Determine where ff is concave up or concave down.
(ii)2 分
Show that ff has the inverse function f−1f^{-1}.
(iii)4 分
Determine the range of ff.
(iv)6 分
Determine the equation of the tangent line to the graph of f−1f^{-1} at (2,f−1(2))(2, f^{-1}(2)).
(v)4 分
Show that 12<f(0)<2\frac{1}{2} < f(0) < 2.
第 2 題10 分
Evaluate the following limits.
(i)5 分
lim⁡x→∞x−sin⁡xx\lim\limits_{x \to \infty} \frac{x - \sin x}{x}.
(ii)5 分
lim⁡k→∞∑i=1k1k+i\lim\limits_{k \to \infty} \sum\limits_{i=1}^k \frac{1}{\sqrt{k+i}}.
第 3 題15 分
(i)5 分
For any x∈Rx \in \mathbb{R}, find the sum of the series ∑k=0∞xk(1+x)k\sum\limits_{k=0}^{\infty} \frac{x^k}{(1+x)^k}.
(ii)10 分
Is the improper integral ∫01ln⁡x1−x2dx\int_0^1 \frac{\ln x}{1-x^2} dx convergent?
第 4 題15 分
Evaluate the following integrals.
(i)7 分
∫0111+xdx\int_0^1 \frac{1}{1+x} dx.
(ii)8 分
∬Ω(x+y)2d(x,y)\iint_\Omega (x + y)^2 d(x,y), where Ω\Omega is the parallelogram bounded by the lines x+y=0x + y = 0, x+y=1x + y = 1, 2x−y=02x - y = 0 and 2x−y=32x - y = 3.
第 5 題15 分
A curve CC in the plane is described by r⃗:[0,π]→R2:r⃗(t)=(−3cos⁡t,2sin⁡t)\vec{r} : [0,\pi] \to \mathbb{R}^2 : \vec{r}(t) = (-3\cos t, 2\sin t).
(i)8 分
Find the area of the region enclosed by the curve CC and the xx-axis.
(ii)7 分
Find the work done by the force F(x,y)=(y,−x)F(x,y) = (y, -x) in moving an object from (−3,0)(-3,0) to (3,0)(3,0) along CC.
第 6 題15 分
Let SS be the circular paraboloid x2+y2−z=1x^2 + y^2 - z = 1. Use the gradient to find
(i)10 分
the direction in which zz increases most rapidly at (1,2)(1,2) and this maximum rate of increase, and
(ii)5 分
the parametric equations of the normal line to SS at (1,2,4)(1,2,4).
第 7 題10 分
Suppose x,y∈(0,1)x, y \in (0,1) and satisfy x+y=1x + y = 1. Use the technique of Calculus to show that 2x+2y<32^x + 2^y < 3.
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