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成功大學 83 年度 微積分

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第 1 題15 分
Let [x][x] denote the greatest integer function on R\mathbb{R}. Show that lim⁡x→∞[x]2x=∞\lim\limits_{x \to \infty} \frac{[x]^2}{x} = \infty. Is ∑n=2∞1nln⁡n\sum\limits_{n=2}^{\infty} \frac{1}{n \ln n} convergent?
(a)8 分
Let [x][x] denote the greatest integer function on R\mathbb{R}. Show that lim⁡x→∞[x]2x=∞\lim\limits_{x \to \infty} \frac{[x]^2}{x} = \infty.
(b)7 分
Is ∑n=2∞1nln⁡n\sum\limits_{n=2}^{\infty} \frac{1}{n \ln n} convergent?
第 2 題15 分
Suppose a>0a > 0. Evaluate ∫0aexex+ea−xdx\int_0^a \frac{e^x}{e^x + e^{a-x}} dx. Suppose a,b>0a, b > 0 and RR is the region {(x,y):x2a2+y2b2≤1}\left\{(x,y): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\right\} in the plane. Evaluate ∬R(x2a2+y2b2)dxdy\iint_R \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right) dx dy.
(a)7 分
Suppose a>0a > 0. Evaluate ∫0aexex+ea−xdx\int_0^a \frac{e^x}{e^x + e^{a-x}} dx.
(b)8 分
Suppose a,b>0a, b > 0 and RR is the region {(x,y):x2a2+y2b2≤1}\left\{(x,y): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\right\} in the plane. Evaluate ∬R(x2a2+y2b2)dxdy\iint_R \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right) dx dy.
第 3 題10 分
Find the limit as xx approaches 00 of the ratio of the area of the triangle to the total shaded area in Figure 1.
第 3 題圖表
第 4 題17 分
Find the points on the paraboloid z=4x2+y2z = 4x^2 + y^2 at which the tangent plane is parallel to the plane x+2y+z=6x + 2y + z = 6. Suppose that a particle moving on a metal plate in the xyxy-plane has velocity v⃗=(1,−4)\vec{v} = (1,-4) (cm/sec) at the point (3,2)(3,2). If the temperature of the plate at points in the xyxy-plane is T(x,y)=y2ln⁡xT(x,y) = y^2 \ln x, (x≥1)(x \geq 1), in degrees Celsius, find dTdt\frac{dT}{dt} at (3,2)(3,2), where tt denotes time.
(a)9 分
Find the points on the paraboloid z=4x2+y2z = 4x^2 + y^2 at which the tangent plane is parallel to the plane x+2y+z=6x + 2y + z = 6.
(b)8 分
Suppose that a particle moving on a metal plate in the xyxy-plane has velocity v⃗=(1,−4)\vec{v} = (1,-4) (cm/sec) at the point (3,2)(3,2). If the temperature of the plate at points in the xyxy-plane is T(x,y)=y2ln⁡xT(x,y) = y^2 \ln x, (x≥1)(x \geq 1), in degrees Celsius, find dTdt\frac{dT}{dt} at (3,2)(3,2), where tt denotes time.
第 5 題15 分
Suppose f:[0,∞)→Rf:[0,\infty) \to \mathbb{R}: f(x)=∫x∞dt1+t4f(x) = \int_x^{\infty} \frac{dt}{\sqrt{1+t^4}}. Show that ff is a one-to-one function. Let f−1f^{-1} denote the inverse of ff. Find (f−1)′(0)(f^{-1})'(0).
(a)5 分
Show that ff is a one-to-one function.
(b)10 分
Let f−1f^{-1} denote the inverse of ff. Find (f−1)′(0)(f^{-1})'(0).
第 6 題15 分
Let Γ\Gamma be the circle with radius 33 and center at the origin. A particle travels once around Γ\Gamma in counterclockwise direction under the force field F⃗(x,y)=(y4,x3+3xy2)\vec{F}(x,y) = (y^4, x^3 + 3xy^2). Use Green's theorem to find the work done by F⃗\vec{F}.
第 7 題13 分
In the plane let LL be a line and Γ\Gamma an ellipse that forms the boundary of a bounded region RR. Use the intermediate-value theorem to show that there is a line parallel to LL that cuts RR into two pieces of equal area.
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