PastExamLabPastExamLab

台灣聯合大學系統 106 年度 微積分 A3/A4/A7

原始 PDF
壓縮檔內：106t/微積分A3A4A7.pdf
中央大學圖書館考古題頁

甲、填充題

共 8 題，每題 8 分，共 64 分。請在答案卷上列出題號依序作答。請注意：本（甲、）部分，共 8 題，命題型態為填充題，不必詳列計算過程，惟若答案被包含在演算過程，將被視為試算流程，無法計分。
第 1 題8 分
How many critical points does the function f(x)=∣x2−1∣f(x) = |x^2 - 1| have?
第 2 題8 分
If lim⁡x→0ax+b−2x=1\lim\limits_{x \to 0} \frac{\sqrt{ax + b} - 2}{x} = 1, then a−b=?a - b = ?
第 3 題8 分
Suppose f′′f'' is continuous on [0,1][0, 1], f(1)=2f(1) = 2, f′(1)=2f'(1) = 2 and the average value of ff on [0,1][0, 1] is 22. Evaluate ∫01x2f′′(x)dx\int_0^1 x^2 f''(x) dx.
第 4 題8 分
Find g′(−12)g'(-\frac{1}{2}), where g(x)g(x) is the inverse of f(x)=x3x2+1f(x) = \frac{x^3}{x^2 + 1}.
第 5 題8 分
Find the directional derivative of f(x,y)=tan⁡−1(xy)f(x, y) = \tan^{-1}(\frac{x}{y}) at P0(1,1)P_0(1, 1) in the direction of v=22i+22j\mathbf{v} = \frac{\sqrt{2}}{2}\mathbf{i} + \frac{\sqrt{2}}{2}\mathbf{j}.
第 6 題8 分
Evaluate ∫06∫x/32xy3+1dydx\int_0^6 \int_{x/3}^2 x\sqrt{y^3 + 1} dy dx.
第 7 題8 分
Let RR be the solid inside x2+y2+z2=1x^2 + y^2 + z^2 = 1, outside z=x2+y2z = \sqrt{x^2 + y^2} and above the xyxy-plane. Find the volume of RR.
第 8 題8 分
Evaluate the line integral ∫C2xydx+(x2+2x)dy\int_C 2xy dx + (x^2 + 2x) dy, where CC: boundary of the region lying between the graphs of the ellipse x25+y24=1\frac{x^2}{5} + \frac{y^2}{4} = 1 and the circle x2+y2=1x^2 + y^2 = 1.

乙、計算、證明題

共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。
第 1 題12 分
Find any extrema of the function f(x,y)=e−xy/4f(x, y) = e^{-xy/4} subject to the constraint x2+y2≤1x^2 + y^2 \leq 1.
第 2 題12 分
For the function
f(x,y)={−5xyx2+y2,if (x,y)≠(0,0)0,if (x,y)=(0,0)f(x, y) = \begin{cases} \frac{-5xy}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases}
show that fx(0,0)f_x(0, 0) and fy(0,0)f_y(0, 0) both exist, but that ff is not differentiable at (0,0)(0, 0).
第 3 題12 分
Determine whether the series ∑n=3∞ln⁡nln⁡ln⁡n\sum\limits_{n=3}^{\infty} \frac{\ln n}{\ln \ln n} converges absolutely or conditionally, or diverges.