103年 - 103 台灣聯合大學系統(清華、交通、陽明、中央四校聯招)學士班轉學生考試_A3、A4、A5、A7：微積分#112765

科目：研究所、轉學考(插大)-微積分 | 年份：103年 | 選擇題數：0 | 申論題數：11

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (11)

1. For what value of the constant m is differentiable at x = 0?

2. Find the horizontal asymptote of the graph of y = x tan.

3. Find the interval of convergence of the power series.

4. The function f(x) = e^x +x, being differentiable and one-to-one, has a differentiable inverse f^-1(x). Find the value of df^-1/dx at the point f(In 2).

5. A passenger ship and an oil tanker left port sometime in the morning; the former headed north, and the latter headed east. At noon, the passenger ship was 40 miles from port and sailing at 30 mph, while the oil tanker was 30 miles from the port and sailing at 20 mph. How fast was the distance between the two ships changing at that time?

6. Evaluate the integral.

7. Find the volume under the surface f(c, y) = e^-x2 , bounded by the xz-plane and the planes y = x and x = 1.

8. Find the work done by the force F = cyi + (y - t)j over the straight line from (1, 1) to (2,3).

1. Suppose that :c and y are related by the equation x = dt.Show that d2y/dx²is proportional to y and find the constant of proportionality.

2. Find the directional derivative of the function h(a, y, z) = cos xy+ e^yz+ ln zx at (1, 0, 1/2) in the direction of u = i+ 2j + 2k.

3. Find the critical point of f(x,u) = xy + 2x - In x²y in the open first quadrant (x ＞ 0, y ＞ 0) and show that f takes on a minimum there.