103年 - 103 國立臺灣大學轉學生招生考試試題_(商、管、農、經學院聯招) ：微積分(C)#113224

1. Find the value of c such that the limit exists and evaluate the limit.

2. Find the extreme values of f(x) = for x ＞ 0.

3. Evaluate the indefnite integral.

4. Find an equation for the line through the origin(原點) tangent to the graph of y =ln x.

5. Find the sum of the series or show that it diverges.

6. (a) Give the defnition of the directional derivative of f(x, y) at (a,b) in the direction of a unit vector u = (u,v), u² + v² = 1.

(b) Show that the directional derivative of the following func- tion exists in every direction u = (u, v) at (0,0), where

(c) Show that the above function f(x, y) is not differentiable at (0,0)

7. Use the transformation z = u/v, y = v to evaluate the double integral

where R is the region in the first quadrant(象限) bounded by the lines y = x, y = 3x, and the hyperbolas xy = 1, xy = 3.

103年 - 103 國立臺灣大學轉學生招生考試試題_(商、管、農、經學院聯招) ：微積分(C)#113224

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