105年 - 105 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)#113221

(a) (5%) The value g(1) must be 2. True or false? Answer:_____

(b) (10%) The derivative g'(1)=_____

(a) (5%) Let s(t) be the distance (in meters) between the two particles in t minutes. s(t)=_____

(b) (10%) Suppose at t= t₀ the two particles are closest to each other. Then t₀=_____

3. (10%) Compute the indefinite integral =_____

4. (10%) Let Ω be a regioin (on the xy-plane) enclosed by , x=0 and y=e. Let S be the solid obtained by revolving Ω about the y-axis. Let V be the volume of S. V=_____

(a) (10%) If the unit vector u (in the xy-plane) at p is the direction (among all directions at p) along which the height (i.e. the value of z) of S increases most rapidly, then u =_____

(b) (10%) Write H for the plane x+2y+3z=1 and the curve C for the intersection S∩H. Let L be the tangent line to C at P and N be the plane perpendicular to L at P. Then the equation of N is _____

(a) (10%) Let A be the area of R. A=_____

(b) (5%) The double integral =_____

7. (15%) Let Ω be the region {(x, y)|(x-1)²+y²＜1}. Evaluate the double integral