102年 - 102 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)#113227

(B). Find = __(2)__ and=__(3)__ at the point x = 1, y = 1 of the curve x³+3x²y+y³=5.

(C). Evaluate=________.

(D). Rotate the region [(x,y) : 0 ≤x ≤π Stand 0 ≤y ≤ sin(x) about the x-axis. Find the volume of the solid obtained.

(E). Let g(x) be the inverse function of the strictly increasing function f(x) = x⁵ + 3x³ + 1. Theng(x)dx = _________.

(F). The coefficient of x⁹ in the Taylor expansion of fo dt is________.

(G). Find the direction derivative of f(x, y) = In(x² + y²) at the point (1,2) in the direction u =. 

 (I). Evaluatedxdy= __(10)__.

 (J). Let f(x,y)=and g(u,u) = f(x(u,U),y(u, U), where x(u,บ) = u² -3uU + 2U² and y(u, ข) = น⁴+ 3นข³ - 4U⁴. Find at the point u = 1 and u= 1. 

(L). Evaluate dxdy, where Ω is the region in the first quadrant bounded by the lines x+y = 1 and x+y = 2. 

(M). Solve the differential equation = 2(1+t)(1+y²)with the initial condition dt y(-1)=0.

(N). Solve the differential equation with the initial condition y(1) =  2. Answer. __(15)__.