100年 - 100 國立臺灣師範大學_轉學生招生考試試題_機電工程學系二年級：微積分#122244

1. In the engine shown in Figure 1, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 rpm (revolutions per minute). Find the velocity (i.e., dx/dt) of the piston when θ = π/3. (15 分)
 Figure 1. The velocity of a piston is related to the angle of the crankshaft.

2. Find the derivative of , x ≠ 2 using logarithmic differentiation technique. (10 分)

3.Evaluate dx. (10 分)

(b) ⋅ ∇f；

(c)curl；

6.Evaluate ∫_Cydx + x²dy, where C is the parabolic arc given by y = 4x - x² from (4, 0) to (1, 3), as shown in Figure 3. (15 分)

(a) Show that vector = (y²cosx + z³)+ (2ysinx - 4)+ (3xz² + 2) is a conservative field. (8 分)

(b) Find the scalar potential function for . (7 分)