103年 - 103 國立臺灣師範大學_學士班轉學生招生考試試題_資訊工程學系二年級：微積分#121937

1. (10 points) Solve the equation
xy' = x² + 3y, x ＞ 0,
given the initial condition y(1) = 2.

(b) (8 points) Find the solution to = 2xy(y² + 1) that satisfies y(0) = 1.

(c) (6 points) Find lim_ₓ→∞.

(a) (8 points) Find the convergence radius and interval of the series

(b) (8 points) Find the Taylor series generated by f(x) = at x = 1.

4. (10 points) The vector r(t) = (2t³ - 3t²)i + (t³ - 12t)j gives the position of a moving particle in the plane at time t. Write an equation for the line tangent to the path of the particle at the point where t = -1.

(a)

(b) ∫dx.

(a) (8 points) Find dy/dx if y =dt.

(b) (8 points) Find d²y/dx² if 2x³ - 3y² = 8.

7. (10 points) The 8 ft wall shown in Fig. 1 stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

103年 - 103 國立臺灣師範大學_學士班轉學生招生考試試題_資訊工程學系二年級：微積分#121937

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