113年 - 113 國立臺北大學_日間學士班轉學生招生考試試題_資訊工程學系日間學士班2年級：微積分#147425

1. (12%) If $F(x) = f(x^3 f(x^3 f(x^3)))$, where f(1) = 2, f(2) = 3, f'(1) = 4, f'(2) = 5, and f'(3) = 6, find F'(1).

(a) Q'(4) and 

(b) C'(4).

3. (12%) Evaluate $\int_0^1 x \sqrt{1-x^3} \, dx$

4. (12%) Calculate $\int_0^1 x \tan^{-1} x \, dx$

5. (13%) Find the solution of the initial-value problem $(sec \, x)y' + y = e^{-\sin x}$, x > 0, and y(0) = 2.

6. (13%) Suppose you start at the point (0, 0, 3) and move 5 units along the curve $x = 3 \sin t$,y = 4t, $z = 3 \cos t$ in the positive direction. Where are you now?

7. (13%) Use spherical coordinates to find the volume of the solid that lies above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere $x^2 + y^2 + z^2 = z$.

8. (13%) Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and v are differentiable, u(1, 2) = 5, $u_0(1, 2) = 4$, $u_0(1, 2) = -3$, v(1, 2) = 7, $v_0(1, 2) = 2$, $v_1(1, 2) = 6$, $G_u(5, 7) = 9$, $G_v(5, 7) = -2$. Find $R_0(1, 2)$ and $R_1(1, 2)$.