所屬科目：研究所、轉學考(插大)-微積分
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 $F'(1)$.
2. (12%) If $f$ and $g$ are the functions whose graphs are shown, let $Q(x) = f(x)/g(x)$ and $C(x) = f(g(x))$. Find (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 $R_1(1, 2)$.
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