104年 - 104 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_資訊工程學士、通訊工程學系學士班2年級：微積分#147542

科目：研究所、轉學考(插大)-微積分 | 年份：104年 | 選擇題數：0 | 申論題數：12

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (12)

1.(10%) The region enclosed by the curves $y = x^3$ and $y = \sqrt{x}$ is rotated about $x = -1$. Let $S$ denotes the volume of the resulting solid. Write the integral required to compute $S$ (no need to evaluate)

(a) with \(y\) as the variable of the integration, i.e write \(S = \int_{1}^{1} \dots dy\) (5%)
(b) with \(x\) as the variable of the integration, i.e write \(S = \int_{1}^{1} \dots dx\) (5%)
2.(20%) The Cobb-Douglas Model for the production function is $P = bL^q K^{1-q}$, where $b$ and $\alpha$ are positive constants and $\alpha < 1$. We want to maximize the production $P$ with the constraint that $mL + nK = q$, where $m$, $n$ and $q$ are positive constants. Show that the maximum production occurs when

$$L = \frac{\alpha q}{m}, \quad K = \frac{(1 - \alpha)q}{n}$$
3.(20%) Let $f(x) = x^2 \ln(1 + x)$. Compute $f^{(x)}(0)$, the $n^{th}$ derivative of $f$ evaluated at $x = 0$, by the following steps.

(a) Find the power series for \( f(x) \). (10%)
(b) Use part (a) to derive \( f^{(10)}(0) \). (10%)
4. (20%)Please evaluate the integrals:

(a) \(\int_0^9\frac{1}{\sqrt{x - 1}} dx\)
(b) \(\int \frac{\sqrt{x^2 - 9}}{x^2} dx\)
(c) \(\int \frac{x^2 - x + 6}{x^3 + 3x} dx\)
(d) \(\int_1^2 (\ln x)^2 dx\)
5. (20%)Please sketch the curve \( y = \frac{x^3 + 4}{x^2} \). (You must describe the domain, intercepts, symmetry, asymptotes, interval of increasing or decreasing, local maximum and minimum values, and concavity and points of inflection.)
6. (5%) Prove \(\lim_{x \to 2} \frac{x^2 + x - 6}{x - 2} = 5\) using the \(\varepsilon, \delta\) definition of a limit.
7. (5%) Find \(\frac{dy}{dx}\) if \(y\cos x = 1 + \sin (xy)\).