115年 - 115 國立臺北大學_日間學士班轉學生招生考試試題_統計學系日間學士班2、3年級：微積分#147568

科目：研究所、轉學考(插大)-微積分 | 年份：115年 | 選擇題數：0 | 申論題數：22

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所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (22)

1. (10%) If the functions are given, answer the following questions:

$$f(x) = x^2 - x - \ln x$$

(a) \((2\%)\) Find the critical numbers of the function.
(b) \((2\%)\) Find the local maximum and minimum values of the function.
(c) \((2\%)\) Find the intervals of concavity and the inflection points.
(d) \((4\%)\) Sketch the curve.
2. (15%) Evaluate the definite integral.

(a) \((5\%)\) \(\int_0^1\cos (\pi x / 2)dx\)
(b) \((5\%)\) \(\int_1^2 (\ln x)^2 dx\)
(c) \((5\%)\) \(\int_0^1\frac{2}{2x^2 + 3x + 1} dx\)
3. (15%) Evaluate the indefinite integral.

(a) \((5\%)\) \(\int e^{x}\sqrt{1 + e^{x}} dx\)
(b) \((5\%)\) \(\int (x^{2} + 2x)\cos xdx\)
(c) \((5\%)\) \(\int e^{2x}\sin (3x)dx\)
4. (8%) Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) \((4\%)\) \(\int_{-\infty}^{\infty}\frac{x^2}{9 + x^4} dx\)
(b) \((4\%)\) \(\int_{1}^{\infty}\frac{x + 1}{\sqrt{x^4 - x}} dx\)
5. (6%) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$$y = \sqrt[3]{x}, \quad y = 0, \quad x = 1 \quad \text{about the y-axis}$$
6. (6%) Determine whether the sequence converges or diverges. If it converges, find the limit.

(a) \((3\%)\) \(a_{n} = \frac{(-1)^{n}}{2\sqrt{n}}\)
(b) \((3\%)\) \(a_{n} = \frac{\cos^{2}n}{2^{n}}\)
7. (6%) Find the radius of convergence and interval of convergence of the series.

(a) \((3\%)\) \(\sum_{n = 0}^{\infty}\frac{(x)^n}{n!}\)
(b) \((3\%)\) \(\sum_{n = 1}^{\infty}n!(2x - 1)^{n}\)
8. (8%) Find the critical points, the local maximum or minimum values, and the saddle points of the function.

$$f(x, y) = x^3 - 12xy + 8y^3$$
9. (6%) Let the equation be \( e^z = xyz \). Use implicit differentiation to find \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
10. \((20\%)\) Calculate the double integral.

(a) \((5\%)\) \(\int_0^1\int_0^{e^y}\sqrt{1 + e^y} dxdy\)
(b) \((5\%)\) \(\iint_{D}x^{3}dA,\quad D = \{(x,y)\mid 1\leq x\leq e,0\leq y\leq \ln x\}\)
(c) \((5\%)\) \(\int_0^4\int_{\sqrt{x}}^2\frac{1}{y^3 + 1} dydx\)
(d) \((5\%)\) \(\iint_{D}xy^{2}dA,\quad D\) is bounded by \(x = 0\) and \(x = \sqrt{1 - y^2}\)