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

(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.

(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 $

(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 $

(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. $$y = \sqrt[3]{x}, \quad y = 0, \quad x = 1 \quad \text{about the y-axis}$$

(a) $ (3\%) $ $ a_{n} = \frac{(-1)^{n}}{2\sqrt{n}} $

(b) $ (3\%) $ $ a_{n} = \frac{\cos^{2}n}{2^{n}} $

(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. $$f(x, y) = x^3 - 12xy + 8y^3$$

(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} $

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

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