107年 - 107 國立臺北大學_日間學士班曁進修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#147566

(a) \( (7\%) \) Find all critical values of \( f \)

(b) \( (6\%) \) Find \( f^{\prime \prime}(x) \) , the second derivative of \( f \)

(c) \( (6\%) \) Find all relative extrema of \( f \)

(d) \( (6\%) \) Find all points of inflection of \( f \)

(a) \( (7\%) \) \( \int_0^\infty x^{\frac{1}{2}}e^{-x}dx \)

(b) \( (8\%) \) \( \int_0^\infty x^{\frac{3}{2}}e^{-x}dx \)

(a) \( (7\%) \) Calculate the upper sum \( U(f,p) \) and the lower sum \( L(f,p) \) of \( f \) over the partition \( P \). Rounded the answers to the third decimal place.

(b) (6%) Approximate \( \int_0^1\frac{1}{1 + x^2} dx \) by the Simpson's rule with \( n = 4 \) and round to the third decimal place.

(c) (7%) Evaluate \( \lim_{n\to \infty}\left[\sum_{i=1}^{n}\frac{n}{n^2 + i^2}\right] \).

(a) \( (7\%) \) Find \( f_{t}(0,0) \) and \( f_{y}(0,0) \).

(b) \( (8\%) \) Explain \( f \) is differentiable at \( (0,0) \) or not.

(a) \( (7\%) \) \( \int_0^\infty \frac{\ln x}{x^2} dx \)

(b) \( (8\%) \) \( \iint_{\mathcal{F}}\frac{y}{1 + x^2} dA \) , where \( R \) is the region bounded by \( y = 0 \) \( y = \sqrt{x} \) \( x = 4 \)

(c) \( (10\%) \) \( \iint_{\mathcal{F}}\frac{xy}{\sqrt{1 + x^2y^2}} dA \) , where \( R \) is the region bounded by \( xy = 1 \) \( xy = 3 \) \( y = x \) \( y = 2x \) by the setting \( u = xy \) $v = \frac{y}{x}$.