106年 - 106 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_資訊工程學系2年級、通訊工程學系2年級：微積分#147503

1. (20%) Find $dy/dx$ where

(a) \( \cos (xy) = 1 + \sin y \)
(b) \( y = \int_0^{x^2 -1}(t + 1)^2 e^{t^2 +2t + 1}dt. \)
(c) \( y = \cos^{-1}(2x + 1) \)
(d) \( y = \tanh (1 + e^{2x}) \)

2. (5%) Find \( (f^{-1})'(2) \) if \( f(x) = 1 + 2x + e^x \).

3. (25%) Sketch the graphs of \( y = \frac{x}{1 - \ln x} \).

(You need to describe (a) domain, (b) intercepts, (c) asymptotes (horizontal and vertical if any), (d) intervals of increasing and decreasing, (e) local maximum and minimum values, (f) interval of concavity, (g) points of inflection and (h) sketch the curve.)

4. (10%) Evaluate the integral \( \int_{1}^{2} (\ln(x + 1))^{2} dx \).

5. (10%) Given the power series \( \sum_{n=1}^{\infty} \frac{1}{2 \cdot 4 \cdot 8 \cdots (2n)} x^n \), (a) find the series' radius and interval of convergence. For what values of \( x \) does the series converge (b) absolutely, (c) conditionally?

6. (15%) Use the method of Lagrange multipliers to find the maximum and minimum values of \( x^{2} + y^{2} \) subject to the constraint \( x^{2} - 2x + y^{2} - 4y = 0 \).

7. (15%) Find the area of the region \( R \) enclosed by the lines \( y = x - 2 \) and \( y = -x \) and the curve \( y = \sqrt{x} \).