109年 - 109 國立臺北大學_碩士班一般入學考試試題_財政學系：微積分#146637

(a) $\lim_{x\to 1}\frac{x^{50} - 1}{x - 1}$.

(b) $\lim_{x\to \infty}\frac{x + \sin x + 2\sqrt{x}}{x + \sin x}$.

2. (10%) Find h'(0) if $h(x) = \left[\frac{g(x) - x}{3 + g(x)}\right]^2$, where $g(0) = 3$$ and $g'(0) = -2$.

3. (10%) Find $$dy/dx$$ for the equation $$xy^3 + (2y)^2 - 5y = \frac{(x^2 - 9)}{\sqrt{x^2 + 2}}$$.

4. (20%) Analyze and sketch the graph of $$f(x) = \frac{2(x^2 - 9)}{x^2 - 4}$$. (Label the intercepts, the coordinates of any local and absolute extreme points, inflection points, and asymptotes if any.)

5. (10%) $$\int_{1}^{2} (\ln(x + 1))^2 dx$$.

6. (10%) Set up the integral $$\int_{R} \int \frac{y}{1 + x^2} dA$$ for both orders of integration and use the more convenient order to evaluate the integral over the region $$R$$ bounded by $$y = 0, y = \sqrt{x}, x = 4$$.

(a) $\sum_{n=1}^{\infty} \frac{4^n}{3^n + 1}$,

(b) $\frac{1}{1} - \frac{2}{1} + \frac{4}{2} - \frac{8}{6} + \frac{16}{24} - \cdots$

8. (10%) Find the Taylor series generated by $f(x) = 2^x$ at $a = 1$.

9. (10%) Assume that $$w = f(s^3 + t^2)$$ and $$f'(x) = e^x$$. Find $$\frac{\partial w}{\partial t}$$ and $$\frac{\partial w}{\partial s}$$.