108年 - 108 國立臺北大學_碩士班一般入學考試試題_財政學系：微積分#146660

1. (16%) (a) Find $\lim_{x\to\infty} \frac{\ln x}{\ln\sqrt{x}+10}$.

(b) Find $\lim_{x\to 1} \frac{x^2-1}{|x-1|}$ if it exists.

2. (14%) (a) What is the definition of the derivative of a function f at a point $x_0$?

(b) Using the definition to calculate the derivatives of the function:
$f(x) = \sqrt{2x+1}.$

3. (10%) Find the constants a and b such that the function $f(x) = \begin{cases} x^2 - 2x, & x \le 1 \\ ax + b, & x > 1 \end{cases} \text{ is differentiable at } x=1.$

4. (10%) Find the integral: $\int \ln x \, dx$.

5. (10%) If $f(x) = \frac{g(x)}{1+x^2e^x}$, where g(1) = g'(1) = e, find f'(1).

6. (10%) Use logarithmic differentiation to find f'(x) if $f(x) = \frac{e^{-3x}\sqrt{2x-5}}{(6-5x)^4}$.

7. (10%) Find the vertical and horizontal asymptotes of the graph of $f(x) = \frac{1-x^2}{8-(\sqrt{2}x)^2}.$

8. (10%) Find the areas of the regions enclosed by the curve $y^2 = 4x+4$ and line $y = 4x-16$.

9. (10%) Evaluate the integral: \( \int_{0}^{2} \int_{0}^{4-x^2} \frac{xe^{2y}}{4-y} \, dy \, dx \).