112年 - 112 國立臺北大學_日間學士班轉學生招生考試試題_經濟學系日間學士班2年級：微積分#147828

1. If $g(x) = xe^{2x+1}$, then  g'(x) = _______ (1) _______.

2. If $g(x) = e^{2\pi i(2x)}$, then  g'(x) = _______ (2) _______.

3. Solve the initial value problem $y' + y = e^x$, $y(0) = 5$. Then, we have y = _______ (3) _______.

4. Suppose $2^x = 3(5^x)$. Then, x = _______ (4) _______.

5. The value of $\int_0^4 \frac{dx}{x^2 - 1}$ is _______ (5) _______.

6. The value of $\lim_{x \to 0} \frac{x^2}{\sin^2 x}$ is _______ (6) _______.

(a) $\sum_{n=1}^{\infty} \frac{n^2 + 1}{5n^2 + 9}$

(b) $\sum_{n=1}^{\infty} (-1)^{n-1} e^{-n}$ (c) $\sum_{n=1}^{\infty} \frac{n!}{n^2}$

8. Consider the conic section $4x^2 + y^2 + 8x - 6y + 4 = 0$. Then its center point (x, y) = _______ (8) _______.

9. Do the Cartesian coordinates $(1, 1)$ match the polar coordinates $(\sqrt{2}, \frac{\pi}{4})$? _______ (9) _______ (Yes or No)

10. If $f(x) = \frac{1}{x}$, then $f''(1) =$ _______ (10) _______.

1. Find the general Taylor polynomial formula for $e^{-x^2}$ around x = 0. _______ (1) _______

2. The average value of function $f(x) = \sin(\pi x)$ over the interval $[0, 1/2]$ is _______ (2) _______.

3. The value of $\int_0^1 \frac{dx}{2x^2 + 5x + 2}$ is _______ (3) _______.

4. The range (or interval) of $x$ that makes series $\sum_{n=0}^{\infty} \frac{(-2)^n (x+4)^n}{n+3}$ convergent is _______ (4) _______.

5. Compute the area enclosed between the graphs $y = 1 - x^2$ and y = 3 - 3x. _______ (5) _______.