111年 - 111 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年級級：微積分#147895

1. The value of $$\sum_{n=1}^{\infty} \frac{20}{n(n+2)}$$ is __(1)__.

2. If $$g(x) = x(e^{x^2})$$, then $$g'(x) =$$ __(2)__.

3. The value of $$\int_{-1}^{1} (x - 2|x - 1|) dx$$ is __(3)__.

4. If $$\int{1}^{3} f(x) dx = 2$$ and $$\int{1}^{3} f(x) dx = -1$$, then $$\int_{1}^{3} [2f(x) - 1] dx =$$ __(4)__.

5. Write down the range of $$a$$, which makes the series $$\sum_{n=1}^{\infty} (-1)^n \frac{a^{n+1}}{3^n}$$ converge. __(5)__

6. If $$\sum{n=1}^{\infty} cn x^n$$ is the Taylor polynomial of order 4 generated by $$f(x) = \cos\left(\frac{x}{2}\right)$$ at 0, then the constant in front of $$x^4$$, $$c_4 =$$ __(6)__.

7. Which of the following 3 series is/are bounded? __(7)__
(a) $$an = \frac{4n+1}{n+1}$$ (b) $$bn = n - \frac{1}{n}$$ (c) $$c_n = \frac{\sin n}{n}$$

8. Which of the following 3 series is/are convergent?__ (8)__

(a) $$\sum{n=1}^{\infty} e^{-n} n!$$ (b) $$\sum{n=1}^{\infty} \left(\frac{-2n}{n+2}\right)^{2n}$$ (c) $$\sum_{n=1}^{\infty} \frac{\sin n}{3^n}$$

9. The domain of $$f(x) = \sqrt{2 - x} + \sqrt{x + 5}$$ is __(9)__.

10. If $$f(x) = x + \cos x$$, then $$\left(f^{-1}\right)'(1) =$$__(10)__.

1. If the integral $$\int_{-\infty}^{\infty} \frac{c}{(1+x^2)} dx = 1$$ and $$c \ge 0$$, then the value of $$c$$ equals ___(1)__.

2. The area of the region bounded by the curves $$x = 1 - y^2$$ and $$x = y^2 - 1$$ is __(2)__.

3. The slope of the tangent line at the point (1, 1) on the graph of $$e^{x^2 - y^2} = 2x^2 - y^2$$ equals __(3)__.

4. The value of $$\lim_{x \to \infty} \left(\sqrt{x^2 + 4x} - x\right)$$ equals__ (4)__.

5. The function $$f(x, y) = 2x^3 + 6xy^2 - 3y^3 - 150x$$ has four stationary points. Among the four, the two saddle points are __(5)__.