105年 - 105 國立臺北大學_日間學士班曁進修學士班轉學生招生考試試題_統計學系、經濟學系日間學士班曁進修學士班3年級：微積分#147562

1. (1) Evaluate $\frac{d}{dx}x^x =$ ________.

(2) Let \( f(x) = \left\{ \begin{array}{ll} \frac{\ln x}{x - 1}, & x > 0, x \neq 1 \\ 1, & x = 1 \end{array} \right. \). Then \( f'(1) = \)________

(3) If \( f(x) = x^{2} + 2x + 1 \), then \( (f^{-1})'(4) = \)________

(4) If \( x > 0 \), then \( \frac{d}{dx}\int_0^G e^{-x^2}dt = \)________

(5) Evaluate \( \lim_{x\to 0}\frac{e^x - 1 - x}{x^2} = \)________

(6) Evaluate \( \int_0^\pi x\sin xdx = \)________

(7) Evaluate \( \int_{x}^{x^2}\frac{1}{x\ln x} dx = \)________

(8) The radius of convergent of the power series \( \sum_{n=1}^{\infty} \frac{x^n}{n \cdot 2^n} \) is ________

(9) Let \( f(x, y) = xe^x - ye^x \). The gradient of \( f \) at \( (0, 0) \) is \( \nabla f(0, 0) = \)________

(10) Evaluate \( \int_0^4\int_0^1\sqrt{1 + y^2} dydx = \)________

(1) \( (8\%) \) Prove that \( 1 \leq a_{n} \leq 2 \), and \( a_{n} \leq a_{n+1} \), \( \forall n \in N \).

(2) \( (6\%) \) Explain why \( \{a_n\} \) is convergent.

(3) \( (6\%) \) Find \( \lim_{n\to \infty}a_n \)

3. (15%) Let z = f(x, y) and $ze^2 - x^2y = 0$.
Prove that $x\frac{\partial z}{\partial x} - y\frac{\partial z}{\partial y} = 1 - \frac{1}{1 + z}$, if $z \neq -1$.

4. (15%) Find all relative maxima, minima, and saddle points of the function :
$f(x, y) = 3x^2y + y^2 - 3xy$.