115年 - 114 國立臺北大學_碩士班一般入學考試試題_統計學系：基礎數學#146989

一、(50%，10% each)Calculus(清楚完整作答，始得完全分數)
1. Find the limit for $\lim_{x\to\infty} (\frac{x+a}{x-a})^{5x} = ?$

2. Find the extreme value for $f(x) = \cos[2\sin(\frac{x}{2})], x \in (0, 2\pi)$.

3. Find the indefinite integral $\int \frac{1}{(1+3^2)^2} dx = ?$

4. Find the definite integral $\int \int_R \frac{1}{\sqrt{x^2+y^2}} dA = ?$ where $R = x^2 + y^2 \leq 9$.

5. 王先生欲以汽車定速在國三高速公路新北市三峽交流道到屏東南州交流道行駛，假設該二交流道間距離為 384 公里(km)。汽車油耗為每小時的汽油使用公升數(litter/hour)，速度為每小時的行駛公里數(km/hour)。若以省油為考量(也就是每公里所使用的油耗越少越好)，假設速度 x 時，耗油為 $f(x) = (x - 90)^2 + 364$，試說明並求導在什麼速度行駛才會最省油呢？

a. (10%) Find the Gram-Schmidt QR factorization of A, i.e. factorize A = QR.

b. (5%) Find the column space C(A) and the left null space $N(A^T)$ of A.

c. (10%) Show that the subspaces C(A) and $N(A^T)$ are orthogonal.

d. (10%) Decompose $\underline{b} = \underline{b}_1 + \underline{b}_2$, where $\underline{b}_1 \in C(A)$ and $\underline{b}_2 \in N(A^T)$, i.e. find $\underline{b}_1$ and $\underline{b}_2$.

e. (5%) Find the projection matrix P such that $\underline{b}_1 = P\underline{b}$.

a. (5%) Find the orthogonal projection of $\underline{x}$ onto the line L, i.e. find the vector $\underline{x}_p$ as shown in the figure below.

b. (5%) Find the projection matrix P that projects $\underline{x}$ onto the line L, i.e. find P such that $\underline{x}_p = P\underline{x}$.