112年 - 112 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#147045

一、(50%) CALCULUS(解答題目須清楚完整，始得完全分數)
1. (10%) Find the limit $$\lim_{x \to 0^+} \left(\frac{1}{x}\right)^x$$

2. (10%) Find the derivative $f'(x)$ of the function $$f(x) = \int_x^{x^2} \sqrt{1 + t^3} \, dt$$

(a) (10%) Evaluate the integral $$\int_{-1}^0 f(x) \, dx$$

(b) (10%) The integral in (a) can be approximated by integrating the linear approximation of the integrand, i.e. let $$f(x) = \frac{1 + x}{\sqrt{1 - x^2}} \approx f(0) + f'(0) x$$ Approximate $\int_{-1}^0 f(x) \, dx$ by evaluating $$\int_{-1}^0 (f(0) + f'(0) x) \, dx$$

4. (10%) Let $$L(\sigma) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2}\left(\frac{x_i - \mu}{\sigma}\right)^2}, \quad \sigma > 0$$ Where $\mu$ and $x_i, i = 1, 2, \dots, n$ are known values. Find the value of $\sigma$ at which $L(\sigma)$ has its maximum. (Skip any derivative test)

(a) (5%) Please write down the \( (i,j) \)-entry of the product \( BCA \) in terms of Sigma notations (i.e. \( \Sigma \)).

(b) \( (5\%) \) Please prove that \( (BC)^{-1} = C^{-1}B^{-1} \).

(a) (5%) Write down the characteristic polynomial of \( A \) and use it to find the eigenvalues.

(b) \( (5\%) \) Find the eigenspaces of \( A \)

(c) (10%) Orthogonally diagonalize the matrix \( A \). (You need to find out an orthogonal matrix \( P \) and a diagonal matrix \( D \) such that \( P^T AP = D \).)

3. (10%) Let $T: V \to W$ be a linear transformation and $B = \{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ be a spanning set for V. Please show that $T(B) = \{T(\vec{v}_1), T(\vec{v}_2), \dots, T(\vec{v}_n)\}$ spans the range of T.

(a) \( (5\%) \) Prove that \( x^{T}Ax = 0 \) for all \( x\in R^n \)

(b) \( (5\%) \) Prove that \( I + A \) is invertible.