111年 - 111 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#147044

1. (10 points) Let $f(x)$ be defined in a neighborhood of the origin. Show that if $f'(0)$ exists, then $$\lim_{h \to 0} \frac{f(h) - f(-h)}{2h} = f'(0).$$ Give a counter example to show that the converse is not true in general, that is, if the above limit exists, then it is not necessary that $f'(0)$ exists.

2. (10 points) Find an equation for the line tangent to the graph of $y = x(\ln x)^2 + \frac{x}{\ln x}$ at the point $(e, 2e)$.

3. (10 points) Let $n, r$ be positive integer. Evaluate $$\lim_{r \to \infty} \left( \frac{1}{r+1} + \frac{1}{r+2} + \dots + \frac{1}{r+nr} \right).$$

4. (10 points) Find the continuous function $f$ and constant $c$ such that $$\int_c^x t f(t) dt = \cos x - \frac{1}{2}, \ \forall x \in R.$$

5. (10 points) Sketch the region of integration for the given integral $\int_0^1 \int_{x^2}^1 x e^{x^2} dy dx$ and evaluate it.

a. \( A - A^{\top} \) is skew-symmetric.

b. the diagonal elements of \( A - A^{\top} \) must be zero.

c. \( \pmb{x}^{\top}(A - A^{\top})\pmb{x} = 0 \) for all \( \pmb{x}\in \mathfrak{R}^n \)

d. \( I_{n} + A - A^{\top} \) is invertible.

a. \( \operatorname{rank}(AB) \leq \operatorname{rank}(B) \)

b. \( \operatorname{rank}(AB) \leq \operatorname{rank}(A) \)

3. (10 points) Find an orthogonal basis for $\Re^3$ that contains the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$.

a. \( det(A) = \lambda_1\lambda_2\dots \lambda_n \)

b. \( tr(A) = \lambda_1 + \lambda_2 + \dots +\lambda_n \)