109年 - 109 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#147039

科目：研究所、轉學考(插大)-基礎數學 | 年份：109年 | 選擇題數：0 | 申論題數：15

試卷資訊

所屬科目：研究所、轉學考(插大)-基礎數學

選擇題 (0)

申論題 (15)

一、(50%) CALCULUS

1. (10%) Let $f(x)$ be a differentiable function as shown in Figure 1. It needs to compute the integral $\int_{a}^{b} f(x)dx$. For some reason, the integration is not possible. We decide to linearize $f(x)$ at $x = a$ and use the trapezoidal area $A$ to approximate the integral, i.e. $\int_{a}^{b} f(x)dx \approx A$. Please find the area $A$.
2. (10%) Sketch the graph of a function $f(x)$ which has all of the following properties:

(a) \(\lim_{x\to 1^{+}}f(x) = \infty ,\lim_{x\to 1^{-}}f(x) = -\infty ,\lim_{x\to -\infty}f(x) = 0\)
(b) \(f(-2) = 3, f(5) = 1, f(0) = 0\)
(c) \(f^{\prime}(5) = 0\) \(f^{\prime}(-2) = 0\)
(d) \( f'(x) > 0 \) if \( x < -2 \) or \( x > 5 \), \( f'(x) < 0 \) if \( -2 < x < 1 \) or \( 1 < x < 5 \).
(e) \( f''(x) > 0 \) if \( x < -3 \) or \( x > 1 \), \( f''(x) < 0 \) if \( -3 < x < 1 \).
3. (10%) Evaluate the integral

$$E = \int_{0}^{L} \frac{\lambda c}{4\pi\delta(x^2 + c^2)^{3/2}} dx$$

Where $c, \lambda, \delta, L$ are constants.
4. (10%) Evaluate the double integral

$$\int_{0}^{2} \int_{x^2}^{4} x e^{-y^2} dy dx$$
5. (10%) Let the function $f(t) = \sin(t^2)$ as shown in Figure 2. Consider two areas $A(x)$ and $B(x)$ for $x < 1.5$, where $A(x)$ represents the triangular area, while $B(x)$ designates the definite integral of $f(t)$ from $t = 0$ to $t = x$. Find

$$\lim_{x \to 0^+} \frac{A(x)}{B(x)}$$
二、(50%) (所有題目請敘述計算過程，無計算過程不給分)

1. Let $A = \begin{bmatrix} 2 & 1 & -1 \\ 1 & 2 & 1 \\ -1 & 1 & 2 \end{bmatrix}$

(a) Write down the characteristic polynomial of \( A \) and use it to find the eigenvalues. (4%)
(b) Find the eigenspaces of \(A\). (5%)
(c) Orthogonally diagonalize the matrix \( A \). (You need to find out an orthogonal matrix \( P \) and a diagonal matrix \( D \) such that \( P^T AP = D \).) (10%)
2. Let $\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k$ be vectors in a vector space V. Please show that

(a) \(\operatorname{span}\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k\}\) is a subspace of V. (10%)
(b) \(\operatorname{span}\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k\}\) is the smallest subspace of \(V\) that contains \(\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k\). (4%)
3. Let \( S = \{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k\} \) be an orthogonal set of nonzero vectors in a vector space \( V \). Please show that the vectors in \( S \) are linear independent. (10%)
4. Let \( T: P_2 \to P_2 \) be the linear transformation defined by

$$T(p(x)) = p(2x - 1).$$

Please find the matrix of T with respect to the basis $\mathcal{E} = \{1, x + 1, x^2\}$. (7%)