引言
2011年的数学三真题是众多考生备考过程中的重要参考。本文将深入解析这一年的数学三真题，帮助考生了解考试趋势，掌握解题技巧，为高效备考提供指导。
一、考试概况
2011年的数学三考试涵盖了高等数学、线性代数和概率论与数理统计三个部分，试题难度适中，考察了考生对基本概念、基本方法和基本技巧的掌握程度。
二、试题解析
1. 高等数学部分
（1）选择题
- 题目：求函数\(f(x)=x^3-3x+1\)的极值点。
- 解析：首先求导\(f'(x)=3x^2-3\)，令\(f'(x)=0\)，得\(x=\pm 1\)。然后判断二阶导数\(f''(x)=6x\)，在\(x=\pm 1\)处符号改变，故\(x=\pm 1\)为极值点。
（2）填空题
- 题目：设\(y=\sin x\)，则\(y'\)的表达式为______。
- 解析：\(y'=\cos x\)。
（3）解答题
- 题目：求曲线\(y=x^2\)在\(x=1\)处的切线方程。
- 解析：求导得\(y'=2x\)，在\(x=1\)处，\(y'=2\)，切线斜率为2。切点为\((1,1)\)，故切线方程为\(y-1=2(x-1)\)，即\(y=2x-1\)。
2. 线性代数部分
（1）选择题
- 题目：设矩阵\(A=\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\)，则\(A\)的逆矩阵为______。
- 解析：计算\(A\)的行列式\(|A|=1\cdot 4-2\cdot 3=4-6=-2\)，\(A\)的逆矩阵为\(\frac{1}{|A|}\begin{bmatrix}4 & -2 \\ -3 & 1\end{bmatrix}=\begin{bmatrix}-2 & 1 \\ \frac{3}{2} & -\frac{1}{2}\end{bmatrix}\)。
（2）填空题
- 题目：设向量\(\boldsymbol{a}=\begin{bmatrix}1 \\ 2\end{bmatrix}\)，\(\boldsymbol{b}=\begin{bmatrix}3 \\ 4\end{bmatrix}\)，则$\boldsymbol{a}\cdot\boldsymbol{b}=______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________