北京大學(xué)考研數(shù)學(xué)試卷

北京大學(xué)考研數(shù)學(xué)試卷一、選擇題

1.設(shè)函數(shù)\(f(x)=e^x-x^2\)，則\(f(x)\)的極值點(diǎn)為：

A.\(x=0\)

B.\(x=1\)

C.\(x=-1\)

D.\(x=2\)

2.設(shè)\(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\)，則\(A^{-1}\)為：

A.\(\begin{bmatrix}4&-2\\-3&1\end{bmatrix}\)

B.\(\begin{bmatrix}1&2\\3&4\end{bmatrix}\)

C.\(\begin{bmatrix}-4&2\\3&-1\end{bmatrix}\)

D.\(\begin{bmatrix}2&-1\\3&-4\end{bmatrix}\)

3.設(shè)\(f(x)=\ln(x)\)，則\(f'(x)\)為：

A.\(\frac{1}{x}\)

B.\(\frac{1}{x^2}\)

C.\(\frac{1}{x^3}\)

D.\(\frac{1}{x^4}\)

4.設(shè)\(f(x)=x^3-3x\)，則\(f'(x)\)為：

A.\(3x^2-3\)

B.\(3x^2+3\)

C.\(3x-3\)

D.\(3x+3\)

5.設(shè)\(f(x)=\int_0^xt^2dt\)，則\(f'(x)\)為：

A.\(x^2\)

B.\(2x\)

C.\(x\)

D.\(0\)

6.設(shè)\(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\)，則\(|A|\)為：

A.\(5\)

B.\(6\)

C.\(7\)

D.\(8\)

7.設(shè)\(f(x)=\frac{1}{x}\)，則\(f''(x)\)為：

A.\(-\frac{1}{x^3}\)

B.\(\frac{1}{x^3}\)

C.\(-\frac{1}{x^2}\)

D.\(\frac{1}{x^2}\)

8.設(shè)\(f(x)=e^x\)，則\(f'(x)\)為：

A.\(e^x\)

B.\(2e^x\)

C.\(e^x+1\)

D.\(e^x-1\)

9.設(shè)\(f(x)=\ln(x)\)，則\(f''(x)\)為：

A.\(\frac{1}{x^2}\)

B.\(-\frac{1}{x^2}\)

C.\(\frac{1}{x^3}\)

D.\(-\frac{1}{x^3}\)

10.設(shè)\(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\)，則\(A^T\)為：

A.\(\begin{bmatrix}1&3\\2&4\end{bmatrix}\)

B.\(\begin{bmatrix}1&2\\3&4\end{bmatrix}\)

C.\(\begin{bmatrix}4&3\\2&1\end{bmatrix}\)

D.\(\begin{bmatrix}4&2\\3&1\end{bmatrix}\)

二、判斷題

1.函數(shù)\(f(x)=e^x\)在其定義域內(nèi)是連續(xù)的。（）

2.一個(gè)二次函數(shù)的圖像始終是一個(gè)開口向上或向下的拋物線。（）

3.柯西中值定理可以用來證明導(dǎo)數(shù)存在的充分必要條件。（）

4.矩陣的行列式為零是矩陣可逆的充分必要條件。（）

5.在實(shí)數(shù)域上，任意兩個(gè)多項(xiàng)式函數(shù)的和也是多項(xiàng)式函數(shù)。（）

三、填空題

1.函數(shù)\(f(x)=x^3-6x+9\)的導(dǎo)數(shù)\(f'(x)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\

四、簡答題

1.簡述拉格朗日中值定理的內(nèi)容，并給出一個(gè)應(yīng)用該定理證明函數(shù)單調(diào)性的例子。

2.如何計(jì)算一個(gè)二次型\(Q(x_1,x_2,...,x_n)=x_1^2+x_2^2+...+x_n^2-2x_1x_2+2x_1x_3-2x_2x_3\)的正慣性指數(shù)？

3.請(qǐng)解釋什么是函數(shù)的泰勒展開式，并說明為什么泰勒展開式在近似計(jì)算中非常有用。

4.簡述矩陣的秩與線性方程組解的關(guān)系，并舉例說明。

5.如何判斷一個(gè)實(shí)二次方程\(ax^2+bx+c=0\)的根是實(shí)數(shù)還是復(fù)數(shù)？請(qǐng)給出相關(guān)的數(shù)學(xué)公式。

五、計(jì)算題

1.計(jì)算定積分\(\int_0^{\pi}\sin^2x\,dx\)。

2.解線性方程組\(\begin{cases}x+2y-z=1\\2x+y+3z=0\\-x+3y+2z=2\end{cases}\)。

3.求函數(shù)\(f(x)=x^3-3x+2\)在點(diǎn)\(x=2\)處的切線方程。

4.計(jì)算矩陣\(A=\begin{bmatrix}1&-2&3\\4&1&2\\3&0&-1\end{bmatrix}\)的行列式\(|A|\)。

5.設(shè)\(f(x)=e^{2x}\sinx\)，求\(f(x)\)的導(dǎo)數(shù)\(f'(x)\)。

六、案例分析題

1.案例分析題：某公司為了評(píng)估其銷售策略的有效性，收集了以下數(shù)據(jù)：銷售額（萬元）、廣告投入（萬元）和銷售人員的數(shù)量。公司希望找到一個(gè)模型來預(yù)測銷售額與廣告投入及銷售人員數(shù)量的關(guān)系。請(qǐng)根據(jù)以下數(shù)據(jù)，利用線性回歸方法建立一個(gè)預(yù)測模型，并分析模型的有效性。

數(shù)據(jù)如下：

|廣告投入|銷售人員數(shù)量|銷售額|

|----------|--------------|--------|

|2|3|5|

|3|4|6|

|4|5|7|

|5|6|8|

|6|7|9|

2.案例分析題：某城市交通管理部門為了提高城市交通效率，決定對(duì)一條主要干道進(jìn)行交通流量調(diào)查。調(diào)查數(shù)據(jù)包括不同時(shí)間段（高峰時(shí)段和非高峰時(shí)段）的車輛流量和道路長度。管理部門希望利用這些數(shù)據(jù)來分析交通流量與道路長度之間的關(guān)系，并據(jù)此提出改善交通擁堵的建議。

數(shù)據(jù)如下：

|時(shí)間段|道路長度（千米）|車輛流量（輛/小時(shí)）|

|--------|------------------|---------------------|

|高峰時(shí)段|2|1200|

|非高峰時(shí)段|2|800|

|高峰時(shí)段|3|1800|

|非高峰時(shí)段|3|1200|

|高峰時(shí)段|4|2400|

|非高峰時(shí)段|4|1600|

請(qǐng)根據(jù)數(shù)據(jù)，建立合適的數(shù)學(xué)模型，并分析模型對(duì)實(shí)際交通管理的指導(dǎo)意義。

七、應(yīng)用題

1.應(yīng)用題：某產(chǎn)品的需求函數(shù)\(Q=10-0.5P\)，其中\(zhòng)(Q\)是需求量，\(P\)是價(jià)格。假設(shè)成本函數(shù)\(C=2Q+10\)，求該產(chǎn)品的利潤函數(shù)\(L(P)\)并求出使得利潤最大化的價(jià)格\(P\)。

2.應(yīng)用題：一個(gè)公司生產(chǎn)兩種產(chǎn)品A和B，產(chǎn)品A的利潤為每單位20元，產(chǎn)品B的利潤為每單位30元。生產(chǎn)產(chǎn)品A需要3小時(shí)，生產(chǎn)產(chǎn)品B需要2小時(shí)。公司每天最多有10小時(shí)的機(jī)器時(shí)間可用。如果每天生產(chǎn)的產(chǎn)品A和產(chǎn)品B的數(shù)量分別為\(x\)和\(y\)，求公司每天的最大利潤，并寫出相應(yīng)的線性規(guī)劃模型。

3.應(yīng)用題：已知函數(shù)\(f(x)=\sqrt{x}\)，求在區(qū)間\([0,4]\)上，函數(shù)\(f(x)\)的平均值，并說明這個(gè)平均值的意義。

4.應(yīng)用題：一個(gè)班級(jí)有30名學(xué)生，其中有15名女生和15名男生。隨機(jī)選擇3名學(xué)生參加比賽，求以下事件的概率：

-事件A：選出的3名學(xué)生都是女生。

-事件B：選出的3名學(xué)生中至少有1名男生。

-事件C：選出的3名學(xué)生中有2名女生和1名男生。

本專業(yè)課理論基礎(chǔ)試卷答案及知識(shí)點(diǎn)總結(jié)如下：

一、選擇題答案：

1.B

2.A

3.A

4.A

5.A

6.A

7.A

8.A

9.A

10.A

二、判斷題答案：

1.對(duì)

2.錯(cuò)

3.錯(cuò)

4.錯(cuò)

5.對(duì)

三、填空題答案：

1.\(f'(x)=3x^2-6x\)

2.\(|A|=2\)

3.\(f''(x)=e^x\)

4.\(f'(x)=3x^2-6x+9\)

5.\(A^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}\)

四、簡答題答案：

1.拉格朗日中值定理：如果函數(shù)\(f(x)\)在閉區(qū)間\([a