大一經(jīng)管類高等數(shù)學試卷

大一經(jīng)管類高等數(shù)學試卷一、選擇題

1.若函數(shù)\(f(x)=x^3-6x^2+9x\)，則\(f'(x)\)等于（）

A.\(3x^2-12x+9\)

B.\(3x^2-12x+3\)

C.\(3x^2-6x+9\)

D.\(3x^2-6x+3\)

2.下列函數(shù)中，可導的函數(shù)是（）

A.\(f(x)=|x|\)

B.\(f(x)=x^2\)

C.\(f(x)=\sqrt{x}\)

D.\(f(x)=\frac{1}{x}\)

3.若\(\int_0^1f(x)\,dx=2\)，則\(\int_0^2f(2x)\,dx\)等于（）

A.4

B.8

C.16

D.0

4.若\(f(x)\)在\(x=1\)處連續(xù)，則\(\lim_{x\to1}(f(x)-f(1))\)等于（）

A.0

B.\(f'(1)\)

C.\(f(1)\)

D.無定義

5.若\(f(x)\)在\(x=a\)處可導，則\(\lim_{x\toa}\frac{f(x)-f(a)}{x-a}\)等于（）

A.0

B.\(f'(a)\)

C.\(f(a)\)

D.無定義

6.若\(f(x)\)在\(x=a\)處可導，則\(\lim_{x\toa}\frac{f(x)-f(a)}{x^2-a^2}\)等于（）

A.0

B.\(f'(a)\)

C.\(f(a)\)

D.無定義

7.若\(f(x)\)在\(x=a\)處連續(xù)，則\(\lim_{x\toa}\frac{f(x)-f(a)}{x-a}\)等于（）

A.0

B.\(f'(a)\)

C.\(f(a)\)

D.無定義

8.若\(f(x)\)在\(x=a\)處可導，則\(\lim_{x\toa}\frac{f(x)-f(a)}{x-a}\)等于（）

A.0

B.\(f'(a)\)

C.\(f(a)\)

D.無定義

9.若\(f(x)\)在\(x=a\)處連續(xù)，則\(\lim_{x\toa}\frac{f(x)-f(a)}{x-a}\)等于（）

A.0

B.\(f'(a)\)

C.\(f(a)\)

D.無定義

10.若\(f(x)\)在\(x=a\)處可導，則\(\lim_{x\toa}\frac{f(x)-f(a)}{x-a}\)等于（）

A.0

B.\(f'(a)\)

C.\(f(a)\)

D.無定義

二、判斷題

1.函數(shù)的可導性意味著它在某一點處連續(xù)。（）

2.定積分的被積函數(shù)必須是有理函數(shù)。（）

3.對于任意函數(shù)\(f(x)\)，若\(\intf(x)\,dx\)存在，則\(\intf(x)\,dx\)必定存在反函數(shù)。（）

4.函數(shù)的導數(shù)表示函數(shù)在某一點處的瞬時變化率。（）

5.如果函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上可導，那么\(f(x)\)在\((a,b)\)內(nèi)必定存在至少一個點\(c\)，使得\(f'(c)=\frac{f(b)-f(a)}{b-a}\)。（）

三、填空題

1.若\(f(x)=x^3-3x+2\)，則\(f'(1)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\

四、簡答題

1.簡述導數(shù)的定義及其幾何意義。

2.舉例說明定積分與不定積分的關(guān)系。

3.解釋什么是函數(shù)的極值，并說明如何求一個函數(shù)的極大值和極小值。

4.簡述微積分基本定理的內(nèi)容及其證明思路。

5.如何判斷一個函數(shù)在某個區(qū)間上是否存在反函數(shù)？請給出一個具體的例子。

五、計算題

1.計算定積分\(\int_0^2(3x^2-4x+2)\,dx\)。

2.求函數(shù)\(f(x)=x^3-6x^2+9x\)的導數(shù)\(f'(x)\)。

3.求函數(shù)\(f(x)=\frac{1}{x^2+1}\)在\(x=0\)處的導數(shù)。

4.計算極限\(\lim_{x\to\infty}\frac{\sinx}{x}\)。

5.求函數(shù)\(f(x)=e^{2x}-e^{-2x}\)的不定積分\(\intf(x)\,dx\)。

六、案例分析題

1.案例背景：某公司生產(chǎn)一種產(chǎn)品，其產(chǎn)量\(Q\)與成本\(C\)的關(guān)系為\(C=10Q+500\)，其中\(zhòng)(Q\)是以百為單位的產(chǎn)品數(shù)量。已知公司固定成本為500元，每生產(chǎn)一單位產(chǎn)品的變動成本為10元。

案例分析：

（1）求公司生產(chǎn)100單位產(chǎn)品的總成本。

（2）求公司生產(chǎn)100單位產(chǎn)品的平均成本和邊際成本。

（3）如果公司希望利潤最大化，那么它應該生產(chǎn)多少單位產(chǎn)品？

2.案例背景：某城市居民用電量與電費之間的關(guān)系可以近似表示為\(y=0.2x+20\)，其中\(zhòng)(y\)是月電費（元），\(x\)是月用電量（度）。

案例分析：

（1）求居民每月用電量為100度時的電費。

（2）求居民用電量為100度時的電費彈性。

（3）假設電費上漲10%，預測居民用電量將如何變化。

七、應用題

1.應用題：已知函數(shù)\(f(x)=x^3-3x+2\)，求\(f(x)\)在區(qū)間\([0,2]\)上的最大值和最小值。

2.應用題：某工廠生產(chǎn)一種產(chǎn)品，其產(chǎn)量\(Q\)與單位產(chǎn)品的生產(chǎn)成本\(C\)的關(guān)系為\(C=50+5Q\)，其中\(zhòng)(Q\)是以百為單位的產(chǎn)品數(shù)量。已知該產(chǎn)品的市場需求函數(shù)為\(P=100-2Q\)，其中\(zhòng)(P\)是單位產(chǎn)品的售價。

（1）求該工廠的總利潤函數(shù)\(L(Q)\)。

（2）求該工廠的最大利潤和相應的產(chǎn)量\(Q\)。

3.應用題：已知某城市居民每月的用水量\(x\)與水費\(y\)的關(guān)系為\(y=0.6x+10\)，其中\(zhòng)(x\)是以立方米為單位的水量，\(y\)是水費（元）。

（1）求居民每月用水量為50立方米時的水費。

（2）求居民用水量的價格彈性。

4.應用題：某公司生產(chǎn)一種產(chǎn)品，其需求函數(shù)為\(P=100-2Q\)，其中\(zhòng)(P\)是單位產(chǎn)品的售價，\(Q\)是銷售量。公司的固定成本為2000元，每生產(chǎn)一單位產(chǎn)品的可變成本為20元。

（1）求公司的總成本函數(shù)\(C(Q)\)。

（2）求公司的總收入函數(shù)\(R(Q)\)。

本專業(yè)課理論基礎(chǔ)試卷答案及知識點總結(jié)如下：

一、選擇題答案

1.A

2.C

3.B

4.A

5.B

6.A

7.B

8.B

9.B

10.B

二、判斷題答案

1.錯誤

2.錯誤

3.錯誤

4.正確

5.正確

三、填空題答案

1.\(f'(1)=0\)

2.\(\intf(x)\,dx=\frac{1}{2}x^2-3x^2+2x+C\)

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

4.\(\lim_{x\to\infty}\frac{\sinx}{x}=0\)

5.\(\intf(x)\,dx=\frac{1}{2}e^{2x}-\frac{1}{2}e^{-2x}+C\)

四、簡答題答案

1.導數(shù)的定義是函數(shù)在某一點處的瞬時變化率，幾何意義上表示曲線在該點的切線斜率。

2.定積分與不定積分的關(guān)系是：定積分可以看作是不定積分的一個特定值，即定積分是積分常數(shù)\(C\)為特定值時的不定積分。

3.函數(shù)的極值是函數(shù)在某個區(qū)間內(nèi)的局部最大值或最小值。求極大值和極小值的方法包括一階導數(shù)法和二階導數(shù)法。

4.微積分基本定理的內(nèi)容是：如果函數(shù)\(f(x)\)在閉區(qū)間\([a,b]\)上連續(xù)，且\(F(x)\)是\(f(x)\)的一個原函數(shù)，那么\(\int_a^bf(x)\,dx=F(b)-F(a)\)。

5.判斷一個函數(shù)在某個區(qū)間上是否存在反函數(shù)的方法是：如果函數(shù)在該區(qū)間上單調(diào)遞增或單調(diào)遞減，并且連續(xù)，那么它在該區(qū)間上存在反函數(shù)。

五、計算題答案

1.\(\int_0^2(3x^2-4x+2)\,dx=\frac{10}{3}x^3-2x^2+2x\bigg|_0^2=\frac{10}{3}\cdot2^3-2\cdot2^2+2\cdot2=\frac{80}{3}-8+4=\frac{68}{3}\)

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

3.\(f'(0)=\lim_{x\to0}\frac{\frac{1}{x^2+1}-1}{x}=\lim_{x\to0}\frac{-x^2}{x(x^2+1)}=\lim_{x\to0}\frac{-x}{x^2+1}=0\)

4.\(\lim_{x\to\infty}\frac{\sinx}{x}=0\)（利用三角函數(shù)的有界性和極限的性質(zhì)）

5.\(\intf(x)\,dx=\frac{1}{2}e^{2x}-\frac{1}{2}e^{-2x}+C\)

六、案例分析題答案

1.（1）總成本\(C(100)=10\cdot100+500=1500\)元。

（2）平均成本\(\frac{C(100)}{100}=15\)元，邊際成本\(C'(100)=10\)元。

（3）利潤最大化時，邊際收益等于邊際成本，即\(100-2Q=10\)，解得\(Q=45\)。

2.（1）總利潤\(L(Q)=(100-2Q)Q-(50+5Q)=50