滄州二中高三數(shù)學(xué)試卷

滄州二中高三數(shù)學(xué)試卷一、選擇題

1.在下列各對函數(shù)中，若函數(shù)\(f(x)\)的圖像是函數(shù)\(g(x)\)的圖像關(guān)于\(y\)軸的對稱圖形，則\(f(x)\)和\(g(x)\)的關(guān)系是（）

A.\(f(x)=g(-x)\)B.\(f(x)=g(x)\)C.\(f(x)=-g(x)\)D.\(f(x)=-g(-x)\)

2.若函數(shù)\(f(x)\)在\((0,+\infty)\)上單調(diào)遞增，且\(f(1)=2\)，則下列各式中，正確的是（）

A.\(f(2)>f(1)\)B.\(f(0)<f(1)\)C.\(f(1)>f(2)\)D.\(f(2)<f(0)\)

3.若函數(shù)\(f(x)=ax^2+bx+c\)的圖像開口向上，且\(f(0)=1\)，\(f(1)=4\)，\(f(2)=9\)，則下列各式中，正確的是（）

A.\(a>0\)B.\(b>0\)C.\(c>0\)D.\(a+b+c>0\)

4.已知函數(shù)\(f(x)=x^3-3x\)，則\(f(x)\)的對稱中心為（）

A.\((0,0)\)B.\((1,0)\)C.\((0,1)\)D.\((1,1)\)

5.若函數(shù)\(f(x)=\sqrt{x^2+1}\)的圖像上任意一點\((x,y)\)到原點的距離為\(d\)，則\(d\)的最小值為（）

A.\(\sqrt{2}\)B.\(1\)C.\(0\)D.無解

6.已知函數(shù)\(f(x)=x^2-2x+1\)，若\(f(x)\)的圖像上任意一點\((x,y)\)到直線\(y=1\)的距離為\(d\)，則\(d\)的最大值為（）

A.\(2\)B.\(1\)C.\(0\)D.無解

7.若函數(shù)\(f(x)=a^x\)的圖像過點\((1,2)\)，則\(a\)的值為（）

A.\(2\)B.\(1\)C.\(\frac{1}{2}\)D.\(-1\)

8.若函數(shù)\(f(x)=\frac{1}{x}\)的圖像上任意一點\((x,y)\)到原點的距離為\(d\)，則\(d\)的最大值為（）

A.\(\sqrt{2}\)B.\(1\)C.\(0\)D.無解

9.已知函數(shù)\(f(x)=x^3-3x\)，則\(f(x)\)的圖像上任意一點\((x,y)\)到直線\(y=0\)的距離為（）

A.\(|x|\)B.\(|x^2|\)C.\(|x^3|\)D.\(|x^2-3x|\)

10.若函數(shù)\(f(x)=\frac{1}{x}\)的圖像上任意一點\((x,y)\)到直線\(y=1\)的距離為\(d\)，則\(d\)的最小值為（）

A.\(\sqrt{2}\)B.\(1\)C.\(0\)D.無解

二、判斷題

1.函數(shù)\(f(x)=ax^2+bx+c\)的圖像是一個拋物線，其中\(zhòng)(a\)、\(b\)、\(c\)是常數(shù)，且\(a\neq0\)。（）

2.如果函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上連續(xù)，且\(f(a)>f(b)\)，那么\(f(x)\)在\([a,b]\)上必定存在一個零點。（）

3.對于任意二次方程\(ax^2+bx+c=0\)，其判別式\(b^2-4ac\)大于零時，方程有兩個不相等的實數(shù)根。（）

4.在直角坐標系中，一個圓的方程可以表示為\(x^2+y^2=r^2\)，其中\(zhòng)(r\)是圓的半徑。（）

5.如果函數(shù)\(f(x)\)在點\(x=a\)處可導(dǎo)，那么函數(shù)\(f(x)\)在點\(x=a\)處必定連續(xù)。（）

三、填空題

1.若函數(shù)\(f(x)=\sqrt{x^2-4}\)的定義域為\(D\)，則\(D=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\

四、簡答題

1.簡述函數(shù)\(f(x)=x^3-3x\)的導(dǎo)數(shù)\(f'(x)\)的求法，并求出\(f'(x)\)。

2.已知函數(shù)\(f(x)=\frac{x^2-1}{x-1}\)，求\(f(x)\)的極限\(\lim_{x\to1}f(x)\)。

3.給定直線\(y=mx+b\)和圓\(x^2+y^2=r^2\)，證明：直線和圓相交的必要條件是\(m^2+1\geq\frac{1}{r^2}\)。

4.簡述二次函數(shù)\(f(x)=ax^2+bx+c\)的圖像與\(x\)軸交點的個數(shù)與判別式\(b^2-4ac\)之間的關(guān)系。

5.設(shè)函數(shù)\(f(x)=\ln(x)\)，證明：對于任意\(x_1>x_2>0\)，有\(zhòng)(f(x_1)-f(x_2)>\frac{x_1-x_2}{x_1x_2}\)。

五、計算題

1.計算定積分\(\int_{0}^{2}(x^2-4x+3)\,dx\)的值。

2.求函數(shù)\(f(x)=e^x-x\)在\(x=0\)處的導(dǎo)數(shù)\(f'(0)\)。

3.解方程組\(\begin{cases}2x+3y=8\\x-y=1\end{cases}\)。

4.計算復(fù)數(shù)\((1+2i)^5\)的值。

5.求函數(shù)\(f(x)=\frac{1}{x^2-4}\)在區(qū)間\([1,3]\)上的定積分\(\int_{1}^{3}f(x)\,dx\)。

六、案例分析題

1.案例背景：某公司計劃生產(chǎn)一批產(chǎn)品，已知生產(chǎn)第\(x\)件產(chǎn)品的成本為\(C(x)=100+2x\)元，其中\(zhòng)(x\)為產(chǎn)品的件數(shù)。又知每件產(chǎn)品的售價為200元，市場需求函數(shù)為\(Q(x)=500-2x\)，其中\(zhòng)(x\)為市場需求量。

案例分析：

（1）求生產(chǎn)\(x\)件產(chǎn)品的利潤函數(shù)\(L(x)\)。

（2）求使得利潤最大化的最優(yōu)生產(chǎn)件數(shù)\(x\)。

（3）根據(jù)市場需求函數(shù)，當\(x=100\)時，市場對這批產(chǎn)品的需求量是多少？

2.案例背景：某班級有30名學(xué)生，其中男生人數(shù)為\(x\)，女生人數(shù)為\(y\)。已知男生平均身高為\(h_1\)，女生平均身高為\(h_2\)，班級總平均身高為\(h\)。

案例分析：

（1）根據(jù)班級總平均身高的定義，寫出\(h\)的表達式。

（2）若已知\(h_1=1.75\)米，\(h_2=1.65\)米，\(h=1.70\)米，求班級中男生和女生的人數(shù)\(x\)和\(y\)。

（3）如果班級中男女生人數(shù)的比例是\(1:1\)，求班級的總平均身高\(h\)。

七、應(yīng)用題

1.應(yīng)用題：某工廠生產(chǎn)一種產(chǎn)品，其固定成本為每天2000元，每生產(chǎn)一件產(chǎn)品的變動成本為10元。如果每件產(chǎn)品的售價為50元，求工廠每天生產(chǎn)多少件產(chǎn)品才能達到盈虧平衡點？

2.應(yīng)用題：已知某函數(shù)\(f(x)=-2x^2+8x+4\)，求函數(shù)\(f(x)\)在區(qū)間\([1,4]\)上的最大值和最小值。

3.應(yīng)用題：一元二次方程\(ax^2+bx+c=0\)的兩個實數(shù)根為\(x_1\)和\(x_2\)，且\(x_1+x_2=-\frac{a}\)，\(x_1x_2=\frac{c}{a}\)。若方程的判別式\(b^2-4ac\)等于0，求方程的解。

4.應(yīng)用題：一輛汽車以60公里/小時的速度行駛，經(jīng)過一段時間后，速度降低到40公里/小時，行駛了相同的時間。假設(shè)汽車減速的過程是勻減速運動，求汽車減速的平均速度。

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

1.A

2.A

3.A

4.A

5.A

6.A

7.A

8.A

9.A

10.A

二、判斷題

1.×

2.×

3.√

4.√

5.√

三、填空題

1.\(\sqrt{x^2-4}>0\)

2.\(\lim_{x\to1}f(x)=2\)

3.\(m^2+1\geq\frac{1}{r^2}\)

4.當\(b^2-4ac>0\)時，有兩個不相等的實數(shù)根；當\(b^2-4ac=0\)時，有兩個相等的實數(shù)根；當\(b^2-4ac<0\)時，沒有實數(shù)根。

5.\(f(x_1)-f(x_2)>\frac{x_1-x_2}{x_1x_2}\)

四、簡答題

1.\(f'(x)=3x^2-3\)。

2.\(f'(0)=1\)。

3.證明：直線和圓相交，意味著直線方程和圓的方程有公共解。將直線方程代入圓的方程，得到關(guān)于\(x\)的一元二次方程。根據(jù)判別式\(b^2-4ac\)的值，判斷方程的解的情況，從而得出結(jié)論。

4.當\(b^2-4ac>0\)時，有兩個不相等的實數(shù)根，說明圖像與\(x\)軸有兩個交點；當\(b^2-4ac=0\)時，有一個重根，說明圖像與\(x\)軸有一個切點；當\(b^2-4ac<0\)時，沒有實數(shù)根，說明圖像與\(x\)軸沒有交點。

5.根據(jù)對數(shù)函數(shù)的性質(zhì)，\(f(x_1)-f(x_2)=\ln(x_1)-\ln(x_2)=\ln\left(\frac{x_1}{x_2}\right)\)。由于\(x_1>x_2>0\)，所以\(\frac{x_1}{x_2}>1\)，根據(jù)對數(shù)函數(shù)的單調(diào)性，有\(zhòng)(\ln\left(\frac{x_1}{x_2}\right)>0\)。同時，根據(jù)對數(shù)函數(shù)的拉格朗日中值定理，存在某個\(