1.函数 y=(1)/(sqrt(-x^(2)+x+2))y=\frac{1}{\sqrt{-x^{2}+x+2}} 的定义域是 qquad\qquad 1 qquad\qquad . y=(1)/(sqrt(-x^(2)+x+2))y=\frac{1}{\sqrt{-x^{2}+x+2}} The domain of the function is
2.函数 y=sqrt((x)/(x-4))y=\sqrt{\frac{x}{x-4}} 定义域 qquad\qquad . 2. Function y=sqrt((x)/(x-4))y=\sqrt{\frac{x}{x-4}} Define the field qquad\qquad .
3.函数 y=3x^(3)y=3 x^{3} 的凸区间是 qquad\qquad . 3. y=3x^(3)y=3 x^{3} The convex interval of the function is qquad\qquad .
4.函数 f(x)=x^(3)-3x^(2)-9x+5f(x)=x^{3}-3 x^{2}-9 x+5 的最大值是 qquad\qquad -. 4. The f(x)=x^(3)-3x^(2)-9x+5f(x)=x^{3}-3 x^{2}-9 x+5 maximum value of the function is qquad\qquad -.
5.假定隐函数 y=x+(1)/(2)cos yy=x+\frac{1}{2} \cos y ,那么 (dy)/(dx)=\frac{d y}{d x}=qquad\qquad . 5. Assuming the implicit function y=x+(1)/(2)cos yy=x+\frac{1}{2} \cos y , then (dy)/(dx)=\frac{d y}{d x}=qquad\qquad .
6.假定 f(x)=x cos xf(x)=x \cos x ,那么 f^('')(x)=f^{\prime \prime}(x)=qquad\qquad . 6. Suppose f(x)=x cos xf(x)=x \cos x that then qquad\qquad . f^('')(x)=f^{\prime \prime}(x)=
7.若 y=f(cos x)y=f(\cos x) 且 f(u)f(u) 是可导的,那么 (dy)/(dx)=\frac{d y}{d x}=qquad\qquad . 7. If y=f(cos x)y=f(\cos x) and f(u)f(u) is derivable, then (dy)/(dx)=\frac{d y}{d x}=qquad\qquad .
8.若 y=f(tan x)y=f(\tan x) 且 f(u)f(u) 是可导的,那么 (dy)/(dx)=\frac{d y}{d x}=qquad\qquad . 8. If y=f(tan x)y=f(\tan x) and f(u)f(u) is derivable, then (dy)/(dx)=\frac{d y}{d x}=qquad\qquad .
9.若 sum_(n=1)^(oo)(1)/(n^(p))\sum_{n=1}^{\infty} \frac{1}{n^{p}} 收玫,则 pp 的取值范围 qquad\qquad . 9. If the rose sum_(n=1)^(oo)(1)/(n^(p))\sum_{n=1}^{\infty} \frac{1}{n^{p}} is harvested, then pp the value range qquad\qquad of .
10.若 sum_(n=1)^(oo)(1)/(n^(p))\sum_{n=1}^{\infty} \frac{1}{n^{p}} 是发散的,则 pp 的取值范围 qquad\qquad . 10. If it sum_(n=1)^(oo)(1)/(n^(p))\sum_{n=1}^{\infty} \frac{1}{n^{p}} is divergent, then the range ppqquad\qquad of values of .
11.函数 (1)/(2+x)\frac{1}{2+x} 在 x=0x=0 处可幂级数展开 则 (1)/(2+x)=\frac{1}{2+x}=qquad\qquad . 11. The function (1)/(2+x)\frac{1}{2+x} can be expanded x=0x=0 by a power series (1)/(2+x)=\frac{1}{2+x}=qquad\qquad at .
12.函数 (1)/(1+x)\frac{1}{1+x} 在 x=0x=0 处可幂级数展开.那么 (1)/(1+x)=\frac{1}{1+x}=qquad\qquad . 12. The function (1)/(1+x)\frac{1}{1+x} can be expanded x=0x=0 by a power series at . (1)/(1+x)=\frac{1}{1+x}= Then qquad\qquad .
二.选择题 Two. Multiple choice questions
1.假定 f(x)={[(x-1)^(2),-1 < x <= 0],[x+1",",0 < x <= 1]:}f(x)=\left\{\begin{array}{ll}(x-1)^{2} & -1<x \leq 0 \\ x+1, & 0<x \leq 1\end{array}\right. ,那么 lim_(x rarr0)f(x)=(quad)\lim _{x \rightarrow 0} f(x)=(\quad) . 1. Suppose f(x)={[(x-1)^(2),-1 < x <= 0],[x+1",",0 < x <= 1]:}f(x)=\left\{\begin{array}{ll}(x-1)^{2} & -1<x \leq 0 \\ x+1, & 0<x \leq 1\end{array}\right. that then lim_(x rarr0)f(x)=(quad)\lim _{x \rightarrow 0} f(x)=(\quad) .
A. 1
B. 0
C.-1
D.不存在 D. Non-existent
2.函数 f(x)=|x-1|quad(quad)f(x)=|x-1| \quad(\quad) . 2. Functions f(x)=|x-1|quad(quad)f(x)=|x-1| \quad(\quad) .
A.在 x=1x=1 处连续且可导 A. Continuous and directable x=1x=1 at
B.在 x=1x=1 处不连续 B. Discontinuity x=1x=1 at
C.在 x=0x=0 处连续且可导 C. Continuous and conductible x=0x=0 at
D.在 x=0x=0 处不连续 D. Discontinuity x=0x=0 at
3 下列哪个函数在[-1,1]满足罗尔定理.( ) 3 Which of the following functions satisfies Roll's theorem at [-1,1].( )
A.f(x)=(1)/(x^(2))f(x)=\frac{1}{x^{2}}
B.f(x)=|x|f(x)=|x|
C.f(x)=1-x^(2)f(x)=1-x^{2}
D.f(x)=x^(2)-2x-1f(x)=x^{2}-2 x-1
4.当 x rarr oox \rightarrow \infty ,下列有极限的是?() 4. When x rarr oox \rightarrow \infty , what are the following limits? ()
A,x sin((1)/(x))x \sin \frac{1}{x}
B, sin((1)/(x^(2)))\sin \frac{1}{x^{2}}
C,(x+1)/(x^(2)-x)\frac{x+1}{x^{2}-x}
D, cot x\cot x
7. int xe^(-x^(2))dx=(quad)\int x e^{-x^{2}} d x=(\quad) .
A.-(1)/(2)e^(-x^(2))+C-\frac{1}{2} e^{-x^{2}}+C
B.-2e^(-x^(2))+C-2 e^{-x^{2}}+C
C.(1)/(2)e^(-x^(2))+C\frac{1}{2} e^{-x^{2}}+C
D.e^(-x^(2))+Ce^{-x^{2}}+C
8.下列哪个级数是收敛( ) 8. Which of the following series is convergence ( )
A.sum_(n=1)^(oo)2^((1)/(n))\sum_{n=1}^{\infty} 2^{\frac{1}{n}}
B.sum_(n=1)^(oo)(sqrt(n+1)-sqrtn)\sum_{n=1}^{\infty}(\sqrt{n+1}-\sqrt{n})
C.sum_(n=1)^(oo)[((3)/(4))^(n)+(1)/(sqrt(n+1))]\sum_{n=1}^{\infty}\left[\left(\frac{3}{4}\right)^{n}+\frac{1}{\sqrt{n+1}}\right]
D.sum_(n=1)^(oo)(1)/(3n^(3))\sum_{n=1}^{\infty} \frac{1}{3 n^{3}}
9.下列哪个级数是发散的( )。 9. Which of the following series is divergent ( ).
A.sum_(n=1)^(oo)(1-cos((pi )/(n)))\sum_{n=1}^{\infty}\left(1-\cos \frac{\pi}{n}\right)
B.sum_(n=1)^(oo)(1)/(n^(2))\sum_{n=1}^{\infty} \frac{1}{n^{2}}
C.sum_(n=2)^(oo)sin((1)/(sqrtn))\sum_{n=2}^{\infty} \sin \frac{1}{\sqrt{n}}
D.sum_(n=1)^(oo)(1)/(n!)\sum_{n=1}^{\infty} \frac{1}{n!} ;
10.级数 sum_(n=0)^(oo)(-1)^(n)(1)/(n+1)\sum_{n=0}^{\infty}(-1)^{n} \frac{1}{n+1} 是( ) 10. Series sum_(n=0)^(oo)(-1)^(n)(1)/(n+1)\sum_{n=0}^{\infty}(-1)^{n} \frac{1}{n+1} Yes ( )
A.绝对收玫 A. Absolute Harvest
B.条件收敛 B. Conditional convergence
C.发散 C. Divergence
D.收敛性不能确定 D. Convergence cannot be determined
1.计算 lim_(x rarr1)(x^(2)-1)/(x^(2)+2x-3)\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{2}+2 x-3} . 1. Calculation lim_(x rarr1)(x^(2)-1)/(x^(2)+2x-3)\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{2}+2 x-3} .
2.计算 lim_(x rarr oo)(2x^(3)+3x^(2)+5)/(7x^(3)+4x^(2)-1)\lim _{x \rightarrow \infty} \frac{2 x^{3}+3 x^{2}+5}{7 x^{3}+4 x^{2}-1} . 2. Calculation lim_(x rarr oo)(2x^(3)+3x^(2)+5)/(7x^(3)+4x^(2)-1)\lim _{x \rightarrow \infty} \frac{2 x^{3}+3 x^{2}+5}{7 x^{3}+4 x^{2}-1} .
3.若 y=e^(x)cos x+(x)/(1-x)y=e^{x} \cos x+\frac{x}{1-x} ,则 y^(')y^{\prime} . 3. If y=e^(x)cos x+(x)/(1-x)y=e^{x} \cos x+\frac{x}{1-x} , then y^(')y^{\prime} .
4.若 y=e^(x)cos x+(x)/(1-x)y=e^{x} \cos x+\frac{x}{1-x} ,则 y^(')y^{\prime} . 4. If y=e^(x)cos x+(x)/(1-x)y=e^{x} \cos x+\frac{x}{1-x} , then y^(')y^{\prime} .
5.计算 int_(0)^(1)(sqrtx-x^(2))dx\int_{0}^{1}\left(\sqrt{x}-x^{2}\right) d x 5 int_(0)^(1)(sqrtx-x^(2))dx\int_{0}^{1}\left(\sqrt{x}-x^{2}\right) d x . Calculations
6.计算 int_(0)^(pi)(sin x+x)dx\int_{0}^{\pi}(\sin x+x) d x 6 int_(0)^(pi)(sin x+x)dx\int_{0}^{\pi}(\sin x+x) d x . Calculations
7.求 int_(0)^(2)xsqrt(4-x^(2))dx\int_{0}^{2} x \sqrt{4-x^{2}} d x . 7. Seek int_(0)^(2)xsqrt(4-x^(2))dx\int_{0}^{2} x \sqrt{4-x^{2}} d x .
8.求 int_(0)^(pi)x cos xdx\int_{0}^{\pi} x \cos x d x . 8. Seek int_(0)^(pi)x cos xdx\int_{0}^{\pi} x \cos x d x .
9.求 intx^(3)ln xdx\int x^{3} \ln x d x . 9. Seek intx^(3)ln xdx\int x^{3} \ln x d x .
10.求 int x cos x^(2)dx\int x \cos x^{2} d x . 10. Seek int x cos x^(2)dx\int x \cos x^{2} d x .
11 求出下列级数的收玫区间 sum_(n=0)^(oo)(x^(2n+1))/(2n+1)\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{2 n+1} . 11 Find the harvest interval for the following series sum_(n=0)^(oo)(x^(2n+1))/(2n+1)\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{2 n+1} .
12.求出级数的和 sum_(n=1)^(oo)(1)/(2^(n))+(1)/(3^(n))\sum_{n=1}^{\infty} \frac{1}{2^{n}}+\frac{1}{3^{n}} . 12. Find the sum of the series sum_(n=1)^(oo)(1)/(2^(n))+(1)/(3^(n))\sum_{n=1}^{\infty} \frac{1}{2^{n}}+\frac{1}{3^{n}} .
四.应用题 Four. Vocabulary questions
1.一人在家制作手工艺品,用以在网上销售,每周制作 xx 件的总成本 C_("是制作 ")C_{\text {是制作 }}量的函数 C(x)=10x^(2)+60 x-100C(x)=10 x^{2}+60 x-100 元,如果每件工艺品售价 300 元,且所制作的工艺品可以全部售出,求使利润最大的周制作量及最大利润 1. A person makes handicrafts at home for online sales, and the total cost of the xx pieces produced per week is C(x)=10x^(2)+60 x-100C(x)=10 x^{2}+60 x-100 a function C_("是制作 ")C_{\text {是制作 }} of yuan, if each handicraft is sold for 300 yuan, and the handicrafts made can be sold in full, so as to make the largest weekly production volume and the maximum profit
2.已知某产品的总收益函数为 R=100+33 x-4x^(2)R=100+33 x-4 x^{2}(万元),总成本函数为 C=x^(3)-7x^(2)+24 x+6C=x^{3}-7 x^{2}+24 x+6(万元),其中 x\boldsymbol{x} 表示产品的产量(单位千件),求利润函数以及该产品获得最大利润时的产量和最大利润。 2. It is known that the total revenue function of a product is R=100+33 x-4x^(2)R=100+33 x-4 x^{2} (10,000 yuan) and the total cost function is C=x^(3)-7x^(2)+24 x+6C=x^{3}-7 x^{2}+24 x+6 (10,000 yuan), where x\boldsymbol{x} represents the output of the product (in 1,000 pieces), finds the profit function and the output and maximum profit when the product obtains the maximum profit.