Homework 08

约 336 个字 5 张图片 预计阅读时间 2 分钟

3.2.5

（a）

\(x_0=0.0\)	1.0000
\(x_1=0.2\)	1.22140	1.1070
\(x_2=0.4\)	1.49182	1.3521	0.61275
\(x_3=0.6\)	1.82212	1.6515	0.7485	0.22625
\(x_4=0.8\)	2.22554	2.0171	0.914	0.27583	0.061975

\[ \begin{aligned} \therefore f(0.05)&=f(0)+1.1070*(0.05-0)+0.61275*(0.05-0)*(0.05-0.2)\\ &+0.22625*(0.05-0)*(0.05-0.2)*(0.05-0.4)\\ &+0.061975*(0.05-0)*(0.05-0.2)*(0.05-0.4)*(0.05-0.6)\approx 1.0513 \end{aligned} \]

（b）

\[ \begin{aligned} f(0.65)&=f(0.8)+2.0171*(0.65-0.8)+0.914*(0.65-0.8)*(0.65-0.6)\\ &+0.27583*(0.65-0.8)*(0.65-0.6)*(0.65-0.4)\\ &+0.061975*(0.65-0.8)*(0.65-0.6)*(0.65-0.4)*(0.65-0.2)\approx 1.91555 \end{aligned} \]

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3.2.13

由 \(f[x_1,x_2]=\frac{f[x_2]-f[x_1]}{x_2-x_1}=10\Rightarrow f[x_1]=3\)

再由 \(f[x_0,x_1,x_2]=\frac{f[x_1,x_2]-f[x_0,x_1]}{x_2-x_0}=\frac{50}{7}\Rightarrow f[x_0,x_1]=5\)

最后由 \(f[x_0,x_1]=\frac{f[x_1]-f[x_0]}{x_1-x_0}=5\Rightarrow f[x_0]=1\)

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3.3.7

（a）

\[ \begin{aligned} H_9(x)&=75x+0.222222x^2(x-3)-0.0311111x^2(x-3)^2-0.00644444x^2(x-3)^2(x-5)\\ &+0.00226389x^2(x-3)^2(x-5)^2-0.000913194x^2(x-3)^2(x-5)^2(x-8)\\ &+0.000130572x^2(x-3)^2(x-5)^2(x-8)^2\\ &-0.0202236x^2(x-3)^2(x-5)^2(x-8)^2(x-13)\\ \end{aligned} \]

代入可得 \(H_9(10)=743,\frac{dH_9}{dx}|_{x=10}=48\)

（b）

\(x\approx 5.6488092\)

（c）

\(\frac{dH_9}{dx}\leq 119.423\)

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3.4.9

\[ \begin{aligned} s_1(3)&=1+b-\frac{3}{4}+d=0\\ s_0(2)&=s_1(2)\Rightarrow 1+B-D=1\\ s_0'(2)&=s_1'(2)\Rightarrow B-3D=b\\ s_0''(2)&=s_1''(2)\Rightarrow -6D=-\frac{3}{2}\\ \therefore B=&D=\frac{1}{4},b=-\frac{1}{2},d=\frac{1}{4} \end{aligned} \]

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3.4.17

\(f(0)=1,f(0.05)=1.1052,f(0.1)=1.2214\)

\[ \therefore F(x)=\begin{cases} \frac{x-0.05}{-0.05}+\frac{x}{0.05}\times 1.1052 & 0\leq x\leq 0.05\\ \frac{x-0.1}{-0.05}\times 1.1052+\frac{x-0.05}{0.05}\times 1.2214 & 0.05\leq x\leq 0.1\\ \end{cases} \]

\(\therefore\int_0^1F(x)dx=0.1107936,\int_0^{0.1}e^{2x}dx=0.1107014\)

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