Homework 01¶
1.2.3¶
（a）\(\text{largest interval}=150\pm 150\times 10^{-3}=150\pm 0.15=[149.85,150.15]\)
（b）\(\text{largest interval}=900\pm 900\times 10^{-3}=900\pm 0.9=[899.1,900.9]\)
（c）\(\text{largest interval}=1500\pm 1500\times 10^{-3}=1500\pm 1.5=[1498.5,1501.5]\)
（d）\(\text{largest interval}=90\pm 90\times 10^{-3}=90\pm 0.09=[89.91,90.09]\)
1.2.11¶
（a）\(\lim\limits_{x\rightarrow 0}f(x)=\lim\limits_{x\rightarrow 0}\frac{-x\sin x}{1-\cos x}=\lim\limits_{x\rightarrow 0}\frac{-\sin x-x\cos x}{\sin x}=\lim\limits_{x\rightarrow 0}\frac{-2\cos x+x\sin x}{\cos x}=-2\)
（b）\(f(0.1)=\frac{0.1\times 0.9950-0.0998}{0.1-0.0998}=-1.500\)
（c）\(f(x)=\frac{x(1-\frac{1}{2}x^2)-(x-\frac{1}{6}x^3)}{x-(x-\frac{1}{6}x^3)}=-2\)
（d）\(\text{rel}_{b}=\frac{-1.5+1.99899998}{1.99899998}=24.96\%\)
\(\text{rel}_c=\frac{2-1.99899998}{1.99899998}=0.05\%\)
1.2.17(b)¶
\(x_1=\frac{1.31\times 4.76-1.93\times 3.24}{4.76-3.24}=\frac{6.236-6.253}{1.52}=-0.011\)
\(x_2=1.31-\frac{(1.93-1.31)\times 3.24}{4.76-3.24}=1.31-\frac{0.62\times 3.24}{1.52}=1.31-\frac{2.009}{1.52}=1.31-1.322=-0.012\)
\(\text{Exact Value}=-0.01157895\)
\(\text{rel}_1=\frac{-0.011+0.01157895}{0.01157895}=5.00\%,\text{rel}_2=\frac{0.012-0.01157895}{0.01157895}=3.64\%\)
\(\therefore\) 第二种更好
1.3.7¶
（a）\(O(h^2)\) （b）\(O(h)\) （c）\(O(h^2)\) （d）\(O(h)\)