EXERCISE SET 2.1

Numerical Analysis數值分析筆記 /  

2.1 The Bisection Method /  

習題組 2.1 /

NO.1.(more...)
use the bisetion method to find the $p_3$ for $f(x)=\sqrt{x}-\cos x $ on $[0,1]$.


NO.2.(more...)
$f(x)=3(x+1)(x-\frac{1}{2})(x-1)$. Use the Bisection method on the following intervals to find $p_{3}$.

• $\left[-2,\ 1.5\right]$ 
• $\left[-1.25,\ 2.5\right]$ 

NO.3.(more...)
use the Bisection method to find solutions accurate to within $10^{-2}$ for $x^{3}-7x^{2}+14x-16=0$ on each interval.

• $\left[0,\ 1\right]$
• $\left[1,\ 3.2\right]$
• $\left[3.2,\ 4\right]$

NO.4.(more...)
use the Bisection method to find solutions accurate to within $10^{-2}$ for $x^{4}-2x^{3}-4x^{2}+x+4=0$ on each interval.

• $\left[-2,\ -1\right]$
• $\left[0,\ 2\right]$
• $\left[2,\ 3\right]$
• $\left[-1,\ 0\right]$

NO.5.(more...)
use the Bisection method to find solutions accurate to within $10^{-5}$ for the following problems.

• $x-2^{-x}=0$ for $0\leq x\leq 1$
• $e^{x}-x^2+3x-2=0$ for $0\leq x\leq 1$
• $x2\cos(2x)-(x+1)^{2}=0$ for $-3\leq x\leq-2$ and $-1\leq x\leq0$
• $x\cos x-2x^{2}+3x-1=0$ for $0.2\leq x\leq0.3$ and $1.2\leq x\leq1.3$

NO.6.(more...) 
use the Bisection method to find solutions accurate to within $10^{-5}$ for the following problems.

• $3x-e^{x}=0$ for $1\leq x\leq2$
• $x+3\cos x-e^{x}=0$ for $0\leq x\leq1$
• $x^{2}-4x+4-\ln x=0$ for $1\leq x\leq2$ and $2\leq x\leq4$
• $x+1-2\sin\pi$x=0 for $0\leq x\leq0.5$ and $0.5\leq x\leq1$

NO.7.(more...)

sketch the graphs of $y=x$ and $y=2\sin x$. use the bisection method to find an  approximation to within $10^{-5}$ to the first positive value of $x$ with $x=2\sin x$.

NO.8.(more...)

sketch the graphs of $y=x$ and $y=\tan x$. use the bisection method to find an  approximation to within $10^{-5}$ to the first positive value of $x$ and $y=\tan x$.

NO.9.(more...)

sketch the graphs of $y=e^{x}-2$ and $y=\cos(e^{x}-2)$.  use the bisection method to find an approximation to within $10^{-5}$ to a value in $\left[0.5,\ 1.5\right]$  with $e^{x}-2=\cos(e^{x}-2)$.

NO.10.(more...)

let $f(x)=(x+2)(x+1)^{2}x(x-1)^{3}(x-2)$, to which zero of $f$ does the bisection method converge when applied on the following intervals?

• $\left[-1.5,\ 2.5\right]$  
• $\left[-0.5,\ 2.4\right]$ 
• $\left[-0.5,\ 3\right]$  
• $\left[-3,\ -0.5\right]$

NO.11.(more...)

let $f(x)=(x+2)(x++1)x(x-1)^{3}(x-2)$, to which zero of $f$ does the bisection method converge when applied on the following intervals?

• $\left[-3,\ 2.5\right]$  
• $\left[-2.5,\ 3\right]$
• $\left[-1.75,\ 1.5\right]$  
• $\left[-1.5,\ 1.75\right]$

NO.12.(more...)

find an approximation to $\sqrt{3}$ correcto within $10^{-4}$ using the bisection algorithm. hint: consider $f(x)=x^{2}-3$.

NO.13.(more...)

find an approximation to $\sqrt[3]{25}$ correct to within $10^{-4}$ using the bisection algorithm.

NO.14.(more...)

use theorem 2.1 to find a bound for the number of iterations needed to achieve an approximation with accuracy $10^{-3}$ to the solution of $x^{3}+x-4=0$ lying in the interval $\left[1,\ 4\right]$, find an approximation to the root with this degree of accuracy.

NO.15.(more...)

use theorem 2.1 to find a bound for the number of iterations needed to achieve an approximation with accuracy $10^{-4}$ to the solution of $x^{3}-x-1=0$ lying in the interval $\left[1,\ 2\right]$, find an approximation to the root with this degree of accurracy.

NO.16.(more...)

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NO.20.(more...)