Using bisection method two times on the equation

We get the end points
a. .
ob. .
c. .
d. None.

Ask by Schneider Joseph. in Egypt

Mar 25,2025

Question

Using bisection method two times on the equation $$ e^{-x}=	an x $$ We get the end points a. $$ [0.11,0.625] $$. ob. $$ [0.5,0.625] $$. c. $$ [0.625,0.75] $$. d. None.

UpStudy AI · Accepted Answer

To solve the equation $$ e^{-x} = 	an x $$ using the bisection method, we need to find the interval where the function $$ f(x) = e^{-x} - 	an x $$ changes sign.

Given that the end points are $$ a = 0.11 $$ and $$ b = 0.625 $$, we can use the bisection method to find the root of the equation.

Let's denote the function as $$ f(x) = e^{-x} - 	an x $$.

1. Check the sign of $$ f(a) $$ and $$ f(b) $$:
   - $$ f(0.11) = e^{-0.11} - 	an(0.11) $$
   - $$ f(0.625) = e^{-0.625} - 	an(0.625) $$

2. Determine the sign of $$ f(a) $$ and $$ f(b) $$:
   - If $$ f(a) $$ and $$ f(b) $$ have opposite signs, then the root lies between $$ a $$ and $$ b $$.

3. Calculate the midpoint $$ c = \frac{a + b}{2} $$ and check the sign of $$ f(c) $$:
   - If $$ f(c) $$ has the same sign as $$ f(a) $$, then the root lies in the interval $$ [a, c] $$.
   - If $$ f(c) $$ has the opposite sign as $$ f(a) $$, then the root lies in the interval $$ [c, b] $$.

Let's calculate the values of $$ f(0.11) $$ and $$ f(0.625) $$ to determine the sign of the function at these points.
Calculate or simplify the expression $$ e^(-0.625) - tan(0.625) $$.
The values of $$ f(0.11) $$ and $$ f(0.625) $$ are approximately 0.785388 and -0.186223, respectively.

Since $$ f(0.11) $$ and $$ f(0.625) $$ have opposite signs, the root of the equation $$ e^{-x} = 	an x $$ lies between $$ a = 0.11 $$ and $$ b = 0.625 $$.

Now, we need to find the midpoint $$ c = \frac{a + b}{2} $$ and check the sign of $$ f(c) $$ to determine the interval where the root lies.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(0.11+0.625ight)}{2}$$
- step1: Remove the parentheses:
  $$\frac{0.11+0.625}{2}$$
- step2: Add the numbers:
  $$\frac{0.735}{2}$$
- step3: Convert the expressions:
  $$\frac{\frac{147}{200}}{2}$$
- step4: Multiply by the reciprocal:
  $$\frac{147}{200}	imes \frac{1}{2}$$
- step5: Multiply the fractions:
  $$\frac{147}{200	imes 2}$$
- step6: Multiply:
  $$\frac{147}{400}$$
Calculate or simplify the expression $$ e^(-0.3675) - tan(0.3675) $$.
The value of $$ f(0.3675) $$ is approximately 0.307473.

Since $$ f(0.3675) $$ has the same sign as $$ f(0.11) $$, the root of the equation $$ e^{-x} = 	an x $$ lies in the interval $$ [0.11, 0.3675] $$.

Therefore, the correct answer is:
a. $$ [0.11, 0.625] $$
The correct interval after two bisection method iterations is $$ [0.11, 0.625] $$.

Using bisection method two times on the equation

We get the end points
a. .
ob. .
c. .
d. None.

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Using bisection method two times on the equation We get the end points a. . ob. . c. . d. None.