Concept explainers
Determine the real root of
(a) Graphically.
(b) Using bisection to determine the root to
(c) Perform the same computation as in (b) but use the false-position method and
(a)
The real roots of the equation
Answer to Problem 3P
Solution:
The approximate real root of the equation is 0.6.
Explanation of Solution
Given Information:
The equation
Calculation:
The graph of the function can be plotted using MATLAB.
Code:
Output:
This gives the following plot:
The roots of an equation can be represented graphically by the x-coordinate of the point where the graph cuts the x-axis. From the plot, the only zeros of the equation can be approximated as 0.6.
(b)
To calculate: The root of the equation
Solution:
The root of the equation can be approximated as 0.59375.
Given Information:
The equation
Formula Used:
A root of an equation can be obtained using the bisection method as follows:
1. Choose 2 values x, say a andb such that
2. Now, estimate the root by
3. If,
Calculation:
For the provided function:
Hence,
Now take the first root to be,
Now,
Thus,
Now, the second root would be:
The approximate error can be computed as:
The approximate relative percentage error is 200%.
Now,
Thus,
Now, the third root would be:
The approximate error can be computed as:
The approximate error is 11.1%.
Now,
Thus,
Now, the fourth root would be:
The approximate error can be computed as:
The approximate error is 5.26%.
As
(c)
To calculate: The root of the equation
Solution:
The root of the equation can be approximated as 0.57956.
Given Information:
The equation
Formula Used:
A root of an equation can be obtained using the false-position method as follows:
1. Choose 2 values x, say a andb such that
2. Now, estimate the root by
3. If,
Calculation:
For the provided function:
Hence,
Now take the first root to be,
Now,
Thus,
Now, the second root would be:
The approximate error can be computed as:
The approximate relative percentage error is 200%.
Now,
Thus,
Now, the third root would be:
The approximate error can be computed as:
The approximate error is 1.29%.
Now,
Thus,
Now, the fourth root would be:
The approximate error can be computed as:
The approximate error is 0.17%.
As
(b)
To calculate: The root of the equation
Answer to Problem 3P
Solution:
The root of the equation can be approximated as 0.59375.
Explanation of Solution
Given Information:
The equation
Formula Used:
A root of an equation can be obtained using the bisection method as follows:
1. Choose 2 values x, say a andb such that
2. Now, estimate the root by
3. If,
Calculation:
For the provided function:
Hence,
Now take the first root to be,
Now,
Thus,
Now, the second root would be:
The approximate error can be computed as:
The approximate relative percentage error is 200%.
Now,
Thus,
Now, the third root would be:
The approximate error can be computed as:
The approximate error is 11.1%.
Now,
Thus,
Now, the fourth root would be:
The approximate error can be computed as:
The approximate error is 5.26%.
As
(c)
To calculate: The root of the equation
Solution:
The root of the equation can be approximated as 0.57956.
Given Information:
The equation
Formula Used:
A root of an equation can be obtained using the false-position method as follows:
1. Choose 2 values x, say a andb such that
2. Now, estimate the root by
3. If,
Calculation:
For the provided function:
Hence,
Now take the first root to be,
Now,
Thus,
Now, the second root would be:
The approximate error can be computed as:
The approximate relative percentage error is 200%.
Now,
Thus,
Now, the third root would be:
The approximate error can be computed as:
The approximate error is 1.29%.
Now,
Thus,
Now, the fourth root would be:
The approximate error can be computed as:
The approximate error is 0.17%.
As
(c)
To calculate: The root of the equation
Answer to Problem 3P
Solution:
The root of the equation can be approximated as 0.57956.
Explanation of Solution
Given Information:
The equation
Formula Used:
A root of an equation can be obtained using the false-position method as follows:
1. Choose 2 values x, say a andb such that
2. Now, estimate the root by
3. If,
Calculation:
For the provided function:
Hence,
Now take the first root to be,
Now,
Thus,
Now, the second root would be:
The approximate error can be computed as:
The approximate relative percentage error is 200%.
Now,
Thus,
Now, the third root would be:
The approximate error can be computed as:
The approximate error is 1.29%.
Now,
Thus,
Now, the fourth root would be:
The approximate error can be computed as:
The approximate error is 0.17%.
As
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