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Welcome to our Math lesson on Exercises involving Iteration, this is the third lesson of our suite of math lessons covering the topic of Iterative Methods for Solving Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
When an Exercise involving Iteration is considered as Completed?
An exercise involving iteration is considered as completed when after applying the iteration (no matter which of the two methods explained earlier you are using) a number of times, the variable is isolated in the desired range.
In all questions solved so far in this tutorial, we found the result by reasoning. Thus, when we were able to limit the solution within the desired borders, the result was considered as found. Let's see another example in this regard.
Example 5
Find one root of the equation
2x4 - 13x3 + 82 = 0
to one decimal place. Start checking from x = 0.
Solution 5
For x = 0, we have
2x4 - 13x3 + 82
= 2 ∙ 04 - 13 ∙ 03 + 82
= 82 (positive)
For x = 1, we have
2x4 - 13x3 + 82
= 2 ∙ 14 - 13 ∙ 13 + 82
= 2 - 13 + 82
= 71 (positive)
For x = 2, we have
2x4 - 13x3 + 82
= 2 ∙ 24 - 13 ∙ 23 + 82
= 2 ∙ 16 - 13 ∙ 8 + 82
= 32 - 104 + 82
= 12 (positive)
For x = 3, we have
2x4 - 13x3 + 82
= 2 ∙ 34 - 13 ∙ 33 + 82
= 2 ∙ 81 - 13 ∙ 27 + 82
= 162 - 351 + 82
= - 107 (negative)
The change in sign of the result means the root we are looking for, lies between 2 and 3. Hence, let's continue with the half - interval method to get as closer to the result as possible, as we did in other examples. Thus, for x = 2.5 we have
2x4 - 13x3 + 82
= 2 ∙ 2.54 - 13 ∙ 2.53 + 82
= 2 ∙ 39.0625 - 13 ∙ 15.625 + 82
= 78.125 - 203.125 + 82
= - 43 (negative)
Therefore, the result is between 2 and 2.5, as in this interval there is a change in sign. Hence, for x = 2.25, we have
2x4 - 13x3 + 82
= 2 ∙ 2.254 - 13 ∙ 2.253 + 82
= 2 ∙ 25.6289 - 13 ∙ 11.3906 + 82
= 51.2578 - 148.0778 + 82
= - 14.82 (negative)
Hence, the result is between 2.0 and 2.25. Now, it is better to change strategy, i.e. to take the values one by one. Thus, for x = 2.1, we have
2x4 - 13x3 + 82
= 2 ∙ 2.14 - 13 ∙ 13 + 82
= 2 ∙ 19.45 - 13 ∙ 9.26 + 82
= 38.9 - 120.4 + 82
= 0.5 (positive)
This is the closest value to zero obtained so far. Therefore, the result is x = 2.1
More Iterative Methods for Solving Equations Lessons and Learning Resources
Equations Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 9.4 | Iterative Methods for Solving Equations | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 9.4.1 | Iterative Methods for Solving Equations | | |
| 9.4.2 | Applications of Iterative Methods | | |
| 9.4.3 | Exercises involving Iteration | | |
| 9.4.4 | Recursive Iteration and Iteration Machines | | |
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