Question

Which of the statements below about the interval method are true? (select all that are true)

a.The interval method may miss roots that are close together, specifically when there are two roots in close proximity and the interval size relative to the root spacing is large
b.The interval method gives both the interval in which the root is located and the actual precise location of the root
c.The interval method will always find every root of the function being examined. It is a great method, but very slow
d.The interval method accuracy improves as the number of intervals being used increases.This improvement in accuracy is directly proportional to the number of intervals.
e.The interval method is a closed method -- which means an upper and lower value are required to bound the range over which a root is sought.
f.If an odd number of roots exist between the lower and the upper limit, at least one interval will be found in which a root is said to exist

Answer 1

The interval method may miss roots when intervals are large (true); it estimates intervals where roots might be but doesn't give exact locations (false); it doesn't always find every root (false); more intervals can improve accuracy but not in direct proportion (partially true); it requires upper and lower bounds (true); and it will find at least one root if an odd number exists within the bounds (true).

Step-by-step explanation:

Among the options provided regarding the interval method in mathematical computations, particularly for locating roots of functions, some are accurate while others are not.

• a. This statement is true. When the interval size is large relative to the spacing between roots, the interval method may indeed miss roots that are close together. Therefore, care must be taken when choosing the size of intervals.
• b. This statement is false. The interval method provides an estimate of the interval within which the root may be located; it does not give the exact location of the root. Precise root-finding requires further numerical methods.
• c. This statement is false. While the interval method can be effective in finding roots, it does not guarantee to locate every root, especially if the function's behavior is not considered when selecting intervals.
• d. This statement is partially true. Increasing the number of intervals can indeed lead to better accuracy in identifying where roots may lie, but the improvement is not strictly 'directly proportional'. Computational complexity and other factors may affect the outcome.
• e. This statement is true. The interval method is indeed a closed method that involves specifying a range defined by upper and lower bounds within which the search for roots is conducted.
• f. This statement is true. If the function examined changes sign across an interval (an indication of a root), and if an odd number of roots exist, at least one interval will show a change in function sign, suggesting the presence of a root.

Therefore, it is clear that statements a, e, and f accurately describe how the interval method functions, with qualifications needed for statement d.