Final answer:
True statements about the incremental search method are that it may miss roots that are close together (A) and that its accuracy improves with more intervals (D), but the method does not necessarily give the precise value of a root (B) or always find every root (C), and the accuracy improvement is not directly proportional to the number of intervals (E).
Step-by-step explanation:
The student asked which statements about the incremental search method for finding roots are true. Here are the details on each:
- A is true. If the interval is too large, the incremental search method may miss roots that are close together.
- B is false. The incremental search method identifies the interval in which a root lies, but doesn't necessarily give the exact value of the root.
- C is false. It's possible for the method to miss roots entirely, especially if the function is not sampled finely enough.
- D is true. As the interval size decreases (meaning more intervals), the accuracy of the incremental search method generally improves as the method can detect changes in the function's sign more reliably.
- E is generally false. The improvement in accuracy is not directly proportional to the number of intervals because the function's behavior and the interval size also play significant roles.
Refining the Search
For better results in the incremental search method, one can decrease the interval size, which increases the number of intervals. The precision of the search is related to the size of these intervals—smaller intervals can lead to more precise detection of roots. However, precision does not improve at a rate directly proportional to the number of intervals, as it also depends on the nature of the function and other factors like computational resources.