You should always perform 3 or 4 iterations of my classroom example(s) BEFORE attempting the homework.

1. Clearly and precisely write the Intermediate Value Theorem.

2. What is the rate of convergence of the bisection method? I.e., is it linear, quadratic, cubic?

3. The function f(x) = 2sin(x) – 1/4*exp(x) has two zeros near x = –5.
   1. Using a starting interval [–7, –5], apply 6 iterations of the bisection method to approximate one the zeros.
   2. Using a starting interval [–5, –3], apply 6 iterations of the bisection method to approximate the other zero.

4. Write this Maple code for the bisection method.
   Then use your working code to do both parts of problem 3 above, and iterate until your approximation has converged to 5 digits.
   Write this code as soon as possible and save it, because you will use it as a template (model) for Programming Assignment 1.

5. See what happens if you rework the example from lecture but using a starting interval [1, 9]. What do the iterates converge to?
   After working this problem, see the answers here.

6. Read and do the Maple example for using Maple's fsolve command to approximate a zero of a function.
   Then use Maple's fsolve command to approximate the zeros in problem 3 above.

7. See the Bisection Method Algorithm (pseudocode).

8. What is the other name of the bisection method?

   
   

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   Do the following AFTER we discuss the error analysis of the bisection method on Thursday.

9. If we use the bisection method to approximate a zero of a function on the interval [–2,3], what is the error bound after 12 iterations?
   Answer:   The error of the 12th iterate x_m is guaranteed to be no greater than 0.001220703... units.

10. If we use interval halving to approximate a zero of a function on the interval [–2,3], how many iterations should be performed to obtain at least 12 decimal place accuracy in our approximation?
    Answer:   44 iterations guarantees at least 12 decimal place accuracy. OR: the 44th iterate is guaranteed to be correct (agrees with the correct value of zero z) to at least 12 decimal places.

11. For the function

    f (x) = e^–x – cos x,

    use the bisection method's error bound formula to determine the number of iterations required to guarantee that we would approximate a zero with an accuracy of at least 7 decimal places if we were to start on the interval [2,7].
    NOTE:   Use the error bound formula only — do NOT actually apply the bisection method to answer the question.
    Answer:   27 iterations guarantees at least 7 decimal place accuracy. OR: the 27th iterate is guaranteed to be correct (agrees with the correct value of zero z) to at least 7 decimal places.