Chapter 2: Worksheet 4 Jack K. Cohen Colorado School of Mines

Intermediate Value Theorem and the Method of Bisection

Suggested Problems Section 2.4: 48, 52, 54

1. Prove that every cubic equation, ax³ + bx² + cx + d = 0    (a≠0), has at least one real root.

2. The folklore says that it takes about 10 bisection steps to gain 3 decimal places of accuracy. For example, if we know the root to within an integer, then 10 bisections will determine 3 places after the decimal point. The folklore is right—explain why.

3. It takes about 4 bisections to gain one decimal place. Given that the asteroid equation, cos(1000/r) = r/(r + 20), has a root in the interval [50008.325, 50008.335], do 4 steps of bisection by hand (by reasoning similar to that in the previous problem, 4 steps is enough to pin down the third digit after the decimal point accurately). You get no credit for a series of cryptic equations! Explain, using clear sentences, your successive calculations and results.

4. We've made available a code called Bisect. Here is the syntax:

   Bisect[f[x], {x, a, b}, eps]
   

   To get that third decimal place in the previous exercise, you would type:

   Bisect[Cos[1000/r] - r/(r + h), {r, 50008.325, 50008.335}, .0005]
   

   This should produce the output:

   50008.33
   50008.3325
   50008.33125
   50008.331875
   50008.3321875
   
   50008.3320312
   

   Using Bisect, obtain the asteroid root r to six places after the decimal point starting from the approximation we have so far (50008.332). Bisect doesn't let you directly specify the folklore 10 iterations, so you have to think about the eps value that will accomplish the task. Tell what your selection for a, b and eps was with your answer.

5. If we knew only that the asteroid root r was in the interval [5000, 100000], would the method of bisection be a practical tool?