Solutions to selected exercises Use the Bisection method to nd solutions accurate to within 102 for x3 7x2 + 14x 6 = 0 on [0,1]. Solution: Let f (x) = x3 7x2 + 14x 6 = 0. Note that f (0) = 6 < 0 and f (1) = 2 > 0, therefore, based on the Intermediate Valu
Math 4650 Homework #3 Solutions 1.3 #7. Find the rates of convergence of the following functions as h 0. 1. lim sin h = 1. h0 h Use sin h = h
+ , and we get
sin h h h3 h2 sin h 1= = + = + . h h 6h 6 So its O(h2 ). 1 cos h = 0. 2. lim h0 h 2 Use cos
Math 4650 Homework #2 Solutions 1.2 #3ab. Suppose p must approximate p with relative error at most 103 . Find the largest interval in which p must lie for each value of p. Solution: The relative error is |p p| = , p
so that we must have (1 )p p (1 + )p 0.
Math 4650 Homework #1 Solutions 1. 1.1 #2. Find intervals containing solutions to the following equations. Solution: For all of these problems, we need to use the intermediate value theorem. So we want to nd numbers a and b such that f (a) > 0 and f (b) <