1 (a) Solve the quadratic equation

by evaluating the quadratic formula using three-digit decimal arithmetic and

rounding. The exact roots rounded to 6 digits are 0.0399675 and 12.3600.

(b) You should observe loss of significance in one of your computed zeros of

part (a). If we consider the 6-digit answers provided as the “true” ones, what is

this solution’s relative error? Following Definition 2.2 in Kharab & Guenther

(p. 33), to how many significant digits does your computed root approximate

the true one? Explain what aspect of the coefficients in the quadratic equation

to be solved led to this loss of significance.

(c) Show that the roots r_{1}, r_{2} of a quadratic equation ax^2+bx+c = 0 satisfy
r_{1}r_{2} = c/a.

(d) While there is loss of significance in one solution of the quadratic
equation in (a),

it does not occur in the other. Write an Octave/Matlab function which accepts

as a vector [ a b c] the coefficients of a quadratic function f (x) = ax^2 + bx +
c,

computes the root that has larger absolute value, and then obtains the other

root using the formula in part (c). You may assume that the roots are distinct

real numbers—i.e., that b^2 - 4ac > 0. (Hand in a printout of your function.)

(e) As in part (a), use 3-digit decimal arithmetic to solve the quadratic equation

whose exact roots rounded to 6 digits are 6.00000 and 6.40000. You should
once

again observe loss of significance. Does our method in part (d) for dealing with

loss of significance help here? If not, what aspect of this situation makes it

different?

2.2.5 If x_{1}, x_{2} are approximations to X_{1}, X_{2} with errors show that
the relative error

of the product X_{1}X_{2} is approximately the sum

2 In class we showed that, under suitable assumptions on f and h, we have

where lies between x and x + h, and lies between x and x - h.

(a) What, exactly, are these assumptions on f and h?

(b) Supposing that f''' is suitably continuous, show that the truncation
error term

may be replaced by a single term
(Hint.

Use the IVT.) In what interval could we assume this new number resides?

9.1.3 Derive the following approximate formulas.

9.1.3 Use Taylor’s series to derive the following approximation formul for
the first derivative

of f .

9.1.9 Show that the approximation formula in Exercise 9.1.8 has an error of O(h^3).