MATH 1070, Exam 3

December 7, 2009

12:00 to 13:50

Instructions

1. You may use a one page formula sheet. Formula sheets may not be shared.

2. Before you begin, enter your name in the space below.

3. Show all your work on the exam itself. If you need additional space, use thebacks of the pages.

4. You may not use books or notes on the exam.

Name

1 1252 1253 1254 125Total 100

I.

1. Consider the equation ex = sin x. Find an interval [a, b] that contains thesmallest positive root. Estimate the number of midpoints c needed to obtainan approximate root that is accurate within an error tolerance of 109.

[a, b] = [0,

2].

| cn| 12n

(b a) = 2n+1

= 109,

n + 1 ln(109)

ln 2

2. Prove that cn cn+1 = 2n2(b0 a0), where cn is the nth computed value ofc in the bisection method.

cn+1 cn =bn+1 an+1

2=

bn an22

=b0 a02n+2

2

II.

1. (a) If the Newtons method is used on f(x) = x3x+1 starting with x0 = 1,what will x1 be?

(b) If the Newtons method is used on f(x) = x5x3 +3 and if xn = 1, whatis xn+1?

(a)

f (x) = 3x2 1, x1 = x0 x30 x0 + 1

3x20 1=

12,

(b)

f (x) = 5x4 3x2, xn+1 = xn x5n x3n + 35x4n 3x2n

= 12.

3

III.

1. (a) If we use the secant method on f(x) = x3 2x + 2 starting with x0 = 0and x1 = 1, what is x2?

(b) If the secant method is used on f(x) = x5 + x3 + 3 and xn2 = 0 andxn1 = 1, what is xn?

(a)

x2 = x1 f(x1)x1 x0

f(x1) f(x0)= 2

(b)

xn = xn1 f(xn1)xn1 xn2

f(xn1) f(xn2)= 3

2

2. (a) If xn+1 = xn+(2exn )(xnxn1)

exnexn1 with x0 = 0 and x1 = 1, what is limn xn?

(b) What is the appropriate formula for finding square roots using the secantmethod?

(a)

f(x) = ex 2, = ln 2

(b)

xn+1 = xn x2n R

xn + xn1

4

IV.

1. Use Aitkens error estimation formula to estimate the error x2 in the fol-lowing iteration

xn+1 =0.5

1 + x4n, x0 = 0.48.

x2 =2

1 2(x2 x1),

x1 =.5

1 + .484= 0.4748,

x2 =.5

1 + x41=

.51 + 0.4748

= 0.4758

2 =x2 x1x1 x0

= 0.1923.

2. What is the order of convergence of the iteration

xn+1 =xn(x

2n + 3a)

3x2n + a

as it converges to the fixed point =a?

3

g(xn) = g() + (xn )g() +12(xn )2g() +

5