7/23/2019 Problem Set 1 for Numerical Methods

1/1

Problem Set #1

1.

Use (a) fixed-point iteration and (b) Newton - Raphson method to determine a root of

5.28.1)( 2

xxxf using 50 x . Perform the computation until a is less than

%05.0s .

2.

Determine the highest real root of 69.109.595.0 23 xxxxf a.

Graphically

b.

Using the secant method 5.35.21 ii xandx

3.

Use the fourth degree Taylor polynomial of x2cos to find the exact value of0

limx

23

2cos1

x

x

4.

Determine the roots of the following simultaneous nonlinear equations using (a) Fixed-point

iteration and (b) the Newton-Raphson method:

2

2

5

75.0

xxyy

xxy

Employ initial guesses of 2.1 yx and discuss the result.5.

An antibiotic decays exponentially in the human body with a half-life of about 2.5 hours.

Suppose a patient takes a 250 mg tablet of the antibiotic every 6 hours.

a.

Write an expression for ,,, 432 QQQ where nQ is the amount (in mg) of the antibiotic in

the body after theth

n tablet is taken. Note that mgQ 2501 .

b.

Write an expression for nQ and put it in closed form.

c.

Assume the antibiotic treatment consists of a total of 28 tablets. Give a numerical

estimate for the amount of antibiotic in the body immediately after the patient takes

the last tablet of the treatment.

6.

Determine the roots of the polynomial 27.11.227.21.2 234

xxxxxf usingBairstows method with initial guesses 0000.2r and 2500.1s .

7.

Many fields of engineering require accurate population estimates. For example, transportation

engineers might find it necessary to determine separately the population growth trends of a city

and adjacent suburb. The population of the urban area is declining with time according to

min,max, utk

uu PePtP u

While the suburb population is growing, as in

tk

OS

S

sS

ePP

PtP

1/1 max,

max,

Where SOSuu kandPPkP ,,, max,max, empirically derived parameters. Determine the time and

corresponding values of tPandtP Su when the suburbs are 20% larger than the city. Theparameters values are 000,75max, uP , yrku /045.0 , peoplePu 000,100min, ,

peoplePs 000,300max, , peoplePO 000,10 , yrkS /08.0 . To obtain your solutions, use

(a) graphical (b) false-position method.