Math5_Exercises_Act 1_3.docx

7/29/2019 Math5_Exercises_Act 1_3.docx

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IPN. ESIA-Zacatenco. Profesores: Mauricio Jess Surez Ledesma y Antonia Ferreira Martnez.

ACTIVIDAD 1.3

1. Trouver la solution des quationssuivantes par les mthodes de bissection, fausseposition, de Newton-Raphson et de la scante.

a) 41pour052 23 xxx b)

04pour013

23 xxx

c) 10pour02 xx x d) 21pour06cos22 xxe xx e) 10pour0232 xxxex 2. Dterminer la valeurnumrique de 2 laide de la mthode de

bissectiondanslintervalle *0,2+ avec 10 itrations. Compareravec la vraievaleur.

3. Soit .14144 2 xxxg Pourquelsintervalles de valeurs de dpart la procdure depointfixe converge-t-elle et diverge-t-elle?

4. Les troisfonctions ci-dessoussont touts des candidates pour faire lapproximation, par lamethode du poin fixe, de 3 21 :

2121

3

21

21

2120

2

4

3

2

3

2

2

1

x

xx

xxg

x

xxxg

xxxg

5. Classer ces fonctions en ordredcroissant de vittesse de convergence de lalgorithme dupointfixe.Astuce:comparer les valeurs de drivesautour du pointfixe.

6. Data lequazionef(x) =0 ove: 22 xexxf x

provare che ammetteununicasoluzionenellintervalloI=[0.5, 1.5].

Eseguiretreiterazioni con lo schema di Newton-Raphsonpartendo dax0=0.5. Utilizzandox3si dica se lo schema di punto fisso

nx

nn exx

21

converge a .

(Usare almeno 7 cifre decimali.)

7. Data lequazione 05186 22 xe x ,siprovicheammetteununicasoluzionenellintervallo I=*0,4+.

Si calcoliil punto iniziale x0effettuandodueiterazioni del mtodo dellebisezioni (odicotomico), a partiredallintervalloassegnato.


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A partire del valore x0calcolato al punto precedente, si calcolinotreiterazioni delmtodo di Newton-Raphson. (Esprimere i risultati in virgolamobilenormalizzata

con almeno 7 cifre decimali)

8. Data lequazione f(x)=0 ove: xxxf 6arctan4

Provare (analticamente o graficamente) che ammette ununica soluzione nellintervallo I=*4,5+.

Dire quante iterazioni sono necessarte col mtodo di bisezione per ottenere lasoluzione con un errore minore di 10

-4.

Eseguir etre iterazioni con lo schema di Newton-Raphson partendo da x0=5. Ulilizzando x3 si dica se il metodo di punto fisso:

4/tan61 nn xx converge a ; in caso affermativocalcolare la soluzione (Esprimere i risultati in

virgolamobilenormalizzata con almeno 7 cifre decimali).

9. Al principio de cada ao un banco deposita 1000 euros en un fondo de inversin y retiraun capital de 6000 euros al final del quinto ao. El tipo medio de inters anual ]1,0[r de

esta inversin es solucin de la ecuacin

5

1

110006000k

kr .

Aplicar los mtodos de biseccin y falsa posicin para obtener r con un error 10-10

.

Cuntas iteraciones son necesarias para el mtodo de bisecciones?

10.Probar que la ecuacin 010cos3 xxx tiene una nica solucin real. Aproximar lasolucin usando el mtodo de Newton-Raphson con 3 iteraciones. Dar una estimacin del

error cometido.

11. Aplicar el mtodo de punto fijo con 10 iteraciones para aproximar la nica solucin mayorque 2 de la ecuacin 0

1

3

2

2

xx

x. Dar una estimacin del error cometido.

12.The volume V for a liquid contained into a spherical tank of radius r, is related with itsheight h trough the next expression:

3)3(2 hrhV

Determine h for r=1 m and V=0.5 m3, using any of the two following arrangements:

r

Vh

h3

33

or 323

Vrhh


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13.For the fluids flow in a piping, the friction is described by a dimensionless number knownas Darcys friction factor (f). The formula to predict f when Re is given, is called Von

Karmans equation:

4.0Relog41 10 ff

The typical Reynolds number values are a) for laminar flows and b) for

transitional flows and c) for turbulent flows .

Determine f when Re=2500.

14.Determine the positive root of the equation by using the fixed-pointiteration method. Carry out the first five iterations.

15.Determine the fifth root of 3512 to four decimal places by finding the numerical solutionof the equation .Use Newtons method. Start at x=30.

16.Determine the positive root of the polynomial .a) Plot the polynomial and choose a point near the root for the first estimate of the

solution. Using Newtons method, determine the approximate solution.

b) From the plot in part (a), choose two points near the root to start the solution processwith the secant method.

17.Given() a) Find all the roots (x1, x2, x3 and x4) of the equation using the bisection method with

a=0.8 and b=1.3. Observe that f(1)=f(2)=0.

b) The same that incised (a), but using regulafalsi method.18.Prove that 09206 35 xxxf has solely one real root (solution) x that is

located between 0 and 1. Find x (to six decimal places) using any method.

19.Use any method for searching roots in order to find the following (to six decimal places):a) The minimum positive tthat satisfies 223.13.19.5 2.2 tet .b) The minimum positive y, implicitly defined by xyeyyx x /5cos 3 , when x=1.c) All the solutions in []for , where ( ).

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Math5_Exercises_Act 1_3.docx