Find all the real solutions to cubic equation, Mathematics

Assignment Help:

Find all the real solutions to cubic equation x^3 + 4x^2 - 10 =0. Use the cubic equation x^3 + 4x^2 - 10 =0 and perform the following call to the bisection method [0, 1, 30]

Use the fixed point iteration to find the fixed point(s) for the function g(x) = 1 + x - x^2/3

Find all the real solutions to cubic equation x^3 +4x^2-10=0. Use the cubic equation x^3 + 4x^2 - 10 =0 and perform the following call to the regulaFalsi [0, 1, 30]

Use newton's method to find the three roots of a cubic polynomial f(x) = 4x^3 - 15x^2 + 17x-6. Determine the Newton-raphson iteration formula g[x] = x - (f(x)/f'(x)) that is used. Show details of the computation for the starting value p0 = 3.

Use the secant method to find the three roots of cubic polynomial f[x]=4x^3 - 16x^2 + 17x - 4. Determine the secant iterative formula g[x] = x - (f[x]/f'[x]) that is used. Show details of the computation for the starting value p0=3 and p1=2.8

Use appropiate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following:

A) f(8.4) if f(8.1)= 16.94410, f(8.3)=17.56492, f(8.6)=18.50515, and f(8.7)=18.82091

B)f(1/3) if f(-0.75)= -0.07181250, f(-0.5) = -0.02475000, f(-0.025) = 0.33493750, and f(0)=18.82091

Use the newton forward divided-difference formula is used to approximate f(0.3) given the following data

X        0.0     0.2     0.4     0.6

F(x)  15.0   21.0   30.0   51.0

Suppose it is discovered that f(0.4) was understand by 10 and f(0.6) was overstated by 5. By what amount should the approximation to f(0.3) be changed?

Using the error formulas

|f(x)-P1(x)| ≤ 1/8 max (f(x))h2, linear interpolation

|f(x)-P2(x)| ≤ 1/9√3 max (f(x))h3, quadratic interpolation

A)  what is an appropriate size for the interpolation table for the function tan x on the interval [0,1] in order that linear interpolation produce an error no larger than 0.5 x 10^6

B)   Answer A)

A) Using taylor series expansions derive the O(h^2) central difference approximation

F'(x)= (f(x+h)-f(x-h))/2h

B)  using richardson extrapolation and taylor series expansions derive the O (h4) derivative approximation

F'(x)= (-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h))/12h

Consider the richardson table for derivatives in the form step size table

Step size                       table

H                                  D(0,0)

H/2                               D(1,0)        D(1,1)

H/2^2                           D(2,0)        D(2,1)        D(2,2)

H/2^3                           D(3,0)        D(3,1)        D(2,3)        D(3,3)

.

.

Where the central difference formula

(h)   = (f(x+h)-f(x-h)) /2h

Is used to construct the first column using

D(n,0)= (h/2^n)

And the following formula

D(n,m)= (4^mD(n,m-1)-D(n-1,m-1))/4^m-1 (use for hand calculations)

D(n,m-1)+((D(n,m-1)-D(n-1,m-1)/(4^m-1)) (use for programming)

Is used, for n≥m, to obtain entries in other columns in terms of the entry to their left and the entry above this entry. For example, D(2,1) is obtained in terms of D(2,0) and D(1,0) and D(3,2) is obtained in terms of D(3,1) and D(2,1)

A) construct the table for the derivative of tan x at x=0.5. Choose an initial step size of h=1 and calculate 4 rows by hand using a calculator

B) use maple procedure richardson in file richardson.txt to calculate 6 rows of the richardson extrapolation table.

----------------------------------------------------------------------------------------------

# lip.txt:

#Symbolic calculation of LIP

#(Lagrange interpolating polynomial)

#

#Arguments

#

#xp   list [x0,x1,....,xn] of nodes

#yp   list[y0,y1,.....,yn] of function values at nodes

#x     symbolic variable for the polynomial

#

#lists xp amd yp have n+1 elements and begin at subscript 1

#so the interpolating polynomial is of degree n

----------------------------------------------------------------------------------------------

lip := proc(xp,yp,x)

         local n,s,p,k,j;

         N := nops(xp) -1; #nops(xp) gives number of elements in xp

         S := 0;

         For k from 0 to n do

                   P := yp[k+1];

                   For j from 0 to n do

                            If j<>k then

                                     P := p*(x-xp[j+1])/(xp[k+1]-xp[j+1]);

                            Fi;

                   Od;

                   S := s=p;

         Od;

         Return s;

End proc:


Related Discussions:- Find all the real solutions to cubic equation

Using calculus method, Sheldon as the day for the challenge gets closer wan...

Sheldon as the day for the challenge gets closer wants to enter the race. Not being content with an equal start, he wants to handicap himself by giving the other yachts a head star

Find distance between points (b + c, Find the distance between the points (...

Find the distance between the points (b + c, c + a) and (c + a, a + b) . Ans : Use distance formula

Find the value of delta, Consider the given graph G below. Find δ( G )=__...

Consider the given graph G below. Find δ( G )=_____ , λ( G )= _____ , κ( G )= _____, number of edge-disjoint AF -paths=_____ , and number of vertex-disjoint AF -paths= ______

Graph and algebraic methods , To answer each question, use the function t(r...

To answer each question, use the function t(r) = d , where t is the time in hours, d is the distance in miles, and r is the rate in miles per hour. a. Sydney drives 10 mi at a c

Precalculuc, evaluate the expression and write the result in the form a + b...

evaluate the expression and write the result in the form a + bi. I^37

Decimals, 0.875 of a number is 2282. What is the number ?

0.875 of a number is 2282. What is the number ?

Define multiplication rule in probability, Q. Define Multiplication Rule in...

Q. Define Multiplication Rule in probability? Ans. A family has two girls, Ann and Barb, and three boys, Carl, David and Earl, in it. In how many ways can the mother pick

Series solutions to differential equation, Before we find into finding seri...

Before we find into finding series solutions to differential equations we require determining when we can get series solutions to differential equations. Therefore, let's start wit

How many different combinations could she form these item, Wendy has 5 pair...

Wendy has 5 pairs of pants and 8 shirts. How many different combinations could she form with these items? Multiply the number of choices for each item to find out the number of

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd