Question

For each equation, write a brief program to compute and print eight steps of Newton's method for finding a positive root (Preferably in Matlab or Python).

a. x=2sinx

b. x^3=sinx+7

c. sinx=1-x

d. x^5+x^2=1+7x^3 for x>=2

Answer #1

A)

clc

clear all

x(1)=1;

iter(1)=0;

for i=1:7

f(i)= x(i)-2*sin(x(i));

der_f(i)=1-2*cos(x(i));

x(i+1)=x(i)-f(i)/der_f(i);

iter(i+1)=iter(i)+1;

end;

fprintf("%i %f", iter, x);

B)

clc

clear all

x(1)=1;

iter(1)=0;

for i=1:7

f(i)= x(i).^3-sin(x(i))-7;

der_f(i)=3*x(i).^2-cos(x(i));

x(i+1)=x(i)-f(i)/der_f(i);

iter(i+1)=iter(i)+1;

end;

fprintf("%i %f", iter, x);

C).

clc

clear all

x(1)=1;

iter(1)=0;

for i=1:7

f(i)= sin(x(i))-1-x(i);

der_f(i)=cos(x(i))-1;

x(i+1)=x(i)-f(i)/der_f(i);

iter(i+1)=iter(i)+1;

end;

fprintf("%i %f", iter, x);

D).

clc

clear all

x(1)=2;

iter(1)=0;

for i=1:7

f(i)= x(i).^5+x(i).^2-1-7*x(i).^3;

der_f(i)=5*x(i).^4+2*x(i)-21*x(i).^2

x(i+1)=x(i)-f(i)/der_f(i);

iter(i+1)=iter(i)+1;

end;

fprintf("%i %f", iter, x);

Each equation has one root. Use Newton’s Method to approximate
the root to eight correct
decimal places. (a) x3 = 2x + 2 (b) ex + x = 7 (c) ex + sin x =
4
**MUST BE DONE IN MATLAB AND NEED CODE

Use
Newton's method to approximate the root of the equation to four
decimal places. Start with x 0 =-1 , and show all work
f(x) = x ^ 5 + 10x + 3
Sketch a picture to illustrate one situation where Newton's
method would fail . Assume the function is non-constant
differentiable , and defined for all real numbers

solve cos(x)=x^2 using Newton's Method accurate to eight decimal
places. Show all steps in solution. How many roots are there?

Use Newton's method to find all solutions of the equation
correct to eight decimal places. Start by drawing a graph to find
initial approximations. (Enter your answers as a comma-separated
list.) x x2 + 1 = 1 − x

Each equation has one real root. Use Newton’s Method to
approximate the root to eight correct decimal places. (a) x5 + x =
1 (b) sin x = 6x + 5 (c) ln x + x2 = 3
**MUST BE DONE IN MATLABE AND SHOW CODE

IN JAVA
1. Write up a small program that accepts two integers from the
user. Print which of the two values is bigger. If they are the
same, print that they are the same.
2. Write a method that accepts three doubles.
Calculate and return the average of the three
passed double values.
3. Write up a method that accepts a score between 0-10. Print
out a message according to the following table. You ay personally
decide the boundaries. i.e.,...

Use Newton's method with the specified initial approximation
x1 to find x3, the third
approximation to the root of the given equation.
x3 + 5x − 2 =
0, x1 = 2
Step 1
If
f(x) =
x3 + 5x − 2,
then
f'(x) = _____ x^2 + _____
2- Use Newton's method to find all roots of the
equation correct to six decimal places. (Enter your answers as a
comma-separated list.)
x4 = 5 + x
.

A graphing calculator is recommended. Use Newton's method to
find all solutions of the equation correct to eight decimal places.
Start by drawing a graph to find initial approximations. (Enter
your answers as a comma-separated list.) -2x^7-4x^4+9x^3+2=0

(Python Programming)
Write a program that prompts a user for a positive integer and
then uses a loop to calculate and display the sum of specific
fractions as follows:
Let's say the user enters 5, then your program will compute: 1/5
+ 2/4 + 3/3 + 4/2 + 5/1 which is 8.7.

11. Write a program to compute the sum of the series
12 + 22 + 32. . . ., such that the sum is doesnot exceed 1000. The
program should display how many terms are used in the sum.
{3 marks}
matlab only
1 to power of 2 , 2 to the power of 2 , 3 to the power
of 3 , not 12 + 22 + 32

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