Question

Use PYTHON to compute the problem please

= Consider the equation f(x) 1/4 – x* (1 – x²) – sin(x) 0 a). Show by a simple test that a root exists between x = 0 and x = 1 b). Perform three iterations beyond the starting values using the secant rule. c). Estimate the derivative at your answer b. Compare to the correct value.

Answer 1

1).ANSWER:

GIVEN THAT:

import numpy as np

def f(x):
return np.pi/4-x*(1-x*x)-np.arcsin(x);
def secant(x1, x2):
ct=0;
n = 0; xm = 0; x0 = 0; c = 0;
if (f(x1) * f(x2) < 0):
while True:
  
# calculate the intermediate value
x0 = ((x1 * f(x2) - x2 * f(x1)) /
(f(x2) - f(x1)));
  
# check if x0 is root of
# equation or not
c = f(x1) * f(x0);
  
# update the value of interval
x1 = x2;
x2 = x0;
  
# update number of iteration
n += 1;
  
# if x0 is the root of equation
# then break the loop
if (c == 0):
break;
xm = ((x1 * f(x2) - x2 * f(x1)) /
(f(x2) - f(x1)));
ct=ct+1;
if(ct==3):
break;
  
print("Root of the given equation =",
round(x0, 6));
return x0;

else:
print("Can not find a root in ",
"the given inteval");

x0=secant(0,1)
Dx=(f(x0+1e-4)-f(x0-1e-4))/(2*1e-4);
print("Dervative is",Dx);