2x^4-3x^2-7x+1=0
Estimate what the real solution will be using Newton's method. Answers must be 4 decimal places. Let the first guess be x=1.75
To estimate the real solution of the equation 2x^4 - 3x^2 - 7x + 1 = 0 using Newton's method, we can follow these steps:
Step 1: Find the derivative of the function f(x) = 2x^4 - 3x^2 - 7x + 1.
f'(x) = 8x^3 - 6x - 7
Step 2: Set up the iteration formula for Newton's method:
x(n+1) = x(n) - f(x(n)) / f'(x(n))
Step 3: Start with the initial guess x(0) = 1.75 and use the iteration formula to find subsequent values of x(n).
Let's apply Newton's method:
For n = 0:
x(0) = 1.75
For n = 1:
x(1) = x(0) - f(x(0)) / f'(x(0))
To find f(x(0)):
f(x(0)) = 2(1.75)^4 - 3(1.75)^2 - 7(1.75) + 1
= 21.453125
To find f'(x(0)):
f'(x(0)) = 8(1.75)^3 - 6(1.75) - 7
= 15.734375
Substituting these values into the iteration formula:
x(1) = 1.75 - 21.453125 / 15.734375
≈ 0.63096 (rounded to 5 decimal places)
For n = 2:
x(2) = x(1) - f(x(1)) / f'(x(1))
To find f(x(1)):
f(x(1)) = 2(0.63096)^4 - 3(0.63096)^2 - 7(0.63096) + 1
≈ -0.65632 (rounded to 5 decimal places)
To find f'(x(1)):
f'(x(1)) = 8(0.63096)^3 - 6(0.63096) - 7
≈ -3.95761 (rounded to 5 decimal places)
Substituting these values into the iteration formula:
x(2) = 0.63096 - (-0.65632) / -3.95761
≈ 0.79015 (rounded to 5 decimal places)
Repeat these steps until you achieve the desired level of accuracy or reach convergence.
Using Newton's method, the estimated real solution of the equation 2x^4 - 3x^2 - 7x + 1 = 0, with the initial guess x = 1.75, is approximately 0.79015.