Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Round your answer to four decimal places.)
x5 − x − 7 = 0, x1 = 1

Question

Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Round your answer to four decimal places.)
x5 − x − 7 = 0,    x1 = 1

amistre64 · Answer

define the tangent line equation at the given point; and find its x intercept.  use that x intercepts as a new guess for X, and repeat

amistre64 · Answer

tangent y = f'(a)(x-a) + f(a)

0 = f'(a)(x-a) + f(a)

0 - f(a) = f'(a)(x-a)

- f(a)/ f'(a) = x-a

a - f(a)/ f'(a) = x

amistre64 · Answer

$$X_2=1-\frac{1^5-1-7}{5(1)-1}$$
$$X_3=X_2-\frac{(X_2)^5-(X_2)-7}{5(X_2)-1}$$

amistre64 · Answer

forgot the ^4  parts

anonymous · Answer

x^4 + x - 4 = 0 ======> f (x) = x^4 + x - 4 

f '(x) = 4x^3 + 1 

lets start with X1 = 1 

Xn+1 = Xn - [ f(x) / f '(x) ] 

Xn+1 = Xn - [ (X^4 + X - 4) / (4X^3 + 1) ] ======> X1 = 1 

X2 = 1 - [ (1^4 + 1 - 4) / (4 * 1^3 + 1) ] ======> X1 = 1.4 

X3 = 1.4 - [ ( (1.4)^4 + 1.4 - 4) / (4 * (1.4)^3 + 1) ] ======> X1 = 1.2963259853039 

X4 = 1.2963259853039 - [ ( (1.2963259853039)^4 + 1.2963259853039 - 4) / (4 * (1.2963259853039)^3 + 1) ] ======> X1 = 1.2839439534061 

X4 = 1.2839439534061 - [ ( (1.2839439534061)^4 + 1.2839439534061 - 4) / (4 * (1.2839439534061)^3 + 1) ] ======> X1 = 1.2837816933800 

X4 = 1.2837816933800 - [ ( (1.2837816933800)^4 + 1.2837816933800 - 4) / (4 * (1.2837816933800)^3 + 1) ] ======> X1 = 1.2837816658635 

THE root is 1.283782 

========= 

2) 

x^3 - x^2 - 1 = 0 ======> f (x) = x^3 - x^2 - 1 

f '(x) = 3x^2 - 2x 

lets start with X1 = 1 

Xn+1 = Xn - [ f(x) / f '(x) ] 

Xn+1 = Xn - [ (X^3 - X^2 - 1) / (3X^2 - 2X) ] ======> X1 = 1 

X2 = 1 - [ (1^3 - 1^2 - 1) / (3 * (1)^2 - 2 * 1) ] ======> X2 = 2 

X3 = 2 - [ (2^3 - 2^2 - 1) / (3 * (2)^2 - 2 * 2) ] ======> X3 = 1.625 

X4 = 1.625 - [ ( (1.625)^3 - (1.625)^2 - 1) / (3 * (1.625)^2 - 2 * 1.625) ] ======> X4 = 1.485785953177 

X5 = 1.485785953177 - [ ( (1.485785953177)^3 - (1.485785953177)^2 - 1) / (3 * (1.485785953177)^2 - 2 * 1.485785953177) ] ======> X5 = 1.46595591974 

X6 = 1.46595591974 - [ ( (1.46595591974)^3 - (1.46595591974)^2 - 1) / (3 * (1.46595591974)^2 - 2 * 1.46595591974) ] ======> X6 = 1.4655713749 

X7 = 1.4655713749 - [ ( (1.4655713749)^3 - (1.4655713749)^2 - 1) / (3 * (1.4655713749)^2 - 2 * 1.4655713749) ] ======> X7 = 1.465571232 

the root is 1.465571

anonymous · Answer

hope that helped