newtons method through a point

Question

Suppose that Newton's method is used to locate a root of the equation f (x) = 0 with initial approximation x₁ = 4. If the second approximation is found to be x₂ = −3, and the tangent line to f (x) at x = 4 passes through the point (14, 4), find f(4).

Answer 1

There's a short cut method to this.

x1 - x2 = f(x1)/f'(x1)

4 - -3 = 7 = f(4)/f'(4)

7f'(4) = f(4)

Since x2 is a more accurate representation of the root, use it for the second point on the tangent line along with (14,4). So the second point is (-3,0).

f'(4) = (4-0)/(14 - -3) = 4/17

7 * 4/17 = 28/17 = f(4)

Answer 2

first equation

-3=4-(f(4)/f'(4)

7f"(4)=f(4)

second equation

f'(4)=(4-f(4))/(14-4)

(10/7)f(4)=4-f(4)

f(4)=28/17

f'(4)=4/17

x=4 really inaccurate initial choice of x for f(x)=0