Chapter 4 | Applications of Derivatives
473
Figure 4.77 The approximations x 0 , x 1 , x 2 ,… approach the actual root x *. The approximations are derived by looking at tangent lines to the graph of f .
Now let’s look at how to calculate the approximations x 0 , x 1 , x 2 ,…. If x 0 is our first approximation, the approximation x 1 is defined by letting ⎛ ⎝ x 1 , 0 ⎞ ⎠ be the x -intercept of the tangent line to f at x 0 . The equation of this tangent line is given by y = f ( x 0 )+ f ′( x 0 )( x − x 0 ). Therefore, x 1 must satisfy f ( x 0 )+ f ′( x 0 )( x 1 − x 0 ) =0. Solving this equation for x 1 , we conclude that x 1 = x 0 − f ( x 0 ) f ′( x 0 ) . Similarly, the point ⎛ ⎝ x 2 , 0 ⎞ ⎠ is the x -intercept of the tangent line to f at x 1 . Therefore, x 2 satisfies the equation x 2 = x 1 − f ( x 1 ) f ′( x 1 ) . In general, for n >0, x n satisfies (4.8) x n = x n −1 − f ( x n −1 ) f ′( x n −1 ) . Next we see how to make use of this technique to approximate the root of the polynomial f ( x ) = x 3 −3 x +1.
Made with FlippingBook - professional solution for displaying marketing and sales documents online