The graph of a function f(x) is given below The graph of a function f(x) is given below. Suppose that Newton's method is used to approximate the roots r and s of the equation f(x) = 0. The blue (1) and red (2) lines are the tangent lines corresponding to initial approximations for finding these roots. In the approximation of the value of r, what is x2? {applet} {image} 1. 2. 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

{image} = 5.16360 {image} = 2.67125 {image} = 1.95360 Use Newton's method with the specified initial approximation {image} to find {image} , the fourth approximation to the root of the given equation. (Give your answer to five decimal places.) {image} {image} = 5.16360 {image} = 2.67125 {image} = 1.95360 {image} = 0.71765 1. 2. 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Use Newton's method to approximate the root of {image} in the interval [1, 2], correct to six decimal places. Use {image} = 1.5 as the initial approximation. x = 1.379411 x = 1.379414 x = 1.379412 x = 1.379415 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50