Explanation Newton’s approximation of is given by, x n = x n − 1 − f ( x n ) f ′ ( x n ) Calculation: The given equation is, f ( x ) = 3 x 4 − 3 x 3 − 11 x 2 − x − 4 … … ( 1 ) Find the values of the equation at the boundary points, x = − 1 and x = − 2 . Substitute x = − 1 in equation (1). f ( − 1 ) = 3 ( − 1 ) 4 − 3 ( − 1 ) 3 − 11 ( − 1 ) 2 − ( − 1 ) − 4 = 3 + 3 − 11 + 1 − 4 = − 8 Substitute x = − 2 in equation (1). f ( − 2 ) = 3 ( − 2 ) 4 − 3 ( − 2 ) 3 − 11 ( − 2 ) 2 − ( − 2 ) − 4 = 48 + 24 − 44 + 2 − 4 = 26 Take the value x 1 = − 1.5 . Step 1: Differentiate the equation (1) with respect to x f ′ ( x ) = 12 x 3 − 9 x 2 − 22 x − 1 … … ( 2 ) Substitute x 1 = − 1.5 in equation (1). f ( − 1.5 ) = 3 ( − 1.5 ) 4 − 3 ( − 1.5 ) 3 − 11 ( − 1.5 ) 2 − ( − 1.5 ) − 4 = 15.1875 + 10.125 − 24.75 + 1.5 − 4 = − 1.9375 Substitute x 1 = − 1.5 in equation (2). f ′ ( − 1.5 ) = 12 ( − 1

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Chapter 24.2, Problem 12E

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The negative root of the equation 3x4−3x3−11x2−x−4 to at least four decimal places by Newton’s method and compare this root with the root obtained by calculator.

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Chapter 24.2, Problem 11E

Chapter 24.2, Problem 13E

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Chapter 24 Solutions

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Ch. 24.1 - Prob. 9E Ch. 24.1 - Prob. 10E Ch. 24.1 - Prob. 11E Ch. 24.1 - Prob. 12E Ch. 24.1 - In Exercises 11–14, find the equations of the...Ch. 24.1 - Prob. 14E Ch. 24.1 - Prob. 15E Ch. 24.1 - Prob. 16E Ch. 24.1 - Prob. 17E Ch. 24.1 - Prob. 18E Ch. 24.1 - Prob. 19E Ch. 24.1 - Prob. 20E Ch. 24.1 - Prob. 21E Ch. 24.1 - Where does the normal line to the parabola y = x —...Ch. 24.1 - Prob. 23E Ch. 24.1 - Prob. 24E Ch. 24.1 - A certain suspension cable with supports on the...Ch. 24.1 - Prob. 26E Ch. 24.1 - Prob. 27E Ch. 24.1 - Prob. 28E Ch. 24.1 - Prob. 29E Ch. 24.1 - Prob. 30E Ch. 24.2 - In Example 1, let x1 = 0.3, and find x2. 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The negative root of the equation 3 x 4 − 3 x 3 − 11 x 2 − x − 4 to at least four decimal places by Newton’s method and compare this root with the root obtained by calculator.

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The negative root of the equation 3x4−3x3−11x2−x−4 to at least four decimal places by Newton’s method and compare this root with the root obtained by calculator.

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Chapter 24 Solutions