Use Newton's method to approximate the indicated root of the equation correct to six decimal places? The root of #f(x) =x^4 − 2x^3 + 3x^2 − 6 = 0# in the interval [1, 2]
This reference for Newtons Method gives us the equation:
where
Substitute equation [2] and [3] into equation [1]:
I recommend that you use an Excel spread sheet.
Enter 1 into cell A1
Enter the following into cell A2:
=A1(A1^42A1^3+3A1^26)/(4A1^36A1^2+6*A1)
Please observe that this is the Excel language equivalent of equation [4].
Copy the contents of cell A2 into cells A3 through A10.
Please observe that the contents of the cells converge on the number 1.596072; this is the root within the specified interval.
By signing up, you agree to our Terms of Service and Privacy Policy
To approximate the root of ( f(x) = x^4  2x^3 + 3x^2  6 = 0 ) in the interval [1, 2] using Newton's method, follow these steps:
 Start with an initial guess, let's say ( x_0 = 1.5 ) (since it lies in the interval [1, 2]).
 Iterate using the formula: [ x_{n+1} = x_n  \frac{f(x_n)}{f'(x_n)} ]
 Calculate ( f(x_n) ) and ( f'(x_n) ) at each iteration.
 Repeat the iterations until the desired accuracy is achieved.
Performing the iterations:

( x_0 = 1.5 ) [ f(x_0) = (1.5)^4  2(1.5)^3 + 3(1.5)^2  6 = 0.375 ] [ f'(x_0) = 4(1.5)^3  6(1.5)^2 + 6(1.5) = 3.75 ] [ x_1 = 1.5  \frac{0.375}{3.75} \approx 1.46667 ]

( x_1 = 1.46667 ) [ f(x_1) = (1.46667)^4  2(1.46667)^3 + 3(1.46667)^2  6 \approx 0.05638 ] [ f'(x_1) = 4(1.46667)^3  6(1.46667)^2 + 6(1.46667) \approx 3.68868 ] [ x_2 = 1.46667  \frac{0.05638}{3.68868} \approx 1.46444 ]

Continuing this process until the desired accuracy of six decimal places is achieved.
After several iterations, the root is approximately ( x \approx 1.464102 ) (correct to six decimal places).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
 The velocity of a particule is v = 2t + cos (2t). When t = k the acceleration is 0. Show that k = pi/4?
 How do you minimize and maximize #f(x,y)=x^3y^2xy# constrained to #xy=4#?
 How do you minimize and maximize #f(x,y)=(xy)/((x2)^2(y4))# constrained to #xy=3#?
 A rancher has 1000m of fencing to enclose two rectangular corrals. The corrals have the same dimensions and one side in common (let that side be x). What dimensions will maximize the enclosed area?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7