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Newton's Method

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Wize University Calculus 1 Textbook > Applications of Differentiation

Newton's Method

5 Activities

Consider the curve generated by the equation
x=−5x2x = - \frac{5}{x^2}
Which polynomial would you use Newton's Method on to estimate a solution to the above equation, and what integer should you choose as a starting point to best estimate the solution?
More Newton's Method Questions:
Newton's Method: Estimate Intersection Point
Use Newton's method to estimate the point of intersection of y=e−x2\displaystyle y=e^{-x^{2}} and y=xy=x.
Newton's Method: Inverse Trigonometric Function
Starting from x0=1x_0=1, Newton's method is used to approximate the solutions of the equation arctan⁡x−12x2=0\displaystyle \arctan x-\frac{1}{2x^2}=0. Find the next approximation x1x_1.
Newton's Method: Estimate Intersection Point
Use Newton's method to estimate the point of intersection of y=e−x2\displaystyle y=e^{-x^{2}} and y=xy=x.
Practice: Approximating a Ln
Q.\textbf{Q.} Approximate ln⁡2\ln{2} to 4 decimal places using Newton's method.
Newton's Method
Use Newton's Method to approximate a solution to f(x)=x4−x=10f(x)=x^4-x=10
Starting with x1=1x_1=1 , x2=?x_2=?
Newton's Method
Using x0=1x_0=1 use Newton's Method to approximate a solution to cos⁡x=x\cos x=x.
Newton's Method
Using Newton's Method with x0=0x_0=0, find x1,…,x4x_1, \dots, x_4 for f(x)=2x3−x+1f(x)=2x^3-x+1.
Newton's Method
Find 23\sqrt[3]{2} to three decimal places.
Newton's Method
Take x0=0x_{0}=0 and find x1,x2,x3x_1,x_2,x_3 for f(x)=x3−3x2+x+1f(x)=x^{3}-3x^{2}+x+1. If necessary, round your answer to the fourth decimal place.
Given y=15x5+4x3+36x−1000y=\dfrac{1}{5}x^5+4x^3+36x-1000, what starting points would make Newton's Method fail? Find a formula for xn+1x_{n+1} in terms of xnx_n.
Practice: Approximating a Ln
Q.\textbf{Q.} Approximate ln⁡2\ln{2} to 4 decimal places using Newton's method.
Newton's Method: Inverse Trigonometric Function
Starting from x0=1x_0=1, Newton's method is used to approximate the solutions of the equation arctan⁡x−12x2=0\displaystyle \arctan x-\frac{1}{2x^2}=0. Find the next approximation x1x_1.
Newton's Method: Estimate Intersection Point
Use Newton's method to estimate the point of intersection of y=e−x2\displaystyle y=e^{-x^{2}} and y=xy=x.
Use one iteration of Newton's method to approximate 93\sqrt[3]{9} where the initial position is x=2.
Newton's Method
Which formula should be used in order to approximate arctan⁡(0.1)\arctan(0.1) using Newton's Method?
Newton's Method
Which of the following points on the following function should you NOT use in order to use Newton's Method?
f(x)=x3−2x+2f(x) = x^3 - 2x + 2
Choose all that apply.
Given y=15x5+4x3+36x−1000y=\frac{1}{5}x^5+4x^3+36x-1000, what starting points would make Newton's method fail? Find a formula for the iterate xn+1x_{n+1} in term of xnx_n.