Newton's method practice problems
Newton's iteration x₁, x₂, … for a root, done exactly. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Use Newton's method on \( \displaystyle f(x) = x^{3} - x - 1 \) with \( \displaystyle x_0 = 2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x - 4 \) with \( \displaystyle x_0 = -2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method to approximate \( \displaystyle \sqrt{10} \), as a root of \( \displaystyle f(x) = x^{2} - 10 \) with \( \displaystyle x_0 = 4 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - 3 x - 3 \) with \( \displaystyle x_0 = 2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - x - 1 \) with \( \displaystyle x_0 = 1 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x + 1 \) with \( \displaystyle x_0 = -2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x + 1 \) with \( \displaystyle x_0 = 2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x - 5 \) with \( \displaystyle x_0 = 2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} + x - 2 \) with \( \displaystyle x_0 = 1 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x - 3 \) with \( \displaystyle x_0 = -1 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.