Newton’s Method and Antiderivatives
MAT1241: Calculus 1
University of Eswatini.
Newton’s Method (Newton-Raphson Method):
Newton’s Method is an iterative numerical technique used to find approximate solutions to equations of
the form (f(x) = 0). It relies on the idea of linear approximation and successive refinements. Here are the
steps:
1. Setup:
○ Given a function (f(x)) and an initial guess .
○ We want to find a better approximation such that .
2. Iteration:
○ Compute the tangent line to the curve at
○ Solve for
○ Repeat the process with the new approximation .
3. Convergence:
○ If the sequence converges, it approaches the root of (f(x)).
Example 1:
Let’s find an approximation for the positive root of .
1. Setup:
○ Initial guess: .
2. Iteration:
○ Compute .
○ Calculate:
3. Repeat:
○ Use as the new guess and repeat the process.
MAT1241: Calculus 1
University of Eswatini.
Newton’s Method (Newton-Raphson Method):
Newton’s Method is an iterative numerical technique used to find approximate solutions to equations of
the form (f(x) = 0). It relies on the idea of linear approximation and successive refinements. Here are the
steps:
1. Setup:
○ Given a function (f(x)) and an initial guess .
○ We want to find a better approximation such that .
2. Iteration:
○ Compute the tangent line to the curve at
○ Solve for
○ Repeat the process with the new approximation .
3. Convergence:
○ If the sequence converges, it approaches the root of (f(x)).
Example 1:
Let’s find an approximation for the positive root of .
1. Setup:
○ Initial guess: .
2. Iteration:
○ Compute .
○ Calculate:
3. Repeat:
○ Use as the new guess and repeat the process.
