Week 10 - Linear Approximation
1. What is Linear Approximation
- We call the tangent line is the linear approximation to the function at .
2. Euler's Method
- AKA, Repeated Linear Approximation
- Definition: Approximate values for the solution of the initial-value problem , , with step size h, at , are
- Let's say , , and h is small number.
- Since this is just an approximation of the derivative, it's better not to pick a point which is all the way on the left hand side of the interval, instead with the middle value:
- Take another example, why is ?
- Since , Let's separate it, to estimate first.
- Set our function , then , so let's start with with step size: 1/4 :
- PS, instead of using , we used , which is more accurate.
- Then use Euler's Method:
- Keep doing this, we got:
- And , pretty close.
3. Newton's Method
- Also called the Newton-Raphson method
- To solve the equation of the form , so the roots of the equation(方程的根) correspond to the x-intercepts of the graph of . The root that we are trying to find is labeled in the figure.
- We start with a first approximation , which is obtained by guessing,
- Consider the tangent line L to the curve at the point and look at the x-intercept of L, labeled .
- The idea behind Newton’s method is that the tangent line is close to the curve and so its x-intercept, , is close to the x-intercept of the curve (namely, the root that we are seeking). Because the tangent is a line, we can easily find its x-intercept.
- To find a formula for in terms of we use the fact that the slope of L is , so its equation is:
- Since the x-intercept of L is , we know that point () is on the line, and so:
- If , we can solve this equation for :
- If we keep repeating this process, we obtain a sequence of approximations as shown:
- In general, if the th approximation is and , then the next approximation is given by:
- If the number become closer and closer to as becomes large, then we say that the sequence converges to and we write
- Sometimes The Newton’s method fails:
- For example, if we choose , then the approximation falls outside the domain of :
- Then we need a better initial approximation .
3.1. Use Newton's Method to Divide Quickly
- Suppose we want to calculate
- Here is one choice: :
- So we can use Newton's Method to find 1/b.