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Linear Approximation




 
Higher-Order Derivatives and Linear Approximation
Using the Tangent Line Approximation Formula
Tangent Line Approximation / Linearization
Example:
Use a linear approximation to approximate the valve of each of the following:
a) sin(18π/17)
b) √15.9 Newton's Method
Basic idea of Newton's Method and how to use it.
One example using Newton's Method to approximate a root. Example:
Find where f(x) = x7 - 1000 intersects the x-axis; find solution correct to 8 decimal places. Newton's Method
Example:
Compute two iterations of Newton's method for the given function and indicated initial guess:
f(x) = x2 - 8, x1



Newton's Method
Example:
Apply two iterations of Newton's method to approximate the x-value of a point of intersection of these two functions using the given initial guess:
f(x) = x2 - 4, g(x) = 2x - 3, x1 = 0
What is the value of x3? Newton's Method - How it Can FAIL
Example:
Given the following equation and initial guess, why would Newton's method fail to approximate a solution?
f(x) = (x - 2)2 - 1; x1 = 2

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