** 17.T Differentials
and linear approximations ** (<-- READ ME)

** Reference:** Stewart 2.9

** Keywords: ** Differentials, linear approximations

** Learning Objectives:** The student will be able to:

- Define the differential of a function
*y=f(x)*:*dy = f '(x)dx* - Use the differential to approximate the change in a function due to a small change in its argument, when the function is given either explicitly, or by a verbal problem description
- Use the linear approximation (also called the tangent line approximation) to approximate a function

**Note:** We will try to start on Newton's method this day, too.

** 17.R Newton's method; Inverse functions and their derivatives **

** Reference: ** Stewart 2.10, 6.1

** Keywords: ** Newton's method,
approximate solution to an equation, iterative technique,
inverse functions, derivatives of inverse functions

** Learning Objectives: ** The student will be able to:

- Use Newton's method to solve
*f(x) = 0* - Explain Newton's method graphically, and derive the formula for implementing it
- Explain the importance of having a sufficiently accurate initial estimate
of the solution to
*f(x) = 0* - Determine if a given function is one-to-one and hence has an inverse
- Find the inverse of a function explicitly, in simple cases
- Graph the inverse of a function from the graph of the function
- Find the derivative of the inverse of a function, at a point, without first obtaining an explicit expression for the inverse function