## Week 17

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