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