This section describes how differentiation can be used to simplify complex calculations and graph functions.
» Session 23: Linear Approximation
» Session 24: Examples of Linear Approximation
» Session 25: Introduction to Quadratic Approximation
» Session 26: Using Quadratic Approximations
» Session 27: Sketching Graphs I - Polynomials and Rational Functions
» Session 28: Sketching Graphs II - General Strategies
» Problem Set 3
In this session you will:
Use Applications of Differentiation (PDF) to do the problems below.
| Section | Topic | Exercises |
|---|---|---|
| 2A | Approximation | 1, 3, 6, 11, 12a, 12d, 12e |
| 2B | Curve sketching | 2a, 2e, 2h, 4, 6a, 6b, 7a, 7b |
Solutions to Applications of Differentiation problems (PDF)
This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.
Linear approximation is a powerful application of a simple idea. Very small sections of a smooth curve are nearly straight; up close, a curve is very similar to its tangent line. We calculate linear approximations (i.e. equations of tangent lines) near x=0 for some popular functions; we can then change variables to get approximations near x=a.
Clip 1: Introduction to Linear Approximation
Clip 2: Linear Approximation to ln x at x=1
Clip 3: Linear Approximation and the Definition of the Derivative
Clip 4: Approximations at 0 for Sine, Cosine and Exponential Functions
Comparing Linear Approximations and Calculator Computations
When using linear approximation, we replace the formula describing a curve by the formula of a straight line. This makes calculation and estimation much easier.
Clip 1: Curves are Hard, Lines are Easy
Clip 2: Linear Approximation of a Complicated Exponential
Clip 3: Question: Can We Use the Original Formula?
Product of Linear Approximations
Clip 1: GPS Time Dilation Example
Linear approximation uses the first derivative to find the straight line that most closely resembles a curve at some point. Quadratic approximation uses the first and second derivatives to find the parabola closest to the curve near a point.
Clip 1: The Formula for Quadratic Approximation
Clip 2: Explaining the Formula by Example
Clip 3: Quadratic Approximation at 0 for Several Examples
Comparing Quadratic Approximations and Calculator Computations
In this session we see some examples of quadratic approximation, then finish compiling a “library” of quadratic approximations of key functions.
Clip 1: List of Approximations
Clip 3: Example: Approximation of a Complicated Function
Clip 1: Approximating ln(1+x) and (1+x)r
Finding a Formula for the Best Degree n Approximation
The first derivative of a function tells us whether its graph slopes up or down or is level. The second derivative tells us how that slope is changing. By combining this information with what we know about asymptotes, intercepts and plotting points we can sketch a pretty good graph of the function.
Clip 1: Introduction to Curve Sketching
Clip 3: Graphing a Rational Function
This session outlines a step by step method of graphing a function then provides an example of this method. Finally, you are asked to turn things around and find the equation of a function given a description of its graph.
Clip 1: General Strategy for Graph Sketching
Graph Features
Use the mathlet below to complete this worked example.
