Estimation and Limits -- Chapter Guide

This chapter looks at two closely related topics -- really two aspects of the same thing -- estimation and limits. In many calculus courses limits are thought of as a very theoretical topic but the other aspect of limits -- estimation -- is very practical and it is this practical aspect that we emphasize.

Contents -- Section Summaries

• Introduction

  This module sets the tone for our treatment of estimation and limits. We look at a model of the pollution in a lake and note that the discriptive

    lim    p(n) = 0
  n --> oo
  

  while true is almost irrelevant -- if we must wait one million years before the pollution level drops to the point where it is safe to swim or to drink the water then we won't be around for the re-opening of the beaches. These leads us to "epsilonics" not because mathematicians require rigor but because practical people need to know when they can swim in the lake.

• Estimation and Limits -- A Descriptive Look

  This module introduces limits and looks at several examples descriptively.

• Estimation and Limits -- Area

  This module looks at the problem of estimating area as one example of estimation and limits.

• Estimation and Limits-- Tangents

  This module looks at the problem of estimating the slope of a tangent to a curve as one example of estimation and limits.

• A More Precise Look

  This module develops the more precise side of estimation and limits motivated by practical problems

• A Graphical Look

  This module looks at estimation and limits graphically.

• Limits as x--> oo

  This module looks at limits of the form

    Lim   f(x)
  x --> oo
  

• One-Sided Limits

  This module looks at limits as x --> a from the left or from the right.

  The next series of five modules (with one interruption for a module on continuity) looks at estimation and limits and at the ways in which more complicated functions are built up from simpler functions. This series of modules can be used in several different ways.

  • As a review of how more complicated functions are built up from simpler ones.

  • Developing the standard facts about limits -- for example

      lim    f(x) + g(x) =  lim  f(x) +   lim   g(x)
    x --> a               x --> a       x --> a
    

  • As a discussion of the way in which errors propogate.

  This series of modules has the usual applied tenor emphasizing practical aspects.

• Multiplying a Function by a Constant

• Adding Functions

• Multiplying Functions

• Dividing Functions

• Continuity

• Composing Functions

• Some Limits Involving Sines and Cosines This module discusses two important limits involving the sine and cosine functions.