Explorations for Calculus I

Janet Beery, University of Redlands

These classroom explorations and exercise sets are designed to help students discover and/or understand important ideas and techniques from calculus.  They assume students have access to a graphing calculator or computer grapher but not necessarily to a computer algebra system (CAS).  I use them with the calculus text by Hughes-Hallett, et al, published by Wiley, and have listed corresponding section numbers from the 4th and 5th editions of this text.  These are listed as (4th / 5th) where they differ.

Riding the Calculus Bus:  Velocities as Limits (2.1)
Slopes as Limits (2.2)
Derivatives as Limits (2.2, 2.3)
Derivatives as Slopes (2.3, preview of graphing and optimization strategies in Chapter 4)
Concavity and First and Second Derivatives (2.5)
Derivatives and Smooth Airplane Take-off (2.6, 3.1)*
Tests for Local Extrema and Concavity (4.1)
Review of Tests for Local Extrema and Concavity (4.1)
Candidates Test for Global Extrema (4.3 / 4.2)
Tests for Global Extrema (4.3, 4.5 / 4.2, 4.4)
Optimization without Calculus (4.3, 4.5 / 4.2, 4.4)
Optimization with Calculus (4.5 / 4.4)
More Optimization (4.5 / 4.4)
Riding the Calculus Bus:  Distances as Areas and Limits (5.1, 7.5)
For introducing the definite integral (Chapter 5) and antiderivatives (Chapter 6):  various explorations from Paul Foerster’s Calculus Explorations, Key Curriculum Press, 1998, especially Exploration 3, Introduction to Definite Integrals, and Exploration 26, A Motion Antiderivative Problem
Modeling the Flight of a Water Balloon (6.3)
Another Fundamental Theorem of Calculus! (6.4)
The Most Ubiquitous Initial Value Problems (11.1)
Population and Food (11.5)
World Population Models (11.7)*

* I rarely have time to use this exploration, but I included it because I sure wish I did!

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