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Resequencing Calculus Gruenwald and Dwyer The Problem Resequencing Calculus Existing Solutions An Early Multivariate Approach Our Solution Revised Sequence Calculus I Calculus II Calculus III Mark Gruenwald and David Dwyer Strengths

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Resequencing Calculus

An Early Multivariate Approach Mark Gruenwald and David Dwyer University of Evansville August 5, 2010

Supported by NSF Grant 0836676

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

The Problem

The standard calculus sequence is a bad fit for many students: Multivariate calculus comes too late for biology, economics, and some chemistry majors. Some students who would benefit from a differential equations course don’t have room for Calc III. Many STEM students need matrices. Calculus has none; linear algebra requires Calc III.

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Existing Solutions

Teach calculus just-in-time within discipline Survey of calculus course or sequence Discipline-specific calculus course or sequence

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Our Solution

Early Multivariate Calculus

Reformulate the 3-semester calculus sequence, primarily by including some multivariate calculus in Calculus II and moving series to Calculus III, so that First two semesters form strong sequence for majors in the life and social sciences and others. Three-course sequence is an even better fit for math, physics, and engineering majors.

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus I

Chapters

1. Functions
2. Limits
3. The Derivative
4. Applications of the Derivative
5. The Integral

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus I

Chapters

1. Functions
2. Limits
3. The Derivative
4. Applications of the Derivative
5. The Integral

Key Changes for Calculus I Incorporation of sequences and their limits

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus I

Chapters

1. Functions
2. Limits
3. The Derivative
4. Applications of the Derivative
5. The Integral

Key Changes for Calculus I Incorporation of sequences and their limits Inclusion of polar and parametric functions

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus I

Chapters

1. Functions
2. Limits
3. The Derivative
4. Applications of the Derivative
5. The Integral

Key Changes for Calculus I Incorporation of sequences and their limits Inclusion of polar and parametric functions Introduction of Taylor polynomials (but not series!)

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus I

Chapters

1. Functions
2. Limits
3. The Derivative
4. Applications of the Derivative
5. The Integral

Key Changes for Calculus I Incorporation of sequences and their limits Inclusion of polar and parametric functions Introduction of Taylor polynomials (but not series!) Reduced emphasis on graphing by hand

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus II

Chapters

6. Techniques of Integration
7. Applications of Integration
8. Differential Equations
9. Matrices and Vectors
10. Functions of Several Variables
11. Double Integrals

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus II

Chapters

6. Techniques of Integration
7. Applications of Integration
8. Differential Equations
9. Matrices and Vectors
10. Functions of Several Variables
11. Double Integrals

Key Changes for Calculus II Inclusion of functions of several variables

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus II

Chapters

6. Techniques of Integration
7. Applications of Integration
8. Differential Equations
9. Matrices and Vectors
10. Functions of Several Variables
11. Double Integrals

Key Changes for Calculus II Inclusion of functions of several variables Double integrals

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus II

Chapters

6. Techniques of Integration
7. Applications of Integration
8. Differential Equations
9. Matrices and Vectors
10. Functions of Several Variables
11. Double Integrals

Key Changes for Calculus II Inclusion of functions of several variables Double integrals Elimination of infinite series

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus III

Chapters

12. Infinite Series
13. Vector-Valued Functions
14. Surfaces, Solids, and Multiple Integrals
15. Vector Analysis
16. Differential Equations Revisted (Optional)

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus III

Chapters

12. Infinite Series
13. Vector-Valued Functions
14. Surfaces, Solids, and Multiple Integrals
15. Vector Analysis
16. Differential Equations Revisted (Optional)

Key Changes for Calculus III Inclusion of infinite series

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Calculus III

Chapters

12. Infinite Series
13. Vector-Valued Functions
14. Surfaces, Solids, and Multiple Integrals
15. Vector Analysis
16. Differential Equations Revisted (Optional)

Key Changes for Calculus III Inclusion of infinite series Fewer required sections

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Strengths

Better fit for life sciences, chemistry, and economics Natural progression of difficulty through the sequence Greater chance of completing vector calculus Facilitates an earlier course in differential equations Allows for calculus-based probability after Calc II

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Challenges

Placing transfer students Placing students with AP (especially BC) credit Overcoming inertia Pacing

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Status

Consulted with (external) math advisory board Consulted with (internal) STEM+ advisory board Completed Calculus I material Calculus II development in progress

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Next Steps

Classroom testing of Calculus I at the University of Evansville (Fall 2010) Classroom testing of Calculus II at UE (Spring 2010) Assessment and modification Classroom testing at other institutions

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Resequencing Calculus Gruenwald and Dwyer The Problem Existing Solutions Our Solution Revised Sequence

Calculus I Calculus II Calculus III

Strengths Challenges Status Next Steps Further Information

Further Information

Mark Gruenwald mg3@evansville.edu David Dwyer dd4@evansville.edu Website resequencingcalculus.com

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 1 - Functions

1.1 Functions and Their Graphs 1.2 Algebra of Functions 1.3 Library of Functions 1.4 Implicit Functions 1.5 Polar Functions 1.6 Parametric Functions

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 2 - Limits

2.1 Limits in Calculus 2.2 Limits: Numerical and Graphical Approaches 2.3 The Precise Definition of a Limit 2.4 Calculating Limits Using Limit Laws 2.5 Limits at Infinity and Limits of Sequences 2.6 Continuity and the Intermediate Value Theorem

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 3 - The Derivative

3.1 Tangents, Velocities, and Other Rates of Change 3.2 Derivatives 3.3 Rules for Differentiation 3.4 Product and Quotient Rules 3.5 Derivatives of Trigonometric Functions 3.6 Chain Rule 3.7 Implicit Differentiation 3.8 Parametric and Polar Differentiation 3.9 Inverse Functions and Their Derivatives 3.10 Logarithmic Functions and Their Derivatives

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 4 - Applications of the Derivative

4.1 Maximum and Minimum Values 4.2 The Mean Value Theorem 4.3 Derivatives and Graphs 4.4 Optimization 4.5 Rates of Change 4.6 L’Hopital’s Rule 4.7 Polynomial Approximations 4.8 Tangent Line Approximations: Differentials and Newton’s Method

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 5 - The Integral

5.1 Antiderivatives and Indefinite Integrals 5.2 Area Under a Curve and Total Change 5.3 The Definite Integral 5.4 The Fundamental Theorem of Calculus 5.5 Integration By Substitution

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 6 - Techniques of Integration

6.1 Advanced Substitution techniques 6.2 Integration by Parts 6.3 Partial Fractions 6.4 Improper Integrals 6.5 Approximating Definite Integrals

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 7 - Applications of Integration

7.1 Average Value and Area Between Curves 7.2 Arc Length 7.3 Volumes 7.4 Solids of Revolution 7.5 Work 7.6 Probability

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 8 - Differential Equations

8.1 Introduction To Differential Equations 8.2 Symbolic Solutions To Differential Equations 8.3 Graphical and Numerical Solutions To Differential Equations 8.4 Exponential Growth and Decay 8.5 Logistic Models 8.6 Linear First Order Differential Equations 8.7 Difference Equations

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 9 - Matrices and Vectors

9.1 Matrices and Vectors 9.2 Determinants and Inverse Matrices 9.3 Vectors: A Geometric Approach 9.4 The Dot Product 9.5 The Cross Product 9.6 Lines and Planes in Space

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 10 - Functions of Several Variables

10.1 Introduction to Functions of Several Variables 10.2 Limits and Continuity 10.3 Partial Derivatives 10.4 Chain Rule 10.5 Directional Derivatives and Gradients 10.6 Tangent Planes and Linear Approximations 10.7 Extrema and the Second Partials Test 10.8 Lagrange Multipliers

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 11 - Double Integrals

11.1 Double Integrals over Rectangles 11.2 Double Integrals over Regions 11.3 Double Integrals in Polar Coordinates 11.4 Applications of Double Integrals

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 12 - Infinite Series

12.1 Sequences of Partial Sums 12.2 Series 12.3 Integral Test 12.4 Comparison Tests 12.5 Alternating Series 12.6 Ratio and Root Tests 12.7 Power Series 12.8 Power Series Representations of Functions 12.9 Taylor Series 12.10 Fourier Series

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 13 - Vector-Valued Functions

13.1 Review of Vectors 13.2 Vector-Valued Functions 13.3 Differentiation and Integration of Vector-Valued Functions 13.4 Arc Length and Curvature 13.5 Motion in Space 13.6 Tangent and Normal Vectors

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 14 - Surfaces and Multiple Integrals

14.1 Conic Sections 14.2 Cylinders and Quadric Surfaces 14.3 Review of Double Integrals 14.4 Surface Area 14.5 Integrals over Surfaces: Triple Integration 14.6 Cylindrical and Spherical Coordinates 14.7 Triple Integrals in Cylindrical and Spherical Coordinates 14.8 Change of Variables: The Jacobian

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 15 - Vector Analysis

15.1 Vector Fields 15.2 Line Integrals 15.3 Conservative Vector Fields 15.4 Green’s Theorem 15.5 Parametric Surfaces 15.6 Surface Integrals 15.7 Divergence Theorem 15.8 Stoke’s Theorem

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Chapter 16 - Differential Equations Revisited

16.1 Exact Differential Equations 16.2 Second-Order Linear Equations: Real Roots 16.3 Second-Order Linear Equations: Complex Roots 16.4 Series Solutions of Differential Equations 16.5 Systems of First-Order Equations

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Resequencing Calculus Gruenwald and Dwyer Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16

Revised Sequence

Course Section Count Typical Majors Calculus I 34 Math, Phys, Engr, Chem, Bio, Econ Calculus II 33 Math, Phys, Engr, Chem, Bio, Econ Calculus III 28 Math, Phys, Engr, Chem (Prof.)