Math 237 Course Notes

Math 237 Course Notes - Calculus 3 Course Notes for MATH...

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Unformatted text preview: Calculus 3 Course Notes for MATH 237 Edition 4.1 J. Wainwright and D. Wolczuk Department of Applied Mathematics Copyright: J. Wainwright, August 1991 2nd Edition, July 1995 D. Wolczuk, 3rd Edition, April 2008 D. Wolczuk, 4th Edition, September 2009 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii To the Student Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Graphs of Scalar Functions 2 1.1 Scalar Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Geometric Interpretation of f : R 2 → R . . . . . . . . . . . . . . . . . 4 2 Limits 9 2.1 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Proving a Limit Does Not Exist . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Proving a Limit Exists . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Continuous Functions 20 3.1 Definition of a Continuous Function . . . . . . . . . . . . . . . . . . . . 20 3.2 The Continuity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Limits revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 The Linear Approximation 29 4.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Second Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 The Tangent Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4 Linear Approximation for f : R 2 → R . . . . . . . . . . . . . . . . . . . 36 4.5 Linear Approximation in Higher Dimensions . . . . . . . . . . . . . . . 39 5 Differentiable Functions 42 5.1 Definition of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Differentiability and Continuity . . . . . . . . . . . . . . . . . . . . . . 47 CONTENTS CONTENTS 5.3 Continuous Partial Derivatives and Differentiability . . . . . . . . . . . 49 5.4 The Linear Approximation Revisited . . . . . . . . . . . . . . . . . . . 52 6 The Chain Rule 55 6.1 Basic Chain Rule in Two Dimensions . . . . . . . . . . . . . . . . . . . 55 6.2 Extensions of the Basic Chain Rule . . . . . . . . . . . . . . . . . . . . 62 6.3 The Chain Rule for Second Partial Derivatives . . . . . . . . . . . . . . 67 7 Directional Derivatives and the Gradient Vector 72 7.1 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.2 The Gradient Vector in Two Dimensions . . . . . . . . . . . . . . . . . 76 7.3 The Gradient Vector in Three Dimensions . . . . . . . . . . . . . . . . 80 8 Taylor Polynomials and Taylor’s Theorem 82 8.1 The Taylor Polynomial of Degree 2 . . . . . . . . . . . . . . . . . . . . 82 8.2 Taylor’s Formula with Second Degree Remainder . . . . . . . . . . . . 85 8.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9 Critical Points...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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Math 237 Course Notes - Calculus 3 Course Notes for MATH...

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