I. Decomposition

Back to: The Principles of Calculus I

I.1. The Algebra of Sets

Linguistic Mapping I.1

Peer Engagement I.1

I.1.1. Setting the Stage

I.1.2. The Language of Set Theory

I.1.3. Unions and Intersections

Knowledge Check I.1

I.2. Intervals and Linear Inequalities

Linguistic Mapping I.2

Peer Engagement I.2

I.2.1. Unions and Intersections of Intervals

I.2.2. Multiple Linear Inequalities

Knowledge Check I.2.1

Knowledge Check I.2.2

I.3. Functions and their Basic Properties

Linguistic Mapping I.3

Peer Engagement I.3

I.3.1. Cartesian Products and Relations

I.3.2 Basic Properties of Functions

I.3.3 Comparing Functions

Knowledge Check I.3.1

Knowledge Check I.3.2

I.4. Functions Given by Simple Formulas

Linguistic Mapping I.4

Peer Engagement I.4

I.4.1. Formulas for Functions

I.4.2. Lines

I.4.3. An Elementary Library

Knowledge Check I.4.1

I.5. Manipulating Functions

Linguistic Mapping I.5

Peer Engagement I.5

I.5.1. Restriction to Subdomains

I.5.2. The Algebra of Functions

I.5.3. Decomposing Functions

I.5.4. Computing the Range of a Function

Knowledge Check I.5.1

Knowledge Check I.5.2

Knowledge Check I.5.3

Knowledge Check I.5.4

I.6. Piecewise Functions

Linguistic Mapping I.6

Peer Engagement I.6

I.6.1. Decomposing Domains

I.6.2. Compound Piecewise Defined Functions

I.6.3. Inequalities Involving Piecewise Defined Functions

Knowledge Check I.6.1

Knowledge Check I.6.2

Knowledge Check I.6.3

Knowledge Check I.6.4

Knowledge Check I.6.5

I.7. Functions on Subsets of the Plane

Linguistic Mapping I.7

Peer Engagement I.7

I.7.1. Functions on the Plane

I.7.2. Level Sets

I.7.3. Single Variable Graphs from Multivariate Functions

Knowledge Check I.7.1

I.8. Linear Systems and Feasible Sets

Linguistic Mapping I.8

Peer Engagement I.8

I.8.1. Systems of Linear Equations

I.8.2. Systems of Linear Inequalities

I.8.3. Expressing Feasible Sets in Set Builder Notation

Knowledge Check I.8.1

Knowledge Check I.8.2

Worksheets

Worksheet I.1: The Axiom of Induction

Worksheet I.2: Visualizing Information about Unions and Intersections of Intervals

Worksheet I.3: Qualitative Analysis of Functions Using Graphical Information

Worksheet I.4A: A Quantitative Analysis of Basic Functions

Worksheet I.4B: Convexity, Concavity, and Basic Estimates

Worksheet I.5: Manipulating Functions

Worksheet I.6A: Decomposing the Domains of Functions

Worksheet I.6B: Using Decomposition to Study Function Composition

Worksheet I.7: Decomposing Functions of Several Variables

Worksheet I.8: Decomposing Regions of the Plane with Piecewise Linear Boundaries

© Copyright 2023 by the POC Writing Team: Bryan Carrillo, Dylan Noack, project PI Yat Sun Poon, and lead author David Weisbart.All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from David Weisbart and the POC Writing Team.

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