Chapter 1: A Beginning Look at Calculus

Welcome to calculus! What is calculus, anyway? This course will be spent answering this question. In the first activity, you and your team will work on a problem that illustrates the kinds of questions calculus can help solve. Many of the mathematical and problem-solving skills you already have will enable you to calculate approximate solutions. As this course progresses, the methods of calculus will enable you to determine precise solutions. You will discover that you may learn better if you show your work clearly and completely. You can explain your ideas in writing, using equations, graphs, tables, and well-labeled calculations. Correctness, completeness, and clarity all count!

Chapter Goals

Build a positive team climate for
learning calculus.

Review precalculus topics.

Calculate volumes of solids
geometrically.

Investigate finite differences.

Explore slope.

Examine the relationship between
position and velocity graphs.

Chapter Outline

Section 1.1

You will start the year exploring area under the curve and slope at a point with your team, both major themes of calculus.

Section 1.2

You will review various types of functions such as piecewise-defined, composite, inverse, and even/odd functions. You will describe their graphs, including stating their domain and range. You will explore the behavior of graphs at the far left and right, as well as any holes or asymptotes within the graph.

Section 1.3

You will study slope by examining finite differences. You will learn to write slope statements and sketch slope functions.

Section 1.4

You will use graphs to analyze motion. You will determine average velocity on distance graphs and velocity graphs. You will discover the relationships among total distance, displacements, velocity, speed, and acceleration.

Section 1.5

You will examine area and slope on distance and velocity graphs to find a relationship between them.