Chapter 2: Rates, Sums, Limits, and Continuity

During Chapter 1 you studied the relationship between velocity and distance. This chapter will continue that focus, leading to new applications of calculus.

In this chapter, you will investigate all three of the core concepts of calculus: limits, rates of change, and area under a curve. The approach statements you wrote in Chapter 1 will become important for the concept of limits.

Finally, instead of examining the end behavior of a function, you will begin to describe a function locally by considering instantaneous velocity and finding local linearizations of functions.

Chapter Goals

Approximate the area under a
curve using Riemann
sums and summation notation.

Predict function behavior with limits.

Formally define continuity.

Discuss local linearity.

Approximate the velocity of an
object at an instant.

Chapter Outline

Section 2.1

You will approximate the area under a curve using Riemann sums and summation notation. You will also use trapezoids to approximate the area.

Section 2.2

You will explore limits through approach statements, graphs, and algebra. You will predict function behavior with limits. You will also use limits to define continuity and see how continuity provides the basis for the Intermediate Value Theorem.

Section 2.3

You will apply your knowledge of rates of change to develop a method to approximate the velocity of an object at an instant. You will explore local linearity concepts and analyze the proofs of limh→0sin⁡(h)h and limh→01−cos⁡(h)h.

Section 2.4

You will complete the development of Riemann sums and use a graphing calculator to investigate using left endpoint, right endpoint, and midpoint rectangles to approximate area under a curve.