Section 2.1The Derivative: “Tangent Lines and

Rates of Change”

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Calculus,10/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

Introduction

Many real-world phenomena involve changing quantities-the speed of a rocket, the inflation of money (currency), the number of bacteria in a culture, the voltage of an electrical signal, etc.

A “derivative” is the mathematical tool for studying the rate at which one value changes relative to another.

Tangent lines relate this change to slope.

Secant Line vs. Tangent Line

As points P and Q on this graph get closer and closer together, the slope of the secant line through P and Q gets closer to the slope of the tangent line which is only at P.

From Geometry, a secant line crossestwice and a tangent

line touches once.

Slope of the Secant Line

The slope of the secant line comes from Algebra I, m = rise/run =

This book uses different notation, but it means the same thing.

Slope of the Tangent Line

The slope of the tangent line is only through one point so we cannot use the same equation. Instead, we must calculate the limit as point Q approaches point P.

Example

We can use this limit to find the slope of the line tangent to the parabola y = x 2 at x = 1.

Velocity

One of the important themes in calculus is the study of motion. To describe motion we discuss speed and direction of travel which, together, comprise velocity.

For this section, we will only consider motion along a line (rectilinear motion).

Average Velocity Example

Instantaneous Velocity

Instead of an average, we often want to determine velocity at a specific instant in time. It is like finding the slope of the tangent line vs. finding the slope of the secant line.

We only have one time, so we cannot subtract. Instead, we must find the limit as we get closer and closer to the exact time we are looking for.

Instantaneous Velocity Example

My brother, cousin and me in Iowa at family reunion.