2 4 Rates of Change and Tangent Lines

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2. 4 Rates of Change and Tangent Lines Copyright © 2007 Pearson Education, Inc.

2. 4 Rates of Change and Tangent Lines Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Average Rates of Change The average rate of change of a quantity over a

Average Rates of Change The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. Also, the average rate of change can be thought of as the slope of a secant line to a curve. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2

Example Average Rates of Change Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Average Rates of Change Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3

Example Average Rates of Change Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Average Rates of Change Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4

Tangent to a Curve In calculus, we often want to define the rate at

Tangent to a Curve In calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. The problem with this is that we only have one point and our usual definition of slope requires two points. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5

Tangent to a Curve The process becomes: 1. Start with what can be calculated,

Tangent to a Curve The process becomes: 1. Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. 2. Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3. Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 6

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 7

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 9

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 10

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 11

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 12

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Tangent to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 13

Slope of a Curve To find the tangent to a curve y = f(x)

Slope of a Curve To find the tangent to a curve y = f(x) at a point P(a, f(a)) calculate the slope of the secant line through P and a point Q(a+h, f(a+h)). Next, investigate the limit of the slope as h→ 0. If the limit exists, it is the slope of the curve at P and we define the tangent at P to be the line through P with this slope. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 14

Slope of a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice

Slope of a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 15

Slope of a Curve at a Point Copyright © 2007 Pearson Education, Inc. Publishing

Slope of a Curve at a Point Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 16

Slope of a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice

Slope of a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 17

Normal to a Curve The normal line to a curve at a point is

Normal to a Curve The normal line to a curve at a point is the line perpendicular to the tangent at the point. The slope of the normal line is the negative reciprocal of the slope of the tangent line. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 18

Example Normal to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson

Example Normal to a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 19

Speed Revisited Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 20

Speed Revisited Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 20

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