Rates of Change and Tangent Lines Chapter 2.4

Average Rates of Change 2

Example 1: Finding Average Rate of Change 3

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Example 2: Growing Drosophila in a Laboratory Experimental biologists often want to know the rates at which populations grow under controlled laboratory conditions. The figure on the next slide shows how the number of fruit flies (Drosophila) grew in a controlled 50-day experiment. The graph was made by counting flies at regular intervals, plotting a point for each count, and drawing a smooth curve through the plotted points (this can be done using a kind of regression known as logistic regression). 6

Example 2: Growing Drosophila in a Laboratory 7

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Instantaneous Rate of Change Suppose that, rather than average rate of change, we want to know the rate of change at on a particular day (say, day 23 from the previous example)? This is the instantaneous rate of change (which was first introduced in Chapter 2.1) In that chapter, we saw how we can find an instantaneous rate of change by starting with an approximation in the form of a secant line, and letting one point get closer to the desired point But we lacked, at that point, a way to formulate this idea mathematically; now we can do this (how?) 11

Instantaneous Rate of Change 12

Instantaneous Rate of Change 13

Example 3: Finding Slope and Tangent Line 14

Example 3: Finding Slope and Tangent Line 15

Example 3: Finding Slope and Tangent Line 16

Example 3: Finding Slope and Tangent Line 17

Slope of a Curve at a Point 18

Slope of a Curve at a Point 19

Example 4: Exploring Slope and Tangent 20

Example 4: Exploring Slope and Tangent 21

Example 4: Exploring Slope and Tangent 22

Example 4: Exploring Slope and Tangent 23

Example 4: Exploring Slope and Tangent 24

Normal to a Curve 25

Example 5: Finding a Normal Line 26

Example 5: Finding a Normal Line 27

Example 6: Finding Instantaneous Rate of Change 28

Example 6: Finding Instantaneous Rate of Change 29

Example 6: Finding Instantaneous Rate of Change 30

Example 7: Investigating Free Fall 31

Example 7: Investigating Free Fall 32

Exercise