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Rates of Change and Tangent Lines Chapter 2.4
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Average Rates of Change 2
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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
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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
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Instantaneous Rate of Change 12
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Instantaneous Rate of Change 13
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Example 3: Finding Slope and Tangent Line 14
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Example 3: Finding Slope and Tangent Line 15
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Example 3: Finding Slope and Tangent Line 16
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Example 3: Finding Slope and Tangent Line 17
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Slope of a Curve at a Point 18
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Slope of a Curve at a Point 19
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Example 4: Exploring Slope and Tangent 20
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Example 4: Exploring Slope and Tangent 21
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Example 4: Exploring Slope and Tangent 22
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Example 4: Exploring Slope and Tangent 23
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Example 4: Exploring Slope and Tangent 24
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Normal to a Curve 25
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Example 5: Finding a Normal Line 26
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Example 5: Finding a Normal Line 27
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Example 6: Finding Instantaneous Rate of Change 28
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Example 6: Finding Instantaneous Rate of Change 29
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Example 6: Finding Instantaneous Rate of Change 30
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Example 7: Investigating Free Fall 31
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Example 7: Investigating Free Fall 32
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Exercise 2.4 33
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