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Chapter 2 2012 Pearson Education, Inc. Section 2.4 Rates of Change and Tangent Lines Limits and Continuity

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Slide 2.4- 2 2012 Pearson Education, Inc. Quick Review

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Slide 2.4- 5 2012 Pearson Education, Inc. Quick Review Solutions

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Slide 2.4- 8 2012 Pearson Education, Inc. What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited …and why The tangent line determines the direction of a body’s motion at every point along its path

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Slide 2.4- 9 2012 Pearson Education, Inc. 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.

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Slide 2.4- 10 2012 Pearson Education, Inc. Example Average Rates of Change

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Slide 2.4- 11 2012 Pearson Education, Inc. 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.

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Slide 2.4- 12 2012 Pearson Education, Inc. 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.

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Slide 2.4- 13 2012 Pearson Education, Inc. Example Tangent to a Curve

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Slide 2.4- 14 2012 Pearson Education, Inc. Example Tangent to a Curve

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Slide 2.4- 15 2012 Pearson Education, Inc. 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.

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Slide 2.4- 16 2012 Pearson Education, Inc. Slope of a Curve

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Slide 2.4- 17 2012 Pearson Education, Inc. Slope of a Curve at a Point

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Slide 2.4- 18 2012 Pearson Education, Inc. Slope of a Curve

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Slide 2.4- 19 2012 Pearson Education, Inc. 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.

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Slide 2.4- 20 2012 Pearson Education, Inc. Example Normal to a Curve

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Slide 2.4- 21 2012 Pearson Education, Inc. Speed Revisited

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Slide 2.4- 22 2012 Pearson Education, Inc. Quick Quiz Sections 2.3 and 2.4

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Slide 2.4- 23 2012 Pearson Education, Inc. Quick Quiz Sections 2.3 and 2.4

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Slide 2.4- 24 2012 Pearson Education, Inc. Quick Quiz Sections 2.3 and 2.4

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Slide 2.4- 25 2012 Pearson Education, Inc. Quick Quiz Sections 2.3 and 2.4

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Slide 2.4- 26 2012 Pearson Education, Inc. Quick Quiz Sections 2.3 and 2.4

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Slide 2.4- 27 2012 Pearson Education, Inc. Quick Quiz Sections 2.3 and 2.4

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Slide 2.4- 28 2012 Pearson Education, Inc. Chapter Test

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Slide 2.4- 34 2012 Pearson Education, Inc. Chapter Test Solutions

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