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Section 2.4 The Second Derivative Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. The First Derivative • • If the first derivative is positive on an interval, the function is increasing on that interval If the first derivative is negative on an interval, the function is decreasing on that interval Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. The First Derivative Recall Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Describe each graph’s direction and concavity … I II III IV I. Increasing and concave up - Increasing at an increasing rate II. Decreasing and concave up - Decreasing at an increasing rate (decreasing magnitude) III.Increasing and concave down - Increasing at a decreasing rate IV.Decreasing and concave down - Decreasing at a decreasing rate (increasing magnitude) The Second Derivative • • The second derivative is the derivative of the derivative function. Thus the second derivative tells us something about the rate of change of the derivative. • If the first derivative is increasing on an interval, then the function is concave up on that interval. And, if the first derivative is increasing on an interval, the second derivative is positive on that interval. • If the first derivative is decreasing on an interval, then the function is concave down on that interval. And, if the first derivative is decreasing on an interval, the second derivative is negative on that interval. • An inflection point is a point at which the concavity changes. The Second Derivative Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. The Second Derivative: Meaning Practical Meaning: The second derivative gives us information about whether the rate is increasing or decreasing. • Distance, Velocity, Acceleration • Price of a stock • Rate of spending The Second Derivative: Meaning Example: The price of a stock is rising faster and faster. Example: The price of a stock is close to bottoming out. The Second Derivative: Meaning Example: An industry is being charged by the Environmental Protection Agency (EPA) with dumping unacceptable levels of toxic pollutants in a lake. Measurements are made of the rate at which pollutants are being discharged into the lake. What does the following graph mean and how would the industry and the EPA interpret the graph. Rate of discharge The EPA will say that the rate of discharge is still rising. The industry will say that the rate of discharge is increasing less quickly, and may soon level off or even start to fall. A year ago Now The Second Derivative: Meaning Rate of discharge Example: An industry is being charged by the Environmental Protection Agency (EPA) with dumping unacceptable levels of toxic pollutants in a lake. Measurements are made of the rate at which pollutants are being discharged into the lake. What does the following graph mean and how would the industry and the EPA interpret the graph. A year ago Now The EPA will say that the rate at which pollutants are being discharged is leveling off, but not to zero—so pollutants will continue to be dumped in the lake. The industry will say that the rate of discharge has decreased significantly. Which is f, f’, and f’’? No pair of graphs are and ′. So, if is the red graph, sketch graphs of ′ and ′′. High School Example A high school principal is concerned about the drop in the percentage of students who graduate from her school, shown in the following table. Year entered school, Percent graduating, 1992 62.8 1995 54.1 1998 48.0 2001 43.5 2004 41.8 1. Calculate the average rate of change of for each of the three-year intervals between 1992 and 2004. Interval, Ave. Rate of Change, 92→95 -2.9 95→98 -2.0 98→01 -1.5 01→04 -0.6 2. Does 2 / 2 appear to be positive or negative between 1992 and 2004? 3. Explain why the values of and / are troublesome to the principal. 4. Explain why the sign of 2 / 2 and the magnitude of / in the year 2001 may give the principal some cause for optimism.