1. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 2.4 The Second Derivative

2. 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

3. The First Derivative

4. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Recall

5. 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) IIIIIIIV Describe each graph’s direction and concavity …

6. 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

7. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. The Second Derivative

8. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example

9. Practical Meaning: The second derivative gives us information about whether the rate is increasing or decreasing. The Second Derivative: Meaning Distance, Velocity, Acceleration Price of a stock Rate of spending

10. Example: The price of a stock is rising faster and faster. The Second Derivative: Meaning Example: The price of a stock is close to bottoming out.

11. 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. The Second Derivative: Meaning Rate of discharge A year agoNow 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.

12. 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. The Second Derivative: Meaning Rate of discharge A year agoNow 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.

13. Which is f, f’, and f’’?

14. 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,19921995199820012004 Percent graduating,62.854.148.043.541.8 Interval,92→9595→9898→0101→04 Ave. Rate of Change,-2.9-2.0-1.5-0.6