1. AP Calculus Unit 5 Day 10

2. Practice: Given f(x) = x 2 – 4 1)Use LRam with 4 equal partitions to estimate the integral from [-4, 4] 2)Use Trapezoidal Rule for same. 3)Find using FTC 4)Find total area between the curve and the x- axis.

3. Part 1: More Practice with yesterday

4. Practice

6. Is F(x) concave up or down at x = 3? Applying Concepts...

8. FTC (Antiderivative Part) States: The interpretation is that the instantaneous rate of change of this accumulating area is in fact the y-value of the curve at the given instant.

9. From a Real-World Application (KNOW this!!!)—Test NEXT Week Similar to the velocity of an object (rate of change of position) Similar to “accumulated” displacement

10. The point ….. Given a rate of change function r(t) Then can be interpreted as an accumulated amount This is important to understand!!!! Study the next two slides.

11. Real Life Applications--Business Rate of Change Function Real Life Meaning Interpret C’(t) Business Application: “Marginal Cost” to produce t units of an itemTotal cost of producing “x” units S’(t) Business Application: Rate at which sales increase where t is measured in days “Accumulated” (total) sales for “x” days “Marginal Cost” change in total cost that arises when the quantity produced changes by one unit

12. Accumulation Example Rate of Change Function Real Life Meaning Interpret N’(t) Industry Rate at which pollutants enter a lake, measured in pounds per month Total number of pounds of pollutants that enter the lake over a period of “x” months The rate at which pollutants enter a lake from a factory is where N is the total number of pounds of pollutants in the lake at time t. How much pollutant enters the lake in 16 months? What is the average rate pollutants enter the lake over the 16 month time period? Include units.

13. Accumulation Example ANSWERS Rate of Change Function Real Life Meaning Interpret N’(t) Industry Rate at which pollutants enter a lake, measured in pounds per month Total number of pounds of pollutants that enter the lake over a period of “x” months The rate at which pollutants enter a lake from a factory is where N is the total number of pounds of pollutants in the lake at time t. How much pollutant enters the lake in 16 months? What is the average rate pollutants enter the lake over the 16 month time period?

14. Part 2: Average Value Theorem

15. Finding the Average y-Value of a Function Let T=f(t) represent temperature at time t, in hours. The temperature is recorded for a 24-hr time period. Below is a graph of the collected temperatures: The “average temperature” could be found for a given 24-hr time period by using the readings at 4-hr intervals.

16. If it is a hot summer day and at 2:00 in the afternoon (hour 14) there is a short thunderstorm that cools the air for an hour between readings. This average temperature would not reflect this dip in temperature. Taking more readings at shorter time intervals would result in a better average value for T.

17. Average Value Theorem: If a function is integrable on [a,b], then the function’s average y-value is:

19. Find the average value