1. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador Calculus 3208 Derivative (18) Unit 4: Chapter # 2 – Section 2.1 (Essential Calculus) Average and Instantaneous Slope

2. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador Average and Instantaneous Slope A Concrete Example of a Rate of Change Average Rate of Change Instantaneous Rate of Change

3. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador An Apple Falls From a Tree Synopsis: An apple falls from atop a 20m tree and is in free-fall as it drops toward the ground. It’s height above the ground (in meters) is a decreasing function of time (in seconds). A Table of Heights and Times: Time (s) 00.511.52 Height (m) 2018.77515.18.9750.4 A Graph of Height vs. Time:

4. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador Average Rate of Change Question: At what average speed is the apple falling between 0.5s and 1.5s after it starts to drop? Avg. Rate of Change = Slope of Secant =

5. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador Instantaneous Rate of Change Question: How fast is the apple falling, exactly 1 second after it starts to drop? Inst. Rate of Change = Slope of Tangent We need to estimate: Where: and Should be as close as possible to the point of tangency! Examples: OR

6. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador Over the span of a month, the depth of the water in a city reservoir varies with time. The table shown gives selected values for the depth of the water (in ft.) as a function of time (in days since the beginning of March). Time (days)210152330 Depth (ft.)1012202127 (A) What is the average rate of change (in ft. per day) of the depth of the reservoir between the 10 th and 30 th of March? (B) What is the approximate instantaneous rate of change of the depth of the reservoir (in ft. per day) on March 23 rd ?

7. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador 1.Let D(t) be the US national debt at time t. The table shown gives approximate values of this function by providing end of year estimates, in billions of dollars, from 1990 to 2010. Approximately how fast was the US national debt rising at the end of 2005?

8. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador The graph at the right represents the population of bacteria in a laboratory culture as a function of time (in minutes). (A) Determine the average growth rate of the bacteria culture over the first three minutes. (B) Estimate the instantaneous growth rate of the bacteria 6 minutes after the culture was started. population time (minutes)

9. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador The graph at the right represents the population of bacteria in a laboratory culture as a function of time (in minutes). (A) Determine the average growth rate of the bacteria culture over the first three minutes. (B) Estimate the instantaneous growth rate of the bacteria 6 minutes after the culture was started. population Time (minutes) - Solution (A) The average rate of change can be calculated by finding the slope of the secant joining (0, 100), and (3, 200). (Shown in Red) bacteria/min (B) The exact instantaneous rate of change can be calculated by finding the slope of the tangent touching the curve at (6, 400). (Shown in Green) We can estimate using the secant joining (6, 400), and (9, 800). Other reasonable estimates are acceptable. bacteria/min

10. Copyright © 2013 All rights reserved, Government of Newfoundland and Labrador The average rate of change of a function on an interval, can be calculated by finding the slope of the secant line joining, and : The instantaneous rate of change of a function at the point, is the slope of the tangent that touches at that point. To estimate the instantaneous rate of change of a function at the point, we can chose a second point, where c is reasonably close to a. We then calculate: