Calculus 3208 Derivative (18)
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1 Calculus 3208 Derivative (8) Unit 4: Chapter # 2 Section 2. (Essential Calculus) Average and Instantaneous Slope
2 Average and Instantaneous Slope A Concrete Example of a Rate of Change Average Rate of Change Instantaneous Rate of Change
3 An Apple Falls From a Tree Synopsis: A Graph of Height vs. Time: 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) Height (m) (0,20) (0.5, 8.775) (. 5.) (.5, 8.975) (2, 0.4)
4 Average Rate of Change Question: At what average speed is the apple falling between 0.5s and.5s after it starts to drop? v avg h t 2 2 h t Avg. Rate of Change = Slope of Secant = m / s y x (0,20) (0.5, 8.775) (. 5.) (.5, 8.975) (2, 0.4)
5 Instantaneous Rate of Change Question: How fast is the apple falling, exactly second after it starts to drop? We need to estimate: Inst. Rate of Change = Slope of Tangent v inst h2 t2 t, h ( ) h t Where: and t 2, h ) ( 2 Should be as close as possible to the point of tangency! Examples: v inst m / s OR v inst m / s (0,20) (0.5, 8.775) (. 5.) (.5, 8.975) (2, 0.4)
6 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) Depth (ft.) (A) What is the average rate of change (in ft. per day) of the depth of the reservoir between the 0 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 . 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 990 to 200. Approximately how fast was the US national debt rising at the end of 2005?
8 population 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. time (minutes)
9 population - Solution 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. (A) The average rate of change can be calculated by finding the slope of the secant joining (0, 00), and (3, 200). (Shown in Red) P t 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. minst bacteria/min Time (minutes)
10 The average rate of change of a function f (x) on an interval a,b, can be calculated by finding the slope of the secant line joining a, f ( a), and b, f ( b) : f b f a m avg b a The instantaneous rate of change of a function f (x) at the point a, f ( a), is the slope of the tangent that touches f (x) at that point. To estimate the instantaneous rate of change of a function f (x) at the point a, f ( a), we can chose a second point c, f ( c), where c is reasonably close to a. We then calculate: m isnt f c f c a a
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