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Section 2.3
Increasing and Decreasing Functions, Average Rate of Change Main Concepts
Increasing/Decreasing Functions
Average rate of change over an interval
Average speed over an interval
Foreshadowing to Calculus
A function, f, is increasing on an interval I if whenever in the interval I.
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A function, f, is decreasing on an interval I if whenever in the interval I.
On what intervals is the following function increasing and on what intervals is it decreasing?
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Average Rate of Change
Average Rate of change of distance = Average speed or velocity = in miles per hour or feet per second
This graph shows s(t)=distance traveled in terms of ‘t’ hours traveled.
What was the average speed for the entire trip?
Over what interval was the speed greatest?
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Average Speed of Falling Object.
Suppose you drop a ball out of a hot air balloon 1600 feet in the air. How fast does it fall?
When does the ball land?
What is the average speed from the time you drop in until it lands?
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Average Speed of Falling Object.
Suppose you drop a ball out of a hot air balloon 1600 feet in the air. How fast does it fall?
What is the average velocity of the ball from 5 seconds until 10 seconds after it is dropped?
From t=9 to t=10?
How fast is the ball falling when it hits the ground?
Average Speed of Falling Object.
Suppose you drop a ball out of a hot air balloon 1600 feet in the air. How fast does it fall?
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What do you do in Calculus?
Find “instantaneous” rate of change instead of average rate of change.
The ball is actually continuing to gain speed until the moment it lands. (Ignoring air resistance)
How do you find approximate the instantaneous speed when it lands? (Similar to example 4 in book)
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Find average rate of change on interval [10­w,10]. This is the last 'w' seconds before the ball lands. Think about 'w' being 0.01 seconds or 0.0001 seconds, getting very very small.
Average Rate of Change= 7
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HW 2.3 #11 State intervals where f(x) is increasing or decreasing.
Use your calculator and look at the graph.
TI­SmartView_TI­84.exe
http://math.hws.edu/xFunctions/
Sec. 2.3 #19 Determine the average rate of change of the function on the given interval.
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• #35 The table shows the number of CD players sold in a small store each year from 1993 until 2003.
• What was average rate of change of sales from 1993 until 2003?
• Between 1993 and 1994?
• Between 1994 and 1996?
HW 2.3 #35 The table shows the number of CD players sold in a small store each year from 1993 until 2003.
• Between which two successive years did CD player sales increase most rapidly?
• Between which two successive years did CD player sales decrease most rapidly?
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Attachments
TI­SmartView_TI­84.exe