Automotive Repair Shop Growth

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Lesson Parent: 
Differentiation V: Derivatives and Rates of Change
Problem: 

Bob's Automotive has experienced repair shop growth given in the graph below. The number of auto shops at the end of each year is listed in the graph. Bob's Automotive growth rate is the average rate of change of Bob's Automotive shops per year. Determine Bob's automotive growth rate for each year.


Bob's Automotive Shop Growth
Answer: 

SAMPLE SOLUTION

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Solution: 

From the graph we see that Bob built his first automotive repair shop in 2005 and by the end of 2011 he had 120 shops. The average rate of change of the number of Bob's shops with respect to year is given by

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where $t_i$ is the year and $N_i$ is the number of shops at the end of year $t_i$.

Year Shop Growth
2005

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2006

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2007

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2008

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2009

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2010

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2011

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Comparing 2009 and 2010 we see that Bob's repair shop growth increased as he went from building 20 shops per year to building 70 shops per year. In 2011 his growth slowed as he built only 20 shops per year. Notice that the slope of the graph is steeper with more shops per year being built. This should reinforce the concept that rates of change are slopes of a graph.