Question

the table shows the number of dvd players sold in a small electronics store in the years 2003 - 2013.
(a) what was the average rate of change of sales between 2003 and 2013?
players/yr
(b) what was the average rate of change of sales between 2003 and 2004?
players/yr
(c) what was the average rate of change of sales between 2004 and 2005?
players/yr
(d) between which two successive years did dvd player sales increase most quickly?
to
between which two successive years did dvd player sales decrease most quickly?
to

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here, \(x\) represents the year and \(y\) represents the number of DVD - players sold.

Step2: Solve part (a)

For the interval \([2003,2013]\), \(x_1 = 2003\), \(f(x_1)=495\), \(x_2 = 2013\), \(f(x_2)=605\).

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Step3: Solve part (b)

For the interval \([2003,2004]\), \(x_1 = 2003\), \(f(x_1)=495\), \(x_2 = 2004\), \(f(x_2)=518\).

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Step4: Solve part (c)

For the interval \([2004,2005]\), \(x_1 = 2004\), \(f(x_1)=518\), \(x_2 = 2005\), \(f(x_2)=410\).

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Step5: Solve part (d)

Calculate the average rate of change for each successive - year interval:

• \(2003 - 2004\): \(\frac{518 - 495}{2004 - 2003}=23\)
• \(2004 - 2005\): \(\frac{410 - 518}{2005 - 2004}=-108\)
• \(2005 - 2006\): \(\frac{402 - 410}{2006 - 2005}=-8\)
• \(2006 - 2007\): \(\frac{520 - 402}{2007 - 2006}=118\)
• \(2007 - 2008\): \(\frac{580 - 520}{2008 - 2007}=60\)
• \(2008 - 2009\): \(\frac{631 - 580}{2009 - 2008}=51\)
• \(2009 - 2010\): \(\frac{719 - 631}{2010 - 2009}=88\)
• \(2010 - 2011\): \(\frac{624 - 719}{2011 - 2010}=-95\)
• \(2011 - 2012\): \(\frac{582 - 624}{2012 - 2011}=-42\)
• \(2012 - 2013\): \(\frac{605 - 582}{2013 - 2012}=23\)

The largest positive value of the average rate of change occurs between \(2006\) and \(2007\). The largest negative value (for decrease) occurs between \(2004\) and \(2005\).

Answer:

(a) \(11\) players/yr
(b) \(23\) players/yr
(c) \(-108\) players/yr
(d) \(2006\) to \(2007\); \(2004\) to \(2005\)