the table shows the number of dvd players sold in a small electronics store in the years 2003 - 2013.\n(a)…

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

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

Answer

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). [ \begin{align*} \text{Average rate of change}&=\frac{605 - 495}{2013-2003}\ &=\frac{110}{10}\ & = 11 \end{align*} ]

Step3: Solve part (b)

For the interval ([2003,2004]), (x_1 = 2003), (f(x_1)=495), (x_2 = 2004), (f(x_2)=518). [ \begin{align*} \text{Average rate of change}&=\frac{518 - 495}{2004 - 2003}\ &=\frac{23}{1}\ &=23 \end{align*} ]

Step4: Solve part (c)

For the interval ([2004,2005]), (x_1 = 2004), (f(x_1)=518), (x_2 = 2005), (f(x_2)=410). [ \begin{align*} \text{Average rate of change}&=\frac{410 - 518}{2005 - 2004}\ &=\frac{- 108}{1}\ &=-108 \end{align*} ]

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)