Study Notes on Car Sales Function and Rate of Change

Cecil's ABC Cars Sales Model

Introduction

• Concept of sales as a function of time.
• The function given is specific to car sales over a one-year period.

Definition of the Function

• The sales function is defined as:
  • $s(t) = 240(6 - 2)$
• Here, ( S ) is the sales over the year, and ( t ) is the time in months (0 ( \leq t \leq ) 12).
• Note that the function may need clarification in mathematical terms as it initially appears unclear mathematically (like not having ( t ) in the primary function form). Assuming some simplification or error, interpret (s(t) = 240(6 - 2)) to other sales metrics.

Average Rate of Cars Sold

• To find the average rate of cars sold over the entire year, calculate:
  • Average Rate = ( \frac{s(12) - s(0)}{12 - 0} )
  • Here, we evaluate the sales function at the endpoints: ( s(0) ) and ( s(12) ).

Sales at Month 0

• Substitute ( t = 0 ) into the sales function to determine initial sales:
  • Assume a base sales:
  • $s(0) = ??$
• Actual calculations depend on further break down or structure of function interpretation from instructor notes.

Sales at Month 12

• Substitute ( t = 12 ) to determine sales at year-end:
  • $s(12) = ??$
• Similar needs as above for breakdown to rationalize outputs.

Find Average Rate

• Form the equation:
  • Average Rate = ( \frac{?? - ??}{12} $$
• Answer needs to input the previous calculations too.

Rate of Change of Car Sales

• Definition of Rate of Change:
  • Derivative of sales function with respect to time.
• Calculate: ( \frac{ds}{dt} ) to find the instant rate of sales.

Equatability of Average Rate and Rate of Change

• Determine during which month the average rate equals the instantaneous rate:
  • Set ( \frac{s(12) - s(0)}{12} = \frac{ds}{dt} ) and solve for ( t ).

Conclusion

• This solves two queries:
  1. Average rate over the year.
  2. The specific month where average rate equates to instantaneous rate.