How do I do this?

Question

Vinyl records (LPs) almost disappeared after CDs were introduced in the early 1980s, but they have been making a comeback. Wagner's Records estimates that t months after the grand opening of the store, the sales of LPs will be S(t) = 5t³ + 4t² + 300t

units. Compute S(6), S'(6), and S''(6), and interpret your results.

s(6)=

s'(6)=

s'''(6)=

Answer 1

I am assuming a typo in the last line of the problem, namely that it should

be s''(t), not s'''(t)

s(t) = 5t³ + 4t² + 300t

s'(t) = 15t² + 8t + 300

s''(t) = 30t + 8

s(6) = 5×6³ + 4×6² + 300×6 = 3024 records

Interpretation:

When the store keeper counts his receipts at the end of the sixth month,

he find that during those first six month he sold a total of 3024 records.

s'(6) = 15×6² + 8×6 = 588 records/month

Interpretation:

At the end of the sixth month the store keeper, unsure how his sales will continue,

assumes that he will be selling records at the same rate as he was at the end of the sixth month.

That is, he assumes that his sales will neither improve nor worsen.

Under that assumption he would sell 588 records the next month,

reaching a total of 3024 + 588 = 3612 records.

Notice that this turns out to be a conservative assumption;

his sales did improve, and by the end of the seventh month he will sell

s(7) = 4011 records.

s"(6) = 30×6 + 8 = 188 records/month²

Interpretation:

At the end of the sixth month the store keeper, unsure how his sales will continue,

allows the possibility that the rate of sales will continue to change,

but will continue to change the same way as it was at the end of the sixth month.

Under that assumption, during the next month the sales would improve by 188 records/month,

reaching 588 + 188 = 776 records/month.

Notice that this turns out to be a conservative assumption;

his sales improved even faster, and by the end of the seventh month he will be selling records at the rate

s'(7) = 791 records/month.