Derivative?

Question

Dorothy Wagner is currently selling 20 "I ♥ Calculus" T-shirts per day, but sales are dropping at a rate of 3 per day. She is currently charging $8.8 per T-shirt, but to compensate for dwindling sales, she is increasing the unit price by $1 per day. How fast and in what direction is her daily revenue currently changing?

Her revenue is decreasing at a rate of ____ per day.

Answer 1

Notice that everything is changing as a function of time. That's a hint that you should set up some equations that express things as a function of time.

One obvious choice is: Cost of a shirt. At time t=0 (that is, "today"), the cost is 8.8. Tomorrow (at time t=1), it will be 9.8. So cost as a function of time is:

C(t) = $8.8 + ($1/day)•t

The next thing that changes as a function of time is: number "N" of shirts sold per day. At t=0 (today), that's 20. Tomorrow (at t=1), it'll be 17. So, number of daily shirts as a function of time is:

N(t) = 20 – (3/day)•t

And now we can put those together to find her daily revenue R as a function of time. Clearly,

daily revenue = (cost per shirt) x (shirts per day). That is:

R(t) = C(t)•N(t)

Now you've got all you need. Just differentiate R with respect to t, and that will give you the rate at which R is changing per day. Hint: this rate will also be a function of t. To find out how fast her daily revenue is changing TODAY, set t=0 in the rate function.