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I'm currently learning Calculus. I would really like to understand the material instead of just mindlessly applying methods. A little dilemma I've come across is I think there seems to be a little hole in my knowledge.

I'm struggling to answer questions relating to real life scenarios.

Example: Let $s(t)$ be a model for DVD sales, in terms of t years. Where $s(t) = \frac{7t}{t^2+1}$.

Q1: Find the rate of change for sales

Q2: When will sales peak?

Q3: How fast will sales increase when the movie is released?

Now, I know how to do the question, since I've watched the lecture. However, I don't really understand what the questions are asking. For Q1, I don't understand what significance finding the rate of change would serve? For example, in a graph with distance against time, I know that by finding the derivative I will be able to find the instantaneous speed at any point in time. But for the example I've presented, what purpose would finding the rate of change serve? What could I possibly find with it?

For Q3, how does the derivative tell us how fast sales will increase? I understand plugging in 0 would be 0 years and in effect when the movie was first released. But, I still don't understand how finding the derivative of s(t) and plugging in 0, will tell us how fast sales will increase.

I've tried wording my questions to the best of my ability, I'm sorry if there is any ambiguity.

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  • $\begingroup$ I don't know how to make a fraction sorry. $\endgroup$ – user666 Oct 23 '16 at 5:07
  • $\begingroup$ @msm I appreciate it $\endgroup$ – user666 Oct 23 '16 at 5:09
  • $\begingroup$ You're welcome. Regarding Q1, it is a rate of change for sale. Note that sale means money. Do you understand the change in rate of that can change the amount of money is earned? It is also important since the people in charge of preparing the DVDs should increase their workload to cope with the rate of sales. The store cannot remain empty ... $\endgroup$ – msm Oct 23 '16 at 5:12
  • $\begingroup$ Regrding Q3, just like a moving object where acceleration is the rate of change of speed, you need to find the second derivative to see how fast the rates are changing. It tells you for instance, the required rate of increase in the workforce to prepare new DvDs. $\endgroup$ – msm Oct 23 '16 at 5:15
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In answer to your question about Q1. The reason why you would want to know the rate of change of your sales, is so that you could predict how much product (in this case DVD's) are needed at a given time. Because you don't want to have too much or too little stock available.

In answer to Q3. The reason why you would want to know how fast the sales increase at release is so that you can once again have just enough product.

The reason why these may seem so confusing as to why we need them, is that this is most likely after the movie has come out. But in reality mathematicians have to create a formula such as your sales formula, based on hype about the movie as well as previous DVD sales of similar movies.

I hope that my reasoning is not too ambiguous.

And I hope that I have got my point across to you.

And I hope this helps you.

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