The figure above shows a picture of a graph which models manufacturing cost per unit. It makes sense to us that because of the cost of raw materials the more you manufacture, the more you spend. So clearly making 3000 units is more expensive than making 2000 units. But, as you can see, it is actually more expensive to make 1000 units than it is to make 2000 units. (This may be because of discounts for in-bulk materials, because of start-up costs in the factory, etc.)
The obvious question for a manufacturer, then, is: how many units should I produce to minimize cost? This scenario is one in the category of “optimization” (as in “what is the optimal solution?”) and is a great application for Calculus! Our senior Calculus students are applying their skills with derivatives to solve problems of this type over the next few weeks. (Other applications: how can I maximize profit? minimize fencing materials to create the largest pen? maximize range of a projectile based on angle of launch?)
A derivative is basically an instantaneous “rate of change”. In this case of cost per unit, we are interested in that moment when the cost is at its lowest point. In other words, we want to know when the cost transitions from going down (a negative rate of change) to going up (a positive rate of change). That moment of transition is the optimal solution. Pop Quiz… if you are looking for a moment when something changes from a negative rate of change to a positive rate of change, what are you looking for? Answer: the moment when the rate of change (or the derivative) is equal to _____. (Answer at end of this post.)
Many Calculus concepts must be mastered in order to fully answer a question of optimization, and since August our WA students have been building idea upon idea to reach this most recent application. (Our WA Calculus class is almost identical to a year in undergraduate Calculus, so the theories we have covered match up very well with what freshman Calculus students have been experiencing at colleges across America.) At this point in the year our students have come a long way! I am so proud of our WA students.
Answer to Pop Quiz: zero! (I.e., the number between positive stuff and negative stuff.)