## 2. Introduction

How long should a bee feed at a flower before moving on to the next flower?

Some critical thinking about this question gives insight into a trade-off situation. Recall that the bee wants to collect as much nectar as possible over the course of a day.

The bee has many options. In one extreme, the bee can choose to remain at one flower for the entire day. In doing so, the bee sacrifices the ease of food collection, but does not "waste" time flying to the next flower. In the other extreme, the bee can choose to remain at any one flower for only a very short time, leaving for the next flower quickly. In this case, although it is easy to collect some nectar from each flower, the bee does not collect much overall since she spends the majority of her time flying between flowers.

It appears that there may be an optimal amount of time for the bee to spend at a flower before moving on to the next. Later we will discuss exactly what optimal refers to.

Most importantly, we must realize that this optimal amount of time is the answer we seek to our question: How long should a bee feed at a flower before moving on to the next?

To begin, consider , the amount of nectar collected by a bee that stays on a flower for time , where is time measured in minutes.

Given what we know about the nectar collection process, the function should have the following properties:

1. The longer the bee stays at a flower, the more nectar the bee collects.
2. If the bee were to stay at the flower indefinitely, it would collect all of the nectar.
3. If the bee does not spend any time at the flower, she will not collect any nectar.
4. The longer the bee stays at the flower, the harder it is to collect nectar.

A reasonable choice for is

In the following question, we will verify that this choice of satisfies the above properties.

Is an increasing or decreasing function?

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is increasing on . This can be verified by using the quotient rule to find the first derivative of :

Since the numerator and denominator of are both positive for all , then for all . That is, is increasing.

Thus, the longer a bee spends at a flower, the more nectar the bee will collect.

Does have a horizontal asymptote?

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Yes, does have a horizontal asymptote for . To see this, we can find the limit of the function as tends to infinity. We have

This limit is of the indeterminate form . We can use L'Hopital's rule to help determine the limit.

We know that

provided the limit on the right exists.

Since

we know . In other words, has horizontal asymptote .

This suggests that if a bee were to stay at a flower indefinitely , the bee would collect all of the nectar.

What are the -intercepts?

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has a -intercept at . This is because

That is, if a bee does not spend any time at the flower , then the bee will not collect any nectar.

Is convex or concave for ?

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is concave on . This is verified by finding the second derivative .

We have already calculated . Applying the quotient rule to gives us :

Since for all , then for all . Thus, is concave on . This suggests that the longer the bee stays at the flower, the harder it is to collect nectar. (The slower the bee collects nectar from the flower, the smaller the . In other words, the smaller the slope of ).

Graph for .

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Thus far, we have found a reasonable function that describes the amount of nectar collected by a bee that stays on a flower for time . In the next section, we will determine what the optimal time refers to.