Calculus Emphasis: Derivative.

Skills Emphasized: First and second derivative.

Note to Student:

There are many instances when a particular amount of substance “surges” in the beginning and then dissipates. For example, when a particular drug is injected into the bloodstream there is an initial concentration surge which decreases over time as the body eliminates the drug. Other phenomena are the amount of radiation after an event, water runoff after a heavy rain, the amount of melatonin your body produces during an average life-time, and the consumer demand for a product after an initial advertising blitz. The graphs of these phenomena have the general shape shown in figure 1 and are modeled by the product of a power function and an exponential function where A, p, and b are all constants relative to the specific phenomena being modeled.

Figure 1

The Problem.

Energy drinks are all the rage these days. A company creates a new energy drink called, “Jolt Juice” which contains an incredibly powerful combination of caffeine, ginseng, and taurine which the manufacture calls the “secret ingredient”. This secret ingredient will pep up the most lethargic individual and is modeled accurately by the surge function

Suppose a college student slams down a 12 oz can of Jolt Juice at 6 am in hopes of getting some extra energy during finals week.

1. Graph the surge function for a 24-hour day.

2. If t is measured in hours, then at what time is the secret ingrediant’s