Me and my partner are doing a group calculus problem and this one question has gotten us both stumped. Any help or way to start the problem would be appreciated.
Biological activity in a pond is reflected in the rate at which carbon dioxide, CO2, is added to or withdrawn from the water. Plants take CO2 out of the water during the day for photosynthesis and put CO2 into the water at night. Animals put CO2 into the water all the time as they respire. Biologists are interested in how the net rate at which CO2 enters the pond varies during the day. Figure 1 below shows this rate as a function of time of day. The rate is measured in millimoles (mmol) of CO2 per liter of water per hour; time is measured in hours past dawn. At dawn, there were 2.600 mmol of CO2 per liter of water. [This problem is about reading the graph below correctly. Be careful to use appropriate notations from calculus in your discussions. For example, use derivative and integral notations in your discussions if appropriate. If any approximations to integrals need be made, use the midpoint method. Also, cite appropriate theorems in your discussion.]
(a) What can be concluded from the fact that the rate is negative during the day and positive at night? Be aware that the graph provides the bombined rate of change of CO2 in the pond due to plants and animals.
(b) Some scientists have suggested that plants respire (breathe) at a constant rate at night, and that they photosynthesize at a constant rate during the day. Does Figure 1 support this view? Be aware that Figure 1 represents the rate at which plant life and animal life put CO2 into the pond. There is only one process (respiration) through which animal life produce CO2, but animals may be more active at certain times of the day than at others. It is reasonable to assume that animal life produces CO2 at a continuous rate but not at a constant rate. You should ask the following question. "If those scientists were correct, how would this affect the graph in Figure 1?"
(c) When was the CO2 content of the water at its lowest? This is a minimization problem, and you should related your answer to a sufficiency test for absolute extrema.
(d) How much CO2 was released into the water during the 12 hours of darkness? Compare this quantity with the amount of CO2 withdrawn from the water during the 12 hours past daylight. How can you tell by looking at the graph whether the CO2 in the pond is in equilibrium?