Student questions - Numerical Integration Lab

If you have a question about the lab, call me at 827-6842 or email jhall@math.umn.edu.

Q: We have a question about the second part of #4. Do you want our answer expressed in written form or math form? We reason that since both models (PC and PL) are approximations of the same function they should be close, and to us that seems logical. However proving that in math form seems to be a bit tricky.

A: You don't have to prove this mathematically (it's quite tricky), but you need to make a good argument why it is true. They suggest using graphical reasoning. I would simplify the problem by looking at a single interval (with data points at each end). Draw the PC and PL models and look at the difference. Now add a new data point in the middle of the interval and look at how the PC and PL models change.

Q: In the first part of question #10 it states that the rate of CO2 production is monotonically decreasing. Does this mean that is never decreases or never increases? We thought that it meant that the CO2 production was decreasing from one hour after sundown untill dawn, but that does not make biological sense because net CO2 production should increase as plants are not producing oxygen after dark. Could you please clarify this?

A: In this context "monotonically" means "always," so monotonically decreasing means always decreasing. The lab data seems to suggest that the CO2 production is decreasing from one hour after sundown until dawn.

What is confusing here is that the data gives the rate at which the CO2 enters the river. This means that if the rate is positive CO2 is entering the river, and if it is negative CO2 is leaving the river. So CO2 is leaving the river during the day (rate is negative) and entering the river during the night (rate is positive). Biologically this makes sense - the plants produce oxygen during the day, taking CO2 out of the river, while during the night they don't produce oxygen and the CO2 builds up again.

It sounds to me like your biological reasoning was correct, you were just a bit unclear about what the data represents. However, this reasoning only explains why the rate is negative during the day and positive during the night. You'll need to do some more thinking to explain why the rate is decreasing during the night.

Q: We have a quick question about #8. For bullet 2 it asks "Think about the piecewise linear and piecewise constant models. What would these models look like had we only sampled twice a day at sunrise and noon?" We were wondering what times on the graph of the data correspond to sunrise and 12:00, since it never says what hour 0 is.

A: t = 0 is sunrise. Also, there is a misprint in this question (see correction sheet). It should read "... had we only sampled twice a day at sunrise and sunset?" The data was collected in a controlled environment with exactly 12 hours of sun and 12 hours of darkness.

Q: On #16 we have y = e^(x^2) and we are trying to find the integral from 0 to 1. We tried taking the ln of both sides and we got ln(y) = x^2. What do we do from this point?

A: e^(x^2) is a function that does not have an elementary antiderivative, so no amount of manipulation will allow you find the exact integral. The point they are making here is that one of the uses of numerical integration is to approximate the definite integral when you can't find an antiderivative. They want you to use the data in the table and the methods discussed in the lab to come up with upper and lower bounds for the integral. Then compute an approximation on your calculator using the "fnInt" button. The number your calculator gives you should be accurate to at least eight decimal places.