I hope you have enjoyed the series and have made some plans to try some new things this year!

As we observed from the original plot, the population change increases as a function of time. If we stare at the above plot, we can observe that the population change increases faster than linearly.

In other words, the above points don't increase as a straight line, but the trail of points tends to bend upward as time increases. Lines are one of the simplest types of equations, so we often want to find a nice simple linear relationship among variables.

But, here, it looks like it won't be able to do that. If all we had were these points and we forgot the experimental system that they came from, we might start employing some more mathematical tricks to find a nice equation for these points. We might think that, since it doesn't look like the points fall on a line, maybe they fall on a parabola.

In fact, we could find an equation for a parabola that gets fairly close to these points. But, before we start going down that road, we need to stop and remember the system we are trying to model.

Our purpose in modeling the data is not simply to find an equation that matches the data points. Our goal is a model that captures the mechanism of the physical system we modeling.

To achieve a model that captures the key features of bacteria population growth, we need to go back and recall what we know about how bacteria reproduce. As mentioned at the outsetbacteria cells reproduce by dividing into two cells, and each cell takes about the same time to mature and divide.

If our snapshots occurred at the same interval as this dividing time, the population size would double at each time step, as it did in the example we used to illustrate dynamical systemscaptured by the bacteria doubling applet.

Looking at the above data, it doesn't appear that the population is doubling at each time step. What went wrong here? Let's imagine the bacteria cells are indeed dividing in half after a certain period of time.

The above snapshots were taken every sixteen minutes. If each bacterium took sixteen minutes to mature and divide, then we'd expect the population to double each time step of sixteen minutes. Since the population growth seems to be slower, it's reasonable to assume each bacterium takes longer than sixteen minutes to mature and divide.

Fitting a linear model to population change versus density If cells are taking longer than a time step to mature and divide, a reasonable assumption is that a certain fraction of the cells divide each time step. We can test this hypothesis by plotting the data a little differently.

As shown in the above plot, the first four data points of population change versus density are pretty much aligned along a line through the origin.

The last point doesn't seem to be exactly in line with the first four, but is a little lower. For now, let's not worry about the last point and focus on the initial growth represented by the first four points.

In the following applet, you can fit the points with a line by moving the red diamond.

Fitting a linear model to bacteria population change as a function of density. Bacteria population density data is plotted by the blue points in the left panel and shown as a chart in the right panel. More information about applet. In the above applet, we forced the line to go through the origin.

Why did we insist the line that goes through the origin? More generally, one may say that a fixed fraction of cells divide every time period.

In this instance, two-thirds of the cells divide every 16 minutes. What about that fifth point? Solving the model More information about video. We can also write out the solution without putting in the value for the initial condition and without multiplying out the solution.

We can also write the solution in terms of original experimental time. You can read more about exponential growth in discrete dynamical systems. How well did we do? The original and computed values are shown in the following table.

Comparison of actual bacterial densities at pH 6.Written by one of the century's truly great thinkers, Ludwig Wittgenstein's Philosophical Investigations is a remarkable--and surprisingly approachable--collection of insights, statements, and nearly displayed thinking habits of the philosopher's work on language, symbols, categories, and a host of other topics.

Organized into nearly short observations, this book is a treasure trove for. Math High School Resources. Click on a section below to view associated resources.

Our Function Table Worksheets & In and Out Boxes Worksheets are free to download, easy to use, and very flexible. These Function Table Worksheets & In and Out Boxes Worksheets are a great resource for children in 3rd Grade, 4th Grade, 5th Grade, 6th Grade, and 7th Grade.

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Algebra 1 answers to Chapter 4 - An Introduction to Functions - Writing a Function Rule - Practice and Problem-Solving Exercises - Page 8 including work step by . Home Textbook Answers Math Algebra Algebra 1 Chapter 4 - An Introduction to Functions - Writing a Function Rule - Lesson Check 1 Algebra 1 by Hall, Prentice.

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Calculus I - Computing Definite Integrals