Informal Calculus with Applications to Biological and Environmental Sciences (Seacrest)

This book is an approachable introduction to calculus with applications to biology and environmental science. For example, one application in the book is determining the volume of earth moved in the 1959 earthquake that created Quake Lake. Another application uses differential equations to model various biological examples, including moose and wolf populations at Isle Royale National Park, ranavirus in amphibians, and competing species of protozoa. The text focuses on intuitive understanding of concepts, but still covers most of the algebra and calculations common in a survey of calculus course.

Algebra Tips and Tricks Part I (Combining Terms, Distributing, Functions, Graphing)

Here are a few algebra tips and tricks to get you started. In later chapters, we will have some “just-in-time” algebra review, so you’ll review an algebra concept just before you need it.

Position to Velocity

The idea of position of an object versus the velocity of an object encompasses all the big ideas of calculus. So that’s where we’ll start!

This graph might represent you walking to a lake two miles away, hanging out for half an hour, then walking home. The first part of the graph that slants upwards represents your walk to the lake, since your distance from home is increasing (higher on the graph). The second part of the graph represents you hanging out at the lake. It’s flat since your distance from home is not changing. Finally, the part of the graph that slants down represents you walking home. Your distance to home is decreasing, so the line goes down on the graph.

What is the slope of the red line? Well, rise over run would be , which is . That’s the same as the velocity graph! Same thing for the green line: it has a slope of zero, and the velocity graph is at zero. Finally, the slope of the blue line is which is , and that is what we have for the velocity graph.

So “slope” and “velocity” are the same thing. But there is another name for this concept that we will use a lot: derivative. Derivative, slope, and velocity all mean the same thing.

Other Examples of Derivatives Let’s see some other examples. Note for each of these, the position graphs is always piecewise linear, or made up of line segments. This makes it easier to find the velocity, or slope.

Integrals

We can also go in the reverse direction: take a velocity graph, and create a position graph. This is called integration or taking an integral. This can be tricky but we can do it at this point if the function is what is called a step function, which is basically a function consisting of a bunch of flat parts.