Example 0.1.
If you have a model for spread of disease where the number of infections doubles every three days, how long can this go on before the model has to change: A few years? A few months? A few weeks? A few days?
Preface Preface: approximations and bounds
This e-textbook is about using math for modeling and coming up with plausible analyses. One of the course goals is number sense. Wikipedia defines this as "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations."
If some kind of answer began to form in your mind without your stopping to get out a calculator, then you have some of the ingredients of number sense already: perhaps you understand exponential growth, perhaps you can remember about how many people there are in the country or the world, perhaps you are familiar with powers of two and know how they relate to this problem. It is useful to be able to think this way. It’s not important whether you use a calculator to answer any given question, but realistically, how often will you stop in casual conversation and whip out a calculator?
In this course, we’ll teach you a number of these ingredients: use of logs, converting to powers of ten, tangent line approximation, Taylor polynomials, pairing off positive and negative summands, approximating integrals with sums and vice versa. Discussions of these will be brief. The point is to use them when you need them, which turns out to be nearly every lesson.
Today we’re going to start with the tangent line approximation. You might think this odd because we haven’t taught you calculus yet. Calculus is a way of computing the slopes of tangent lines to graphs. But conceptually, understanding the tangent line approximation takes knowledge only of algebra and geometry, not calculus. So, we’ll preview the idea now, and in fact several more ideas from the course, and then later see how to use calculus to do these analyses more methodically.
