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1: Power functions as building blocks

Last updated

Jun 21, 2023
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- Preface
- 1.1: Power Functions

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Like tall architectural marvels that are made of simple units (beams, bricks, and tiles), many interesting functions can be constructed from simpler building blocks. In this chapter, we study a family of simple functions, the power functions - those of the form $f(x)=x^{n}$ .

Our first task is to understand properties of the members of this "family". We will see that basic observations of power functions such as $x^{2}, x^{3}$ leads to insights into a biological problem of why the size of living cells is limited. Later, we use power functions as "building blocks" to construct polynomials, and rational functions. We then develop important approaches to sketch the shapes of the resulting graphs.

1.1: Power Functions
1.2: How big can a cell be? A model for nutrient balance
In this section, we follow a reasonable set of assumptions and mathematical facts to explore how nutrient balance can affect and limit cell size.
1.3: Sustainability and energy balance on Earth
The sustainability of life on Planet Earth depends on a fine balance between the temperature of its oceans and land masses and the ability of life forms to tolerate climate change. As a follow-up to our model for nutrient balance, we introduce a simple energy balance model to track incoming and outgoing energy and determine a rough estimate for the Earth’s temperature.
1.4: First steps in graph sketching
1.5: Rate of an Enzyme-Catalyzed Reaction
Rational functions often play a role in biochemistry. Here we discuss two such examples and the contexts in which they appear. In both cases, we consider the initial rise of the function as well as its eventual saturation.
1.6: Predator Response
1.7: Summary
1.8: Exercises

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