Introduction
197
- 3.1The Chain Rule
- 3.2Implicit Differentiation; Derivatives of the Inverse Trigonometric Functions
- 3.3Derivatives of Logarithmic Functions
- 3.4Differentials; Linear Approximations; Newton’s Method
- 3.5Taylor Polynomials
- 3.6Hyperbolic Functions
- Chapter Review
- Chapter Project
World Population Growth
In the late 1700’s Thomas Malthus predicted that population growth, if left unchecked, would outstrip food resources and lead to mass starvation. His prediction turned out to be incorrect, since in 1800 world population had not yet reached one billion and currently world population is in excess of seven billion. Malthus’ most dire predictions did not come true largely because improvements in food production made his models inaccurate. On the other hand, his population growth model is still used today, and population growth remains an important issue in human progress.
CHAPTER 3 PROJECT
In the Chapter Project on page 253 we explore a basic model to predict world population and examine its accuracy.
In this chapter, we continue exploring properties of derivatives, beginning with the Chain Rule, which allows us to differentiate composite functions. The Chain Rule also provides the means to establish derivative formulas for exponential functions, logarithmic functions, and hyperbolic functions.
We also use the derivative to solve problems involving relative error in the measurements of two related variables, a way to approximate the real zeros of a function, and the approximation of functions by polynomials.