IN THIS CHAPTER we study the differential calculus of functions of one variable.
- SECTION 2.1 introduces the concept of function and discusses arithmetic operations on functions, limits, one-sided limits, limits at \(\pm\infty\), and monotonic functions.
- SECTION 2.2 defines continuity and discusses removable discontinuities, composite functions, bounded functions, the intermediate value theorem, uniform continuity, and additional properties of monotonic functions.
- SECTION 2.3 introduces the derivative and its geometric interpretation. Topics covered include the interchange of differentiation and arithmetic operations, the chain rule, one-sided derivatives, extreme values of a differentiable function, Rolle’s theorem, the intermediate value theorem for derivatives, and the mean value theorem and its consequences.
- SECTION 2.4 presents a comprehensive discussion of L’Hospital’s rule.
- SECTION 2.5 discusses the approximation of a function \(f\) by the Taylor polynomials of \(f\) and applies this result to locating local extrema of \(f\). The section concludes with the extended mean value theorem, which implies Taylor’s theorem.