# 2: Differential Calculus of Functions of One Variable

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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.

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