CBSE Class 11 — Notes, Chapters & Practice Quizzes

Chapter 12: Limits and Derivatives

Summary

This chapter introduces calculus, the study of change. Starting from average velocities of a falling body, it builds an intuitive idea of the derivative as an instantaneous rate of change. The limit \(\lim_{x\to a}f(x)\) is the value \(f(x)\) approaches as \(x\) approaches \(a\); it exists only when the left-hand limit \(\lim_{x\to a^-}f(x)\) and right-hand limit \(\lim_{x\to a^+}f(x)\) coincide. Algebra of limits lets the limit of a sum, difference, product or quotient be computed from the limits of the parts; for polynomials and rational functions the limit is found by substitution (where the denominator is non-zero). Two standard trigonometric limits are \(\lim_{x\to0}\dfrac{\sin x}{x}=1\) and \(\lim_{x\to0}\dfrac{1-\cos x}{x}=0\). The derivative of \(f\) at \(x\) is defined from first principles as \(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\). The algebra of derivatives gives sum, difference, product and quotient rules, and yields standard results such as \(\dfrac{d}{dx}(x^n)=nx^{n-1}\), \(\dfrac{d}{dx}\sin x=\cos x\) and \(\dfrac{d}{dx}\cos x=-\sin x\).

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