Let , where for any integer…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let h(x)=x1−cos⁡xh(x)=\dfrac{x}{1-\cos x}, where x≠2nπx\neq2n\pi for any integer nn. (i) Find h′(x)h'(x). (ii) Show that h′(x)=(1−cos⁡x)+xsin⁡x(1−cos⁡x)2h'(x)=\dfrac{(1-\cos x)+x\sin x}{(1-\cos x)^2}. (iii) Simplify the expression obtained in part (ii) for small positive values of xx by taking the first non-zero terms of the series expansions of sin⁡x\sin x and cos⁡x\cos x. What approximate value does this give for h′(0.1)h'(0.1) (correct to four decimal places)?

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