Which of the following best explains why the slope of the graph of increases without…

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

Which of the following best explains why the slope of the graph of y=tan⁡xy=\tan x increases without bound as xx approaches π2\dfrac{\pi}{2} from the left? (a) Because cos⁡x\cos x approaches 0+0^{+}, the reciprocal sec⁡x\sec x grows without bound; since the derivative is sec⁡x\sec x, the slope diverges. (b) Because sin⁡x\sin x approaches 11, the derivative sin⁡2x\sin^{2}x approaches 11, so the slope remains finite. (c) Because cos⁡x\cos x approaches 0+0^{+}, the expression sec⁡2x=1cos⁡2x\sec^{2}x=\dfrac{1}{\cos^{2}x} grows without bound, so the derivative diverges. (d) Since cos⁡x\cos x changes sign near π2\dfrac{\pi}{2}, the factor sec⁡2x\sec^{2}x becomes negative, causing the derivative to approach −∞-\infty.

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