Derivative as a Rate Measurer — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsDerivative as a Rate Measurer3 Marks
Question
Find an angle $\theta$ Which increases twice as fast as its cosine.
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Answer
Let $\text{x}=\cos\theta$ Differentiating both sides with respect to t, we get $\frac{\text{dx}}{\text{dt}}=\frac{\text{d}(\cos\theta)}{\text{dt}}$ $=-\sin\theta\frac{\text{d}\theta}{\text{dt}}$ But it is given that $\frac{\text{d}\theta}{\text{dt}}=2\frac{\text{dx}}{\text{dt}}$ $\Rightarrow\frac{\text{dx}}{\text{dt}}=-\sin\theta\Big(2\frac{\text{dx}}{\text{dt}}\Big)$ $\Rightarrow\sin\theta=-\frac{1}{2}$ $\Rightarrow\theta=\pi+\frac{\pi}{6}=\frac{7\pi}{6}$ Hence, $\theta=\frac{7\pi}{6}.$
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