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Chapter 6: Problem 13
If \(x=\tan \left(\frac{y}{2}\right)-\left(\frac{(1+\tan (y / 2))^{2}}{\tan (y / 2)}\right)\), prove that \(2 \frac{d y}{d x}=-\sin y(1+\sin y+\cos y)\)
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If \(f(x)=a x+b, x \in[-2,2]\), then the point \(c \in(-2,2)\) where $$ f(c)=\frac{f(2)-f(-2)}{4} $$ (a) does not exist (b) can be any \(c \in(-2,2)\) (c) can be only 1 (d) can be only \(-1\).
If \(y=\left(C_{1}+C_{2} x\right) \sin x+\left(C_{3}+C_{4} x\right) \cos x\), show that \(\frac{d^{4} y}{d x^{4}}+2 \frac{d^{2} y}{d x^{2}}+y=0\)
If \(y=\tan ^{-1} x\), prove that \(\left(1+x^{2}\right) y_{2}+2 x y_{1}=0\)
If \(y=\cos ^{-1}\left(\sqrt{\frac{\cos 3 x}{\cos ^{3} x}}\right)\), prove that \(\frac{d y}{d x}=\sqrt{\frac{6}{\cos 2 x+\cos 4 x}}\)
If \(y=\cos ^{-1}\left(\frac{5 t+12 \sqrt{1-t^{2}}}{13}\right)\) and \(x=\cos ^{-1}\left(\frac{1-t^{2}}{1+t^{2}}\right)\), find \(\frac{d y}{d x}\)
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