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Chapter 6: Problem 5
Express \(3^{x}, x^{\pi},\) and \(x^{\sin x}\) using the base \(e\).
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Assume that \(y>0\) is fixed and that \(x>0 .\) Show that \(\frac{d}{d x}(\ln x y)=\frac{d}{d x}(\ln x) .\) Recall that if two functions have the same derivative, then they differ by an additive constant. Set \(x=1\) to evaluate the constant and prove that \(\ln x y=\ln x+\ln y\).
Show that \(\cosh ^{-1}(\cosh x)=|x|\) by using the formula \(\cosh ^{-1} t=\ln (t+\sqrt{t^{2}-1})\) and by considering the cases \(x \geq 0\) and \(x<0.\)
Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=e^{x}, y=0, x=0,\) and \(x=2\) revolved about the \(x\) -axis
Use l'H么pital's Rule to evaluate the following limits. $$\lim _{x \rightarrow 1^{-}} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}$$
Let \(R\) be the region bounded by the curve \(y=\sqrt{x+a}(\text { with } a>0),\) the \(y\) -axis, and the \(x\) -axis. Let \(S\) be the solid generated by rotating \(R\) about the \(y\) -axis. Let \(T\) be the inscribed cone that has the same circular base as \(S\) and height \(\sqrt{a} .\) Show that volume \((S) /\) volume \((T)=\frac{8}{5}\).
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