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Chapter 6: Problem 24
Approximate the solution of the equation \(x=\cos x\), accurate to within six decimals.
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Given that the restriction of the tangent function tan to \(I:=(-\pi / 2, \pi / 2)\) is strictly increasing and that \(\tan (I)=\mathbb{R}\), let Arctan: \(\mathbb{R} \rightarrow \mathbb{R}\) be the function inverse to the restriction of \(\tan\) to \(I\). Show that Arctan is differentiable on \(\mathbb{R}\) and that \(D \operatorname{Arctan}(y)=\left(1+y^{2}\right)^{-1}\) for \(y \in \mathbb{R}\).
Determine where each of the following functions from \(\mathbb{R}\) to \(\mathbb{R}\) is differentiable and find the derivative: (a) \(f(x):=|x|+|x+1|\), (b) \(g(x):=2 x+|x|\) (c) \(h(x):=x|x|\), (d) \(k(x):=|\sin x|\),
Evaluate the following limits: (a) \(\lim _{x \rightarrow 0} \frac{\operatorname{Arctan} x}{x}(-\infty, \infty)\), (b) \(\lim _{x \rightarrow 0} \frac{1}{x(\ln x)^{2}} \quad(0,1)\), (c) \(\lim _{x \rightarrow 0+} x^{3} \ln x \quad(0, \infty)\) (d) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x}}(0, \infty)\).
If \(r>0\) is a rational number, let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x):=x^{r} \sin (1 / x)\) for \(x \neq 0\), and \(f(0):=0\). Determine those values of \(r\) for which \(f^{\prime}(0)\) exists.
Show that if \(x>0\), then \(1+\frac{1}{2} x-\frac{1}{8} x^{2} \leq \sqrt{1+x} \leq 1+\frac{1}{2} x\).
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