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Chapter 6: Problem 6
Use Taylor's Theorem with \(n=2\) to obtain more accurate approximations for \(\sqrt{1.2}\) and \(\sqrt{2}\).
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Assume that there exists a function \(L:(0, \infty) \rightarrow \mathbb{R}\) such that \(L^{\prime}(x)=1 / x\) for \(x>0 .\) Calculate the derivatives of the following functions: (a) \(f(x):=L(2 x+3)\) for \(x>0\), (b) \(g(x):=\left(L\left(x^{2}\right)\right)^{3}\) for \(x>0\), (c) \(h(x):=L(a x)\) for \(a>0, x>0\), (d) \(k(x):=L(L(x))\) when \(L(x)>0, x>0\).
Approximate the real zeros of \(h(x):=x^{3}-x-1\). Apply Newton's Method starting with the initial choices (a) \(x_{1}:=2\), (b) \(x_{1}:=0\), (c) \(x_{1}:=-2\). Explain what happens.
The equation \(\ln x=x-2\) has two solutions. Approximate them using Newton's Method. What happens if \(x_{1}:=\frac{1}{2}\) is the initial point?
Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be differentiable on \((0, \infty)\) and assume that \(f^{\prime}(x) \rightarrow b\) as \(x \rightarrow \infty\). (a) Show that for any \(h>0\), we have \(\lim _{x \rightarrow \infty}(f(x+h)-f(x)) / h=b\). (b) Show that if \(f(x) \rightarrow a\) as \(x \rightarrow \infty\), then \(b=0\). (c) Show that \(\lim _{x \rightarrow \infty}(f(x) / x)=b\).
Let \(g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(g(x):=x+2 x^{2} \sin (1 / x)\) for \(x \neq 0\) and \(g(0):=0 .\) Show that \(g^{\prime}(0)=1\), but in every neighborhood of 0 the derivative \(g^{\prime}(x)\) takes on both positive and negative values. Thus \(g\) is not monotonic in any neighborhood of \(0 .\)
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