91Ó°ÊÓ

(a) Bound the error in the approximation $$ \sin (x) \doteq x \quad|x| \leq \delta $$ (b) For small values of \(\delta\), measure the relative error in \(\sin (x) \doteq x\) by using $$ \frac{\sin (x)-x}{\sin (x)} \doteq \frac{\sin (x)-x}{x} \quad x \neq 0 $$ Bound this modified relative error for \(|x| \leq \delta\). Choose \(\delta\) to make this error less than .01, corresponding to a 1 percent error.

Using Taylor's theorem for functions of two variables, find linear and quadratic approximations to the following functions \(f(x, y)\) for small values of \(x\) and \(y\). Give the tangent plane function \(z=p(x, y)\) whose graph is tangent to that of \(z=f(x, y)\) at \((0,0, f(0,0))\). (a) \(\sqrt{1+2 x-y}\) (b) \(\frac{1+x}{1+y}\) (c) \(x \cdot \cos (x-y)\) (d) \(\cos \left(x+\sqrt{\pi^{2}+y}\right)\)

Convert the following numbers to their decimal equivalents. (a) \((10101.101)_{2}\) (b) \((2 A 3 . F F)_{16}\) (c) \((.101010101 \ldots)_{2}\) (d) \((. A A A A \ldots)_{16}\) (e) \((.00011001100110011 \ldots)_{2}\) (f) \((11 \ldots 1)_{2}\) with the parentheses enclosing \(n 1\) s.

Use Taylor approximations to avoid the loss-of-significance error in the following computations. (a) \(f(x)=\frac{e^{x}-e^{-x}}{2 x}\) (b) \(f(x)=\frac{\log (1-x)+x e^{x / 2}}{x^{3}}\) In both cases, what is \(\operatorname{Limit}_{x \rightarrow 0} f(x)\) ?

Consider evaluating \(\cos (x)\) for large \(x\) by using the Taylor approximation \((1.1 .5)\) $$ \cos (x) \doteq 1-\frac{x^{2}}{2 !}+\cdots+(-1)^{n} \frac{x^{2 n}}{(2 n) !} $$ To see the difficulty involved in using this approximation, use it to evaluate \(\cos (2 \pi)=1 .\) Determine \(n\) so that the Taylor approximation error is less than .0005. Then repeat the type of computation used in (1.4.17) and Table 1.2. How should \(\cos (x)\) be evaluated for larger values of \(x ?\)