91Ó°ÊÓ

Use the error term of a Taylor polynomial to estimate the error involved in using \(\sin x \approx x\) to approximate \(\sin 1^{\circ}\).

Use the 64 -bit long real format to find the decimal equivalent of the following floating-point machine numbers. a. \(0 \quad 10000001010 \quad 1001001100000000000000000000000000000000000000000000\) b. \(1 \quad 10000001010 \quad 1001001100000000000000000000000000000000000000000000\) c. \(0 \quad 01111111111 \quad 0101001100000000000000000000000000000000000000000000\) d. \(0 \quad 01111111111 \quad 0101001100000000000000000000000000000000000000000001\)

Use a Taylor polynomial about \(\pi / 4\) to approximate \(\cos 42^{\circ}\) to an accuracy of \(10^{-6}\).

Find the \(n\)th Maclaurin polynomial \(P_{n}(x)\) for \(f(x)=\arctan x\).

a. Show that the polynomial nesting technique described in Example 6 can also be applied to the evaluation of $$ f(x)=1.01 e^{4 x}-4.62 e^{3 x}-3.11 e^{2 x}+12.2 e^{x}-1.99 $$ b. Use three-digit rounding arithmetic, the assumption that \(e^{1.53}=4.62\), and the fact that \(e^{n x}=\left(e^{x}\right)^{n}\) to evaluate \(f(1.53)\) as given in part (a). c. Redo the calculation in part (b) by first nesting the calculations. d. Compare the approximations in parts (b) and (c) to the true three-digit result \(f(1.53)=-7.61\).

A rectangular parallelepiped has sides of length \(3 \mathrm{~cm}, 4 \mathrm{~cm}\), and \(5 \mathrm{~cm}\), measured to the nearest centimeter. What are the best upper and lower bounds for the volume of this parallelepiped? What are the best upper and lower bounds for the surface area?

Prove the Generalized Rolle's Theorem, Theorem \(1.10\), by verifying the following. a. Use Rolle's Theorem to show that \(f^{\prime}\left(z_{i}\right)=0\) for \(n-1\) numbers in \([a, b]\) with \(a

Suppose that \(f l(y)\) is a \(k\)-digit rounding approximation to \(y\). Show that $$ \left|\frac{y-f l(y)}{y}\right| \leq 0.5 \times 10^{-k+1} $$

The error function defined by $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ gives the probability that any one of a series of trials will lie within \(x\) units of the mean, assuming that the trials have a normal distribution with mean 0 and standard deviation \(\sqrt{2} / 2 .\) This integral cannot be evaluated in terms of elementary functions, so an approximating technique must be used. a. Integrate the Maclaurin series for \(e^{-x^{2}}\) to show that $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k+1}}{(2 k+1) k !}$$ b. The error function can also be expressed in the form $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} e^{-x^{2}} \sum_{k=0}^{\infty} \frac{2^{k} x^{2 k+1}}{1 \cdot 3 \cdot 5 \cdots(2 k+1)}$$ Verify that the two series agree for \(k=1,2,3\), and 4. [Hint: Use the Maclaurin series for \(e^{-x^{2}}\).] c. Use the series in part (a) to approximate erf(1) to within \(10^{-7}\). d. Use the same number of terms as in part (c) to approximate erf(1) with the series in part (b). e. Explain why difficulties occur using the series in part (b) to approximate \(\operatorname{erf}(x)\).

A function \(f:[a, b] \rightarrow \mathbb{R}\) is said to satisfy a Lipschitz condition with Lipschitz constant \(L\) on \([a, b]\) if, for every \(x, y \in[a, b]\), we have \(|f(x)-f(y)| \leq L|x-y|\). a. Show that if \(f\) satisfies a Lipschitz condition with Lipschitz constant \(L\) on an interval \([a, b]\), then \(f \in C[a, b]\) b. Show that if \(f\) has a derivative that is bounded on \([a, b]\) by \(L\), then \(f\) satisfies a Lipschitz condition with Lipschitz constant \(L\) on \([a, b]\). c. Give an example of a function that is continuous on a closed interval but does not satisfy a Lipschitz condition on the interval.