91Ó°ÊÓ

The Trapezoidal rule applied to \(\int_{0}^{2} f(x) d x\) gives the value 4, and Simpson's rule gives the value 2 . What is \(f(1) ?\)

Use the following data and the knowledge that the first five derivatives of \(f\) are bounded on \([1,5]\) by \(2,3,6,12\) and 23 , respectively, to approximate \(f^{\prime}(3)\) as accurately as possible. Find a bound for the error. $$ \begin{array}{l|l|l|l|l|l} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 2.4142 & 2.6734 & 2.8974 & 3.0976 & 3.2804 \end{array} $$

Find the degree of precision of the quadrature formula $$ \int_{-1}^{1} f(x) d x=f\left(-\frac{\sqrt{3}}{3}\right)+f\left(\frac{\sqrt{3}}{3}\right) $$.

In Exercise 26 of Section 1.1, a Maclaurin series was integrated to approximate erf (1), where erf \((x)\) is the normal distribution error function defined by, $$ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t $$ Approximate erf(1) to within \(10^{-7}\).

Show that the error \(E(f)\) for Composite Simpson's rule can be approximated by $$ -\frac{h^{4}}{180}\left[f^{m \prime}(b)-f^{\prime \prime}(a)\right] $$ \(\left[\right.\) Hint: \(\sum_{j=1}^{n / 2} f^{(4)}\left(\xi_{j}\right)(2 h)\) is a Riemann Sum for \(\int_{a}^{b} f^{(4)}(x) d x\).].

In multivariable calculus and in statistics courses it is shown that $$ \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2 \pi}} e^{-(1 / 2)(x / \sigma)^{2}} d x=1 $$ for any positive \(\sigma\). The function $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(1 / 2)(x / \sigma)^{2}} $$ is the normal density function with mean \(\mu=0\) and standard deviation \(\sigma .\) The probability that a randomly chosen value described by this distribution lies in \([a, b]\) is given by \(\int_{a}^{b} f(x) d x\). Approximate to within \(10^{-5}\) the probability that a randomly chosen value described by this distribution will lie in a. \(\quad[-\sigma, \sigma]\). b. \(\quad[-2 \sigma, 2 \sigma]\) c. \([-3 \sigma, 3 \sigma]\)

Derive an \(O\left(h^{4}\right)\) five-point formula to approximate \(f^{\prime}\left(x_{0}\right)\) that uses \(f\left(x_{0}-h\right), f\left(x_{0}\right), f\left(x_{0}+h\right)\), \(f\left(x_{0}+2 h\right)\), and \(f\left(x_{0}+3 h\right) .\) [Hint: Consider the expression \(A f\left(x_{0}-h\right)+B f\left(x_{0}+h\right)+C f\left(x_{0}+\right.\) \(2 h)+D f\left(x_{0}+3 h\right)\). Expand in fourth Taylor polynomials, and choose \(A, B, C\), and \(D\) appropriately.]

Derive Simpson's rule with error term by using $$ \int_{x_{0}}^{x_{2}} f(x) d x=a_{0} f\left(x_{0}\right)+a_{1} f\left(x_{1}\right)+a_{2} f\left(x_{2}\right)+k f^{(4)}(\xi) $$ Find \(a_{0}, a_{1}\), and \(a_{2}\) from the fact that Simpson's rule is exact for \(f(x)=x^{n}\) when \(n=1,2\), and 3 . Then find \(k\) by applying the integration formula with \(f(x)=x^{4}\).

In Exercise 10 of Section \(3.4\) data were given describing a car traveling on a straight road. That problem asked to predict the position and speed of the car when \(t=10 \mathrm{~s}\). Use the following times and positions to predict the speed at each time listed. $$ \begin{array}{l|l|r|r|r|r|r} \text { Time } & 0 & 3 & 5 & 8 & 10 & 13 \\ \hline \text { Distance } & 0 & 225 & 383 & 623 & 742 & 993 \end{array} $$