Chapter 10

Estimate the values of the following integrals within \(.05 .\) (a) \(\int_{0}^{1} e^{-x^{2}} d x\) (b) \(\int_{0}^{1} \frac{\sin x}{x} d x\) (c) \(\int_{1}^{2} \frac{d x}{1+\log x}\)

Let \(0 \leq A \leq 1\) and define a sequence \(\left\\{x_{n}\right\\}\) by $$ x_{0}=0 \quad x_{n+1}=x_{n}+\frac{1}{2}\left(A-\left(x_{n}\right)^{2}\right) $$ (a) Prove lim \(x_{n}=\sqrt{A}\). (b) Estimate the rate of convergence.

Use the method suggested in Formula \((10-28)\) to find the minimum value of \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(3 x+5 y=8\)

Use Simpson's rule to estimate the value of \(\int_{0} d x /(1+x)\), using the values of \(f\) at the points 0 , \(.25, .50, .75,1.0\)

Find the minimum value of \(F(x, y)=3\left(x+2 y-\frac{3}{2}\right)^{2}+4 x^{3}+12 y^{2}\) on the rectangle \(|x| \leq 1\), \(0 \leq y \leq 1\)

(a) Determine the number of real roots of the equation $$ \sin x+\frac{1}{6} x=1 $$ (b) Find each root, accurate to 0001 .

Show that $$ \iint_{D} \sqrt{4 x^{2}-y^{2}} d x d y<\frac{\sqrt{15}}{6} $$ where \(D\) is the triangle with vertices at \((0,0),(1,0),(1,1)\).

Find the maximum and minimum values of \(f(x, y, z)=x y+z\) subject to the constraints \(x \geq 0, y \geq 0, x z+y=4\), and \(y z+x=5\)

Let \(A>0\) and consider the following two sequences: $$ \begin{array}{ll} x_{0}=A & x_{n+1}=\frac{1}{2}\left[x_{n}+A /\left(x_{n}\right)^{2}\right] \\ y_{0}=A & y_{n+1}=\frac{1}{3}\left[2 y_{n}+A /\left(y_{n}\right)^{2}\right] \end{array} $$ (a) Prove that \(\lim x_{n}=\lim y_{n}=\sqrt[3]{A}\). (b) Which of these is a better algorithm for calculating \(\sqrt[3]{A} ?\) Explain why.

Show that \(\int_{0}^{\pi} e^{-x} \sqrt{\sin x} d x<1\)

Solve the system $$ \left\\{\begin{array}{l} \sin (y)=x \\ \cos (x)=y \end{array}\right. $$ by fixed point methods.

Suppose that \(f(0)=1, f(1)=4, f(2)=4, f(3)=3\), and that \(f^{\prime \prime}(x) \leq 0\) for \(0 \leq x \leq 3 .\) What is the best estimate you can give for \(\int_{0}^{3} f ?\)

What is the maximum value of \(x-2 y+2 z\) among the points \((x, y, z)\) with \(x^{2}+y^{2}+z^{2}=9\) ?

Define a sequence of functions \(\left\\{f_{n}\right\\}\) by \(f_{0}(x)=\) any function of class \(C^{x}\) on \(-\infty

Estimate the value of $$ \iiint_{D} \frac{d x d y d z}{5+x+y+z} $$ where \(D\) is the unit cube with opposite vertices at \((0,0,0)\) and \((1,1,1)\), using a decomposition of \(D\) into 8 subcubes and the trapezoidal method.

Find the minimum of \(x y+y z\) for points \((x, y, z)\) which obey the relations \(x^{2}+y^{2}=2, y z=2\).

Sketch the set of points \((A, B)\) for which the equation \(x^{5}-5 x^{3}+A x+B=0\) has \((a)\) exactly one real root; \((b)\) exactly three real roots; \((c)\) exactly five real roots.

Estimate \(\int_{0}^{2} \sin (1 / x) d x\) to within .01.

What is the volume of the largest rectangular box with sides parallel to the coordinate planes which can be inscribed in the ellipsoid \((x / a)^{2}+(y / b)^{2}+(z / c)^{2}=1 ?\)

Use the method of Gauss \((10-41)\) to verify the following computation: $$ \int_{0}^{\pi / 2} \frac{d \theta}{\sqrt{5-\sin ^{2} \theta}}=.74220624 $$

Given that \(f(.5)=2.0, f(.6)=2.3, f(.8)=2.7\), and \(\left|f^{\prime \prime \prime}(x)\right| \leq 4\), estimate the value of \(f^{\prime}(.6)\) and \(f^{\prime \prime}(.6)\) with error terms.

Obtain an approximate formula for \(f^{\prime}(a)\) and \(f^{\prime \prime}(a)\), making use of the five values \(f(a+k h)\) for \(k=0, \pm 1, \pm 2\), and estimate its accuracy in terms of the maximum value of \(\left|f^{(5)}(x)\right|\) or \(\left|f^{(6)}(x)\right|\) on the interval \([a-2 h, a+2 h]\).