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Math textbook answers & solutions

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Most popular questions for Math Textbooks

Find the area enclosed by the curve \(x y^{2}=a^{2}(a-x)\) and \(\mathrm{y}\)-axis.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function that is continuous on \((a, b)\). If the range of \(f\) is contained in \((\alpha, \beta)\) and \(\phi:[\alpha, \beta] \rightarrow \mathbb{R}\) is a convex function that is continuous at \(\alpha\) and \(\beta\), then show that $\operatorname{Av}(f) \in(\alpha, \beta)\(, the function \)\phi \circ f:[a, b] \rightarrow \mathbb{R}$ is integrable, and \(\phi(\mathrm{Av}(f)) \leq \mathrm{Av}(\phi \circ f)\).

Use the DFT and Corollary \(10.8\) to find the trigonometric interpolating function for the following data: \begin{tabular}{r|rrr|rrr|rrr|r} \(t\) & \(x\) & & \(t\) & \(x\) & & \(t\) & \(x\) & & \(t\) & \(x\) \\ \hline 0 & 0 & & 0 & 1 & & 0 & \(-1\) & & 0 & 1 \\ \(\frac{1}{4}\) & 1 & (b) & \(\frac{1}{4}\) & 1 & (c) & \(\frac{1}{4}\) & 1 & (d) & \(\frac{1}{4}\) & 1 \\ \(\frac{1}{2}\) & 0 & & \(\frac{1}{2}\) & \(-1\) & & \(\frac{1}{2}\) & \(-1\) & & \(\frac{1}{2}\) & 1 \\ \(\frac{3}{4}\) & \(-1\) & & \(\frac{3}{4}\) & \(-1\) & & \(\frac{3}{4}\) & 1 & & \(\frac{3}{4}\) & 1 \end{tabular}

Die Weinmenge, die von einer automatischen Abfüllanlage in eine \(0.75\)-l-Flasche abgefüllt wird, sei aus mancherlei Gründen als eine Zufallsvariable aufzufassen, deren Erwartungswert gleich \(0.72\) und deren Standardabweichung gleich \(0.01\) beträgt. 1\. Wie gro \(B\) ist die Wahrscheinlichkeit mindestens, dass in eine Flasche zwischen \(0.71\) und \(0.91\) abgefüllt werden? 2\. Wie groB ist höchstens die Wahrscheinlichkeit, dass in eine Flasche weniger als \(0.71\) abgefüllt werden, wenn die Verteilung der von der Abfullanlage abgegebenen Menge symmetrisch ist?

The function \(\int_{0}^{x} \frac{1}{\sqrt{2 \pi}} e^{-t^{2} / 2} d t\) is related to the distribution function of the standard normal random variable, a very important distribution in probability and statistics. Often times we want to solve equations like $$ \int_{0}^{x} \frac{1}{\sqrt{2 \pi}} e^{-t^{2} / 2} d t=z $$ for \(x,\) where \(z\) is some real number between 0 and \(1 .\) This can be done by using Newton's method to solve the equation \(f(x)=0\) where $$ f(x)=\int_{0}^{x} \frac{1}{\sqrt{2 \pi}} e^{-t^{2} / 2} d t-z $$ Note that from Fundamental Theorem of Calculus, \(f^{\prime}(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} .\) Newton's method will require the calculation of $$ \int_{0}^{p_{k}} \frac{1}{\sqrt{2 \pi}} e^{-t^{2} / 2} d t $$ where \(p_{k}\) is a Newton iterate. This integral can be computed using numerical quadrature. Write a Python code that takes \(z\) as its input, and outputs \(x,\) such that Equation (4.14) holds. In your code, use the Python codes for Newton's method and the composite Simpson's rule you were given in class. For Newton's method set tolerance to \(10^{-5}\) and \(p_{0}=0.5,\) and for composite Simpson's rule take \(n=10,\) when computing the integrals (4.15) that appear in Newton iteration. Then run your code for \(z=0.4\) and \(z=0.1,\) and report your output.
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