# subnetting exercises and solutions

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Calculation(1). $\dps{\int_a^b \frac{\rd x}{\sqrt{(x-a) (b-x)}}\ (b>a)}$.(2). $\dps{\int_{-1}^1 \frac{\rd x}{(a-x) \sqrt{1-x^2}}\ (a>1)}.$Answer:(1). $$\beex \bea \mbox{original points}=\int_{-\frac{b-a}{2}}^\frac{b-a}{2} \frac{\rd t}{\sqrt{\sex{t+\frac{b-a}{2}}\sex{ \frac{b-a}{2}-t}}}\\ =\int_{-\frac{b-a}{2}}^{\frac{b-a}{2}} \frac{\rd t}{\sqrt{\sex{\frac{b-a}{2}}^2-t^2}}\\ =\INT_{-\FRAC{\PI}{2}}^\FRAC{\PI}{2} \frac{1}{\frac{b-a}{2}\cos \tt}\cdot \frac{b-a}{2}\cos \tt\rd \tt\\ = \pi. \eea \eeex ### [Typical problems and methods in mathematical analysis of Periven exercises reference Solutions]4.5.11 Set f (x)  is a nonnegative continuous function on the 0\leq x(1). There is a bounded derivative f ' (x)  on the 0\leq x(2). \dps{\int_0^\infty f (x) \rd xProof: Set |f ' |\leq m, then by the Lagrange mean value theorem,$$\bex |f (x)-F (Y) |=|f ' (\xi) |\cdot |x-y|\leq m|x-y|. \eex$$and f  Lipschitz continuous, consistent continuous. The conclusion is established by an example of 4.5.24.[Typical problems and methods in mathematical analysis of Periven ### [Typical problems and methods in mathematical analysis of Periven exercises reference Solutions]4.5.3 \dps{\int_0^\infty f (x^p+x^{-p}) \frac{\ln x}{1+x^2}\rd x} (function f (x)  continuous) Answer:$$\beex \bea \mbox{original points} =\int_0^1+\int_1^\infty f (x^p+x^{-p}) \frac{\ln x}{1+x^2}\rd x\\ =\int_\infty^1f ( t^{-p}+t^p) \frac{-\ln T}{1+\frac{1}{t^2}}\cdot \sex{-\frac{1}{t^2}}\rd t +\int_1^\infty f (x^p+x^{-p}) \frac{\ln x}{1+ X^2}\rd x\quad\sex{x=\frac{1}{t}}\\ =0. \eea \eeex$$[Typical problems and methods in mathematical analysis of Periven exe ### Learning reinforcement Learning (with Code, exercises and Solutions) __reinforcement algorithms using Python,openai Gymand. I separated them into chapters (with brief summaries) and exercises, and solutions so, can use them to supplement T He theoretical material above.all of the ' is ' in the Github repository. Some of the more time-intensive algorithms are still work and progress. I ' ll update this post as I implement them. Table of Contents Introduction to RL problems, OpenAI gym MDPs ### [Typical problems and methods in mathematical analysis of Periven exercises reference Solutions]4.3.20 Set a >0, function f (x)  on [0,a] continuous micro, Proof:$$\bex |f (0) |\leq \frac{1}{a}\int_0^a |f (x) |\rd x+\int_0^a |f ' (x) |\rd x. \eex$$Huazhong Normal UniversitySolution: by$$\beex \bea \int_0^a f (x) \rd x=\int_0^a \sez{f (0) +\int_0^x F ' (t) \rd t}\rd x\\ =af (0) +\int_0^a \int_t^a F ' (t) \rd x\rd t\\ =af (0) +\int_0^a F ' (t) (a-t) \rd t \eea \eeex$$known$$\beex \bea |f (0) |=\sev{\frac{1}{a}\int_0^a f (x) \rd x-\int_0^a F ' (t) \sex{1-\frac{t}{a}}\rd t}\\ \leq \frac{1}

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### [Typical problems and methods in mathematical analysis of Periven exercises reference Solutions]5.1.1

Set $k, i,j$ are natural numbers, and $k =i+j$, try to find the series $\dps{\vsm{n}\frac{1}{(Kn-i) (kn+j)}}$.Solution: Former $N$ for the original series and for $$\beex \bea \sum_{n=1}^n \frac{1}{(kn-i) (kn+j)} =\frac{1}{k}\sum_{n=1}^n \sex{\frac{1}{kn-i}-\ FRAC{1}{KN+J}} =\frac{1}{k}\sez{\sum_{n=1}^n \frac{1}{kn-i}-\sum_{n=1}^n \frac{1}{k (n+1)-i}}\\ =\frac{1}{k}\ Sez{\frac{1}{k-i}-\frac{1}{k (n+1)-i}}. \eea \eeex$$ so the series and for $\dps{\frac{1}{k (k-i)}}$.[Typical problems and method

### [Typical problems and methods in mathematical analysis of Periven exercises reference Solutions]4.5.7

{2k+\frac{1}{4}},\quad b_k=f^{-1}\sex{2k+\frac{3}{4}, \eex$$have$$ \bex A_k\leq X\leq B_k\ra 2k+\frac{1}{4}\leq f (x) =x+\frac{1}{x}\leq 2k+\frac{3}{4} \ra \sin^2\sez{\pi\sex{x+\frac{1}{x }}}\geq \frac{1}{2}, \eex\beex \bea \int_{a_k}^{b_k}\sin^2\sez{\pi\sex{x+\frac{1}{x}}}\rd x \geq \frac{1}{2} (B_k-a_k) \ =\frac{1}{2}\sez{f^{-1} (Z_k)-f^{-1} (y_k)}\quad\sex{z_k=2k+\frac{3}{4},\ y_k=2k+\frac{1}{4}}\\ =\frac{1}{4}\sez{z_k-y_k+\sqrt{z^2-4}-\sqrt{y_k^2-4}}\\ \geq \frac{1}{4} (z_k-y_k) \ = \

### Learning reinforcement Learning (with Code, exercises and Solutions) __reinforcement

algorithms using Python,openai Gymand. I separated them into chapters (with brief summaries) and exercises, and solutions so, can use them to supplement T He theoretical material above.all of the ' is ' in the Github repository. Some of the more time-intensive algorithms are still work and progress. I ' ll update this post as I implement them. Table of Contents Introduction to RL problems, OpenAI gym MDPs

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