Articles of calculus

About the relation between $\int_{0}^{\infty}\frac{\cosh((\alpha-\beta)x)}{(\cosh(x))^{\alpha+\beta }} \,dx$ and $B(\alpha,\beta)$

How do I get from $$B(\alpha,\beta)=\int_{0}^{1}{ t^{ \alpha -1} (1-t)^{ \beta -1} \,dt}$$ to: $$2^{2- \alpha – \beta } \int_{0}^{ \infty}{ \frac{ \cosh (( \alpha – \beta )x)}{( \cosh(x))^{ \alpha + \beta }} \,dx}$$ ? I have tried different change of variables such as $t=1- \frac{1}{ \cosh (x)}$ and got $\int_{0}^{ \infty}{ \frac{ \sinh (x) […]

How to calculate complicated geometrical series?

I have a geometrical series (I don’t know if its geometrical series or not): $$ \sum_{n=1}^{\infty }n\rho ^{n}(1-\rho) $$ how can I simplify it ? ( assume that $ 0 \le \rho \le 1$ ) The last answer in my calculatio should be $\frac{\rho}{1-\rho}$. But I really don’t know how ?

Newton's method problem

I know how this works, I know what it does and what the goal of it is but I don’t understand what this homework problem wants me to do and why I don’t get the correct answer. I am supposed to used newton’s method with the specified initial approximation x1 to find x3, the third […]

Is distribution theory necessary for most users of the Dirac delta function?

What would be wrong with defining the Dirac delta function as $$ \int_{-\infty}^{\infty} \delta^{(n)}(f(x)) g(x) \,dx := \lim_{h\rightarrow 0}\int_{-\infty}^{\infty} \delta_h^{(n)}(f(x)) g(x) \,dx $$ for a suitable nascent delta function $\delta_h(x)$? For example, the rectangular pulse, hat function, and normal distribution nascent delta functions are all suitable in the case $n=0.$ For $n\leq1,$ the hat function […]

Ei Approximation

I’m working with the function $F(x)=e^{-k(x+1)}\int_1^x\frac{N^2}{t(N-t)}e^{kt}dt$. Breaking it down into into single fractions helps a little, yielding: $F(x)=Ne^{-k(x+1)} \int_1^x [\frac{1}{t} + \frac{1}{N-t}]e^{kt}dt$. If you toss that into Wolfram Alpha (without the limits), you’ll get the antiderivative as $Ne^{-k(x+1)}[Ei(kt)-e^{-kN}Ei(k(t-N))]_{1}^x$. I’d like to approximate this value, or at least bound its value. So far, I have that […]

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] – 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $$$$ I really have no clue as to how to go about this as I’ve never encountered such questions before. I would be truly grateful if somebody would please show me […]

Question about limit with exponents

I tried to solve: $$\lim\limits_{x \to \infty} e^{x-x^2}$$ but I can’t get it done. I’ve tried to use Hopital’s rule after rewriting it to: $$\lim\limits_{x \to \infty} \frac{e^x}{e^{x^2}}$$ but this does not lead to a solition. Anyone with a good hint?

Limit question as $x$ and $y$ approach infinity?

I have to prove that the limit of the function $\frac{x^2}{x^2+y^2}$ as $x$ approaches infinity and as y approaches infinity does not exist. I thought about finding the side limits, and if they are not equal, bam! I have solved it. But what should I take as side limits here? $+$ and $-$ infinity? Thank […]

why is it possible to factor a limit?

I am confused by the examples in calculus textbook where they factor a polynomial to find the limit. I don’t understand how the limits of $\frac{1}{x-3}$ and $\frac{x+3}{x^2-9} $ are the same. The only thing the book I read did was factor the polynomial, but it doesn’t explain why that is possible. I have thought […]

Help deriving an function from (Lagrangian?) properties

I’m trying to derive a function f(x) that has the following properties: $$f_x-\frac{f}{x}=g_x$$ (You might call this the Lagrangian?) where $$\begin{align*} f&=f(x)\\ f_x&=\frac{df}{dx}\\ g_x&=\frac{d}{dx}\left(\frac{f}{x}\right) \end{align*}$$ By rearranging I’ve gotten to the point where: $$\frac{f}{f_x}=\frac{x^2-x}{x+1}.$$ But can’t figure out what function satisfies this. Thanks