Articles of calculus

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

Any example of a function which is discontinuous at each point in a deleted neighborhood of a point at which that function is differentiable?

Let $f$ be a real (or complex) function defined on the segment $(a, b)$ of the real line, and let $p \in (a, b)$. If $f$ is differentiable at $p$, then of course $f$ is continuous at $p$ as well. (The converse may not hold.) But can we have an example where $f$ is differentiable […]

Suppose that $f(x)= xg(x)$ for some function $g$ which is continuous at $0$. Prove that $f$ is differentiable at 0 , and find $f'(0)$ in terms of $g$.

Suppose that $f(x)= xg(x)$ for some function $g$ which is continuous at $0$. Prove that $f$ is differentiable at 0 , and find $f'(0)$ in terms of $g$. Okay, so far I have: $$f(x) = x g(x) \quad \Longrightarrow \quad g(x) = \frac{f(x)}{x}$$ So, $f(x) = \frac{x \cdot f(x)}{x}$. Since $g$ is continuous at $0$, […]

By using d'Alembert's formula, substitute $P(z)$ and $Q(z)$ into the general solution to obtain an expression for $u(x, t)$

We will define a motion that satisfies the equation: $$u_{tt} = c^2u_{xx}\qquad x ∈ (0, 1),\: t > 0$$We have the displacement of a string being $u(x, t)$, at position $x$ and time $t$, which is stretched between two fixed points at $x = 0$ and $x = 1$. where $c$ is a real positive […]