# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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### A piecewise function inserted into an integral

Let $f : \mathbb{R}^{+} \times \mathbb{R}^{+} \to \mathbb{R}$ be a nice function, and let's say we define a function $g: \mathbb{R} \times \mathbb{R}^{+} \to \mathbb{R}$ as: $g(x,t) := f(x,t)$ for ...
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### Calculation of $\frac{1}{\sqrt{2\pi}\sigma(x)}\int_{-\infty}^{\infty}|u|\exp\left(-\frac{u^2}{2\sigma{^2}(x)}\right)\ du$

How to show that $$\dfrac{1}{\sqrt{2\pi}\sigma(x)}\int_{-\infty}^{\infty}|u|\operatorname{exp}\left(-\dfrac{u^2}{2\sigma{^2}(x)}\right)\mathop{du}=\sqrt{\dfrac{2}{\pi}}\sigma(x)$$ I think it has to ...
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### Points on a normal to a superellipse at distance $d$ from the curve

Given a point P0 $(x0, y0)$ lying on a super ellipse, $(x/a)^n + (y/b)^n = 1$, where $2 <= n <= 5$, I'm trying to derive an equation to describe the point P1 (x1, y1) lying on the normal through ...
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### Confusing use of the chain rule

Let $f$ be a function of three real variables, $(x, y, z)$ where $z$ is a function of $(x,y)$. By the chain rule on $f(x,y,z(x,y))$: ...
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### Evaluate $\int_{0}^{1} \int_{3y}^{3} e^{x^2} \, dx \, dy$

I have to evaluate the integral $$\int_{0}^{1} \int_{3y}^{3} e^{x^2} \, dx \, dy.$$ Can you give me a hint because I can't figure it out how to integrate $\int e^{x^2} dx$.
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### What's exactly the deal with differentials? (Confessions of a desperate calculus student)

So I don't know if I'm the only one to feel this, but ever since I was introduced to Calculus, I've had a slight (if not to say major) aversion to differentials. This sort of "phobia" started from ...
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### Linear equation and exponential

I have an equation I need to figure out. I apologize for my lack of knowledge and improper use of terms. My math skills are quite rusty. Using the equation $A=P\left(1+\frac i{100}\right)^n$, a) If ...
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### Specific question about investment returns that needs someone smart to answer it! [on hold]

OK, so I'm trying to work out the answer to a very specific question and for me it's complex maths. I'm sure for you mathematicians out there it's fairly everyday. If anyone can help I'd be very ...
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### The value of the limit $\ln(1+2^x)\ln(1+3/x)$ [on hold]

How to find the value of the limit $$\lim _{x\rightarrow +\infty}\left (\ln (1+2^x)\ln (1+\frac{3}{x})\right )$$
Assume that f is a function with $|f^{(n)}(x)| \le 11,$ for all n and all real x. Let $T_n(x)$ denote the Taylor polynomial of degree n for f(x) about the point $x=0$. What is the least integer n ...