Questions tagged [calculus]

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

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Integrate the product of a heaviside step and the absolute value?

I have a rather tricky integral here: $$\underbrace{\int_0^R r_0\Theta(R-r_0)|r-r_0|dr_0}_{(1)} - \underbrace{\int_0^R r_0\Theta(R-r_0)|r+r_0|dr_0}_{(2)} \ \ \ \ \cases{0\le r < \infty \\ R=1}$$ ...
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Determining the significance of a curve's factors

Given the equation $x^2+x+1$ you could easily determine that $x^2$ will have the greatest overall impact on the curve--then $x$ and finally $1$. And this holds true for any coefficients present as the ...
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Why can the Binomial Distribution be Approximated by a Normal Distribtuion?

As a practice problem, I am trying to prove the relationship between the Normal Distribution and the Binomial Distribution. I have seen several proofs of this before (e.g. Justifying the Normal Approx ...
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Concavity where second derivative is undefined at a point but negative everywhere else

I want to say that $y=-|x|^{1.5}$ must be strictly concave since $y'$ is continuous and $y''=-.75/\sqrt{|x|}<0$ everywhere except at $x=0$ where it's undefined. (I'm working in real numbers only.) ...
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What is the sign of $I_n = \int _{0}^{1}\frac{x^{2n+1}}{x^{2}+1}dx$ [closed]

I was given an exercice to calculate $I_0$ and then $I_0 + I_1$ and then deduce $I_1$, and then asks the sign of $I_n$, can someone help? I tried deductive reasoning but I don't know how to complete ...
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Differential Equations riddle: $f=(f’)(f’’)(f’’’)(f’’’’)\dots$ [closed]

$$f=(f’)(f’’)(f’’’)(f’’’’)\dots$$ I found this question someone posted in a group chat, and no one has solved it yet
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Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

I am trying to reason why for any $2$ polynomials $P(x)$ and $Q(x)$ defined over the reals, $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$. This assertion was made in this answer which I am ...
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1 vote
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Stokes Theorem on standard simplices

I was studying the de Rham homomorphism, and more specifically Stokes theorem for chains, and I'm wondering whether one could avoid talking about manifolds with corners and just give a definition of ...
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Integral of Poisson Kernel

This doubt comes from Dupaigne's book named stable solutions of elliptic partial differential equations. The Poisson Kernel is P(x,y)=\frac{\partial G(x,y)}{\partial n_{y}}=\frac{1-|x|...
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Why do you need the arc length element in surface area integrals? [duplicate]

From ChatGPT: "When you revolve a curve $y=f(x)$ around the x-axis, you are generating a surface composed of infinitesimally thin strips that are themselves small frustums (truncated cones). Each ...
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The trailing cone of a guided long range torpedo is to be conical with slant edge s cm. [closed]

The trailing cone of a guided long range torpedo is to be conical with slant edge s cm. Radius is r and height is h. The cone is hollow and must contain the maximum possible value of fuel. Find the ...
1 vote
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Two similar formulas for ellipse circumference?

Does $$\int_{0}^{2\pi}\sqrt{a^2\cos^2x+b^2\sin^2x}dx= \int_{0}^{2\pi}\sqrt{a^2\sin^2x+b^2\cos^2x}dx$$ The right expression is the ellipse circumference.
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