# Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.

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### Double integral with unknown exponents

The problem is: Compute $I = \int \int _G \dfrac{1}{(1+3x^2+6xy+5y^2)^\alpha} dx dy$ for $\alpha > 2$ and $G =\{ (x,y): -y \leq x \leq y \}$. I have already tried plotting some examples to ...
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### Newton Method in a proof question

I just find out the function of the tangent line and the relationship between $x_n$ and $x_{n+1}$ ,but how can I prove questions $(1),(3)$ and $(4)$?
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### The continuity of a two-variable function and its derivative

In this question, There's a corresponding value of f at (0,1), so we can say that f has a limit at (0,1) does it correct? f is not continuous at (0,1) so we can't do derivative,so (B) & (C) are ...
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### Is there a nice expansion for $\int e^{\cos(x)}dx$? [duplicate]

This indefinite integral has no closed form $$I(x)=\int_0^xe^{\cos(t)}dt$$ The usual expansions of $I (x)$ are generally obtained by expanding $e ^ x$ and integrating $\int \cos ^ \alpha (x) dx$,...
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### Show that a function is radial iff it is rotation invariant

Let $d\in\mathbb N$, $\Omega\subseteq\mathbb R^d$ and $f:\Omega\to\mathbb R$. Remember that $f$ is called radial if $f(x)=f(y)$ for all $x,y\in\Omega$ with $\|x\|=\|y\|$. Let $O(d)$ denote the ...
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### Minimum and Maximum of $f(x,y)=\sin(y-x)-\sin(y)+\sin(x)$

I need to identify the minimum and maximum of $f(x,y)=\sin(y-x)-\sin(y)+\sin(x)$ with $0 \le x \le y \le 2\pi$. First I calculated $$\nabla f(x,y)=(\cos(x)-\cos(y-x),\cos(y-x)-\cos(y)).$$ The ...
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### Two different definitions of Gateaux differentiability

My textbook defines Gateaux derivative as the following: (Defnition) A mapping $F: D\subset \mathbb{R^n} \to \mathbb{R^m}$ is Gateaux-differentiable at an interior point $x$ of of D if there eixsts ...
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### Proof of Fermat's Theorem for a Local Minimum

Fermat's Theorem for Local Extrema states that if a function $f(x)$ has a local extremum at $c$ and $f'(c)$ exists, then $f'(c)=0$. I saw a textbook proof for the local maximum case that used the ...
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### Effect of Rectifying Function on Fourier Transform

Let there be some rectifying function, e.g. $$r(x) = \max (x,0).$$ What would the effect of this filter be on the Fourier transform of some generic function $f(x)$, in particular its Laplacian? I have ...
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### Can the product of two functions with no inflection points have an inflection point?

If $f(x)$ and $g(x)$ are functions with no inflection points, can $h(x)=f(x)\cdot g(x)$ have an inflection point? Edit: I experimented a bit with a few functions (like $x^2\cdot x^2$) in a graphing ...
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### Evaluate $f'(1)$ and $f'(6)$ if $f'(x)=5x^\frac45 -6x^\frac67$.

Suppose that $f(x)=5x^\frac45 -6x^\frac67$ . Evaluate each of the following: $f'(1)$, $f'(6).$ Should I plug in the values, use the product rule and then differentiate?... Or should I differentiate ...
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### Completeness of a dual space

I am working on a problem and i have pretty much a solution to every part ( total of three parts ). I would appreciate it, if you look at the solutions and let me know if they are acceptable. If not, ...
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### If $6=x\sin\alpha+y\sin55^\circ$ and $x\cos\alpha=y\cos55^\circ$, find $\alpha$ for which $x$ and $y$ are as small as possible

Given equations, $$6 = x\,\sin\alpha + y\,\sin55^\circ \tag{1}$$ and $$x\,\cos\alpha = y\,\cos55^\circ \tag{2}$$ Find the value of $\alpha$ for which $x$ and $y$ are as small as possible. If the ...
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### Integration with respect to a discontinuous function

I know that a function is differentiable iff it follows the first theorem of differentiability i.e. $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ so if a function is discontinuous at some points it is non ...
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### Let $f(x) = (x+1)\ln(x+1)−x\ln x−\ln(2x+1)$ for $x > 0$. Show that f is strictly increasing on (0, ∞). [closed]

Let $f(x) = (x+1)\ln(x+1)−x\ln x−\ln(2x+1)$ for $x > 0$. Show that f is strictly increasing on (0, ∞). Further, show that the sequence $\frac{ (n+1)^{n+1}} {(n^n) (2n+1)}$ is strictly ...
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### If $u=x\cos y$ and $v=x\sin y$, then near $(x_0,y_0)$, with $x_0\neq 0$, then $(x,y)$ can be written as a differentiable function of $(u,v)$

Consider the following equations $$u = x \cos(y)\ \ \text{and}\ \ v=x \sin(y)$$ Show that near to $(x_0,y_0)$, with $x_0\neq 0$, $(x,y)$ can be written as a differentiable function of $(u,v)$. It's ...
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### The reduction formula doesn't work for $\frac{d}{dx}\int_{2x-\frac{\pi}{2}}^{\frac{\pi}{2}+3x} \sin^{2021}{t}\,dt$

$$\frac{d}{dx}\int_{2x-\frac{\pi}{2}}^{\frac{\pi}{2}+3x} \sin^{2021}{t}\,dt$$ I used the reduction formula for this problem, but I couldn't get the result. Do you know any other ways I can use to ...
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### If $f \in \mathcal{C}([a,b])$ exactly one local minimum $x^*$, then $a \le x_1 < x_2 \le x^* \implies f(x_1) > f(x_2)$

Show that if a continuous function $f: [a,b] \rightarrow \mathbb{R}$ has exactly one local minimum, then $$a \le x_1 < x_2 \le x^* \implies f(x_1) > f(x_2)$$ and x^* \le x_1 < x_2 \le b \...
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### Stuck on a polar coordinates integral calculus question

Once every century the Klingon moon Bosok completely eclipses the moon Yan-ki. As seen from the city Mitek, the two moons appear as circular discs, with radii respectively $\sqrt{2},1$ (unit: dorks). ...