# Questions tagged [calculus]

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

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### Not using Stokes theorem for line integral

I have a question for the integral of $f(x,y,z)=(y-1)dx+z^2dy+ydz$ over the curve of intersection between the surface $x^2+y^2=z^2/2$ and the plane $z=y+1$. I know that I can do this using Stokes ...
4answers
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### Range of $f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$

I am trying to find the range of this function: $$f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$$ So I think that means I have to find minima and maxima. Using partial derivatives gets messy, so I was ...
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### Proving discontinuity of a function defined by a double integral

Show that $$f(\alpha)=\int_{\alpha}^1 dx \int_{\alpha}^1\dfrac{(x-y)}{(x+y)^3} dy$$ is not continuous at $\alpha=0$. I have found out the iterated integral at $\alpha=0$ which is $1/2$. But after ...
3answers
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### Calculus for the Practical Man: Chapter 4, Problem 7

A man standing on a wharf is hauling in a rope attached to a boat at the rate of four feet a second. If his hands are nine feet above the point of attachment, how fast is the boast approaching the ...
0answers
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### How do we calculate $\int_{i}^{f} (L_1 - L_2) dt$ where $L_1 = f(t)$ and $L_2 = f(t+dt)$? Thanks. [on hold]

I have two functions f(u) and g(u). I have to use these two functions to calculate a function $h(u) = \int_{i}^{f} (L_1 - L_2) du$ where $L_1 = func(f(u),g(u))$ and $L_2 = func(f(u+du),g(u+du))$.
1answer
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### Proving limit of bounded convergent sequence is bounded

Question: Show that if $a \leq x_n \leq b$ for every $n$ and $x_n \rightarrow x$, then $a \leq x \leq b$. Proof: Let $\epsilon>0$. By assumption $a_n \leq x_n \leq b$ for all $n$. By definition ...
0answers
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### Solve: $2y'' = e^y$ [on hold]

I wish to find the solution of the differential equation: $$2 y^{\prime \prime}=\mathrm{e}^{y}$$
4answers
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### Strange! Right solution of equation is partially wrong.

Let $0<a<1$ be arbitrary but fixed. The equation $$\frac{x^2 (1+y)^2}{\left(a y + \sqrt{1+(1-a^2)y}\right)^2} = 1$$ in $y$ has according to straight-forward calculus and Mathematica two ...
2answers
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### find total surface area of solid using formula for surface area using integration

find the total surface area of solid which is formed when the region enclosed by $x^2+y^2=4$ in the first quadrant is rotated about $y=-1$. Edit: I'm getting that on rotating this solid I'm getting ...
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### What, if anything, is fundamentally special about plugging certain functions into certain ODEs?

While studying differential equations, I have noticed that several solution techniques are nearly identical. Reduction of order, variation of parameters (first-order version), and transforming ...
2answers
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### Is there a basis which spans the real numbers?

Is there a finite set of real numbers $S=\{a_1, a_2, ..., a_n \}$ such that every real number can be written as a linear combination (with integer coefficients) of the elements of $S$? If no, is there ...
1answer
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### If $f: R \to R$ is one one and differentiable function and graph of $y = f(x)$ is symmetrical about the point $(4,0)$ then…

If $f: R \to R$ is one one and differentiable function and graph of $y = f(x)$ is symmetrical about the point $(4,0)$ then, If $f'(-100) >0$ then roots of $x^{2} - f'(10)x - f'(10) = 0$ may be non ...
5answers
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### Proving the Continuity of $e^x$

Can one say that $e^x$ is the sum of an infinite number of terms (Taylor expansion), every term being a continuous polynomial in itself, the sum of all the terms is continuous and so $e^x$ is ...
2answers
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### find largest $x$ based on $f(x,y)=c$

Consider the following equation: $$\frac{p}{\left(a-x\right)^2+y^2}+\frac{1-p}{\left(b-x\right)^2+y^2}=1$$ where $0<p<1$ and $a,b\in\mathbb{R}$. I want to look for the largest $x$ satisfying ...
1answer
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1answer
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