# Questions tagged [calculus]

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

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### Theorem 5.13 in “Principles of Mathematical Analysis” by Walter Rudin L'Hospital's Rule L'Hopital's Rule

I am reading "Principles of Mathematical Analysis" by Walter Rudin. Thank you Saaqib Mahmood. I copied and pasted your text Theorem 5.13 on p.109: Suppose $f$ and $g$ are real and ...
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### Determining continuity in multivariable calculus

If f is a function defined throughout a disk centred at $(x_0, y_0)$, and if $f_x(x_0, y_0)$ and $f_y(x_0, y_0)$ both exist, then f is continuous at $(x_0,y_0)$. I know to be a function to be ...
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### Why doesn't the graph of $x^2-\cos x$ look wiggly?

When I use Desmos to draw the graph of the function $f$ defined by $f(x):=x^2-\cos x$, the graph looks very similar to a quadratic function. Unlike the graph of, say, $x-\cos x$, it does not have any ...
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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 ...
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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 $L_2 = func(f(u+du),g(u+du))$.
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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 ...
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### Related rates problem: How rapidly is the area of an equilateral triangle increasing at the instant when each side is $69.28$ inches?

A metal plate in the shape of an equilateral triangle is being heated in such a way that each of the sides is increasing at the rate of ten inches per hour. How rapidly is the area increasing at the ...
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### Solve: $2y'' = e^y$ [on hold]

I wish to find the solution of the differential equation: $$2 y^{\prime \prime}=\mathrm{e}^{y}$$
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### Find inflection points of $e^{\cos x}$

I got a function $$f(x) = e^{\cos x}$$ I would like to find inflection points of the function above within range $[-2\pi, 2\pi]$ Find first derivative $$f'(x) = e^{\cos x}(-\sin x)$$ Find second ...
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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 ...
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### “every regular functions is differentiable infinitely” [on hold]

Is there a theorem that states something like "every regular functions is differentiable infinitely"? I want to prove that $$\frac{1}{1+x^2}$$ (or similar "simple" functions) has a second derivative ...
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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 ...
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### Values of θ for which r is minimized in a circle?

So, I've been stuck trying to figure out how to find the value(s) of θ where r is minimized in a circle? The circle's equation is r = 6 sin θ with an interval of 0≤θ≤2π. I believe the answer to be ...
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### Which of given condition follows from mean value theorem?

Let $f:R \to R$ be differentiable which of the following follows from mean value theorem : (a) For all $a,b \in R$ if $c \in (a,b)$, then $\dfrac{f(b) - f(a)}{b-a} = f'(c)$ (b) For some $a,b \in R$ ...
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