Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

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

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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The Supremum and Bounded Functions

I'm trying to show that this is true: Let $X$ be a set and suppose $f$ and $g$ are bounded (real-valued) functions defined on $X$. Then, $$ \sup_{x \in X}|f(x)g(x)| \leq \sup_{x \in X}|f(x)|\sup_{x \in X}|g(x)| $$ I think I'm pretty close but I'm…
AFX
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Is there a difference between $y=\frac{\sqrt{1-x}}{\sqrt{1+x}}$ and $y=\sqrt{\frac{1-x}{1+x}}$

Is there a difference between $$y=\frac{\sqrt{1-x}}{\sqrt{1+x}}$$ and $$y=\sqrt{\frac{1-x}{1+x}}$$ If there is a difference, why when I give the square for both equations, they will be equal.
user189855
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Little o(h) limit about h=0

I understand that generally if a function $f(h)$ is described as $o(h)$ that $f(h)$ has a smaller rate of growth than $h$ (like it would have to be $\sqrt{h}$). i.e. $\sqrt{h} = o(h)$, just like (for example) $4h=o(h^2)$. The notes I'm reading…
Mauro Augusto
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Understanding the proof that $L_\infty$ norm is equal to $\max\{f(x_i)\}$

Attached is a proof I found. It is probably very basic, but I can not understand the marked thing. Why this term is zero? I hope someone can explain it for me. Edit(elaboration): A Norm is a function that takes a function $f$ and returns a number.…
Artium
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How to compute the limit $\lim_{n \to \infty} (1 + 2^n + 3^n +4^n+5^n)^{1/n}$?

How to compute the limit $\lim_{n \to \infty} (1 + 2^n + 3^n +4^n+5^n)^{1/n}$? My partial solution: $(1 + 2^n + 3^n +4^n+5^n)^{1/n} \leq (5 \times 5^n)^{1/n}$. Therefore $\lim_{n \to \infty} (1 + 2^n + 3^n +4^n+5^n)^{1/n} \leq \lim_{n \to \infty}…
LJR
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Find the sign of $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$

I'd love your help with finding the sign of the following integral: $$\int_{0}^{2 \pi}\frac{\sin x}{x} dx$$ I know that computing it is impossible. I tried to use integration by parts and maybe to learn about the sign of each part and conclude…
Jozef
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The limit of $\tan(n)$

I know that the $\lim_{x\rightarrow\infty}\tan(x)$ does not exist. Now my question: Is the sequence $\tan(n)$ convergent for $n\in\mathbb N$?
user108209
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Why is $e^{-x^2}$ such a big deal?

It seems like integrating it is a big deal and everything, but I don't understand why. My teacher is making it seem really important in my calculus class for seemingly no reason. Help would be appreciated.
louie mcconnell
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Convergence/Divergence of $\sum_{n=1}^{\infty} \sin(1/n)$

it is a question Convergence/Divergence of calculus II! Please give me a hand! Determine convergence or divergence using any method covered so far. $$\sum_{n = 1}^{\infty} \sin (1/n)$$
None
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Is a differentiable function always continuous?

Continuous Functions are not Always Differentiable. But can we safely say that if a function f(x) is differentiable within range $(a,b)$ then it is continuous in the interval $[a,b]$ . If so , what is the logic behind it ?
Neer
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Evaluation of $\int\frac{\ln(\cos x+\sqrt{\cos 2x})}{\sin^2 x}dx$

Evaluation of $\displaystyle \int\frac{\ln(\cos x+\sqrt{\cos 2x})}{\sin^2 x}dx$ My Try:: Let $\displaystyle I = \int \frac{\ln(\cos x+\sqrt{\cos 2x})}{\sin^2 x}dx = \int \ln(\cos x+\sqrt{\cos 2x})\cdot \csc^2 xdx$ So $\displaystyle I =…
juantheron
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Evaluation of $ \int \frac{x^2+n(n-1)}{(x\cdot \sin x+n\cdot \cos x)^2}dx$

Evaluation of $\displaystyle \int \frac{x^2+n(n-1)}{(x\cdot \sin x+n\cdot \cos x)^2}dx$ $\bf{My\; Solution:}$ Using $\displaystyle (x\cdot \sin x+n\cdot \cos x) = \sqrt{x^2+n^2}\left\{\frac{x}{\sqrt{x^2+n^2}}\cdot \sin…
juantheron
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Composing a smooth even function and square root

Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $f(-x)=f(x)$ for all x. Define $g:[0,\infty)\to\mathbb{R}$ by $g(x)=f(\sqrt{x})$. Is $g$ necessarily smooth at $0$? I guess the answer is positive. $f$ being an even function implies that all its…
Amitai Yuval
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Meaning behind differentials

So I think I understand what differentials are, but let me know if I'm wrong. So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in terms of the differentials $\text{dy}$ and…
Hola_Mundo
  • 71
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Maximum area of a triangle inside an ellipse.

The question says to find the maximum area of a triangle formed by joining the points $A,B$ and origin $O$, where $A$ and $B$ are points of intersection of an arbitrary line passing through $(4,5)$ with the ellipse…
user1001001
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