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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integrate $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}{\ln{(\ln{\tan{x}})}dx} $

Evaluate this integrate $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}{\ln{(\ln{\tan{x}})}dx} $$ My friend tian_275461 proposed this integrate,but I have no idea about it.
pxchg1200
  • 2,050
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$\sum _{k=1}^{n-1}k^p \lt \frac {n^{p+1}} {p+1} \lt \sum _{k=-1}^{n}k^p$

I'm going through a proof of the theorem that says $\int_0^bx^pdx = \frac {b^{p+1}}{p+1}$, and it begins with the inequality. $\sum _{k=1}^{n-1}k^p \lt \frac {n^{p+1}} {p+1} \lt \sum _{k=-1}^{n}k^p$ What I'm having trouble understanding where this…
AlexHeuman
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Transform $\prod_{k=0}^{n} (1+x^{2^{k}})$ to $\sum_{k=0}^{n} c_{k}x^{k}$

I want to transform the following $$\prod_{k=0}^{n} (1+x^{2^{k}})$$ to the canonical form $\sum_{k=0}^{n} c_{k}x^{k}$ This is what I got so far \begin{align*} \prod_{k=0}^{n} (1+x^{2^{k}})= \dfrac{x^{2^{n}}-1}{x-1} (x^{2^{n}}+1) \\ \end{align*} but…
Friedrich
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Proving all Solutions of a Polynomial Cannot all be Real

If, a, b, c, d and e are all real numbers how could I prove that the 5 solutions of the equation: $$f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e == 0$$ cannot all be real valued if: $$2a^2 < 5b$$ Any assistance is appreciated.
Nate222
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Show this equation has at least one root in $(0,1)$

Let $ax^2+bx+c=0$ be a quadratic equation, where $a,b,c\in\mathbb{R}$. If $2a+3b+6c=0$, then show that this equation will have atleast one root in $(0,1)$. I think it involves either Rolle's Theorem or Lagrange's Mean Value Theorem, but can't think…
Ashish Gaurav
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Improper integral $ \int\limits_0^{\infty} \frac{ x^2 \arctan x }{x^4 + x^2 + 1 } dx $

Im trying to find $$ \int\limits_0^{\infty} \frac{ x^2 \arctan x }{x^4 + x^2 + 1 } dx $$ Thoughts: My first thought was to write the denominator as $(x^2+1)^2 -x^2 $ and then by difference of squares we have $(x^2+1-x)(x^2+1+x)$. Then write the…
James
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Differentiability vs Having a Derivative

In some calculus texts, one comes across the following statement - "A function is differentiable, if it has a derivative and a function has a derivative, if it is differentiable.". Two such texts are - A treatise on Advanced Calculus by Philip…
Nikhil Panikkar
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If $f(\frac{x}{y})=\frac{f(x)}{f(y)} \, , f(y),y \neq 0$ and $f'(1)=2$ then $f(x)=$?

If $f(\frac{x}{y})=\frac{f(x)}{f(y)} \, , f(y),y \neq 0$ and $f'(1)=2$ then $f(x)=$? I am not sure where to begin, any hints on starting and steps is apreciated. Thank you
Ashish Gaurav
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If a first derivative doesn't exist at a certain point, is it not a critical point?

Say $f'(x)$ of a function was $(x+2) \over (x+3)$. If $x = -2$ then $f'(x)$ = 0, which means that at this point, there would be a local min or max. But what if $x = -3$? It doesn't exist for $f'(x)$, so do we just ignore it? State it DNE at $x = -3$…
ming
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How to solve $\int \ln(x)\cos(x)\: \mathrm{d}x$?

I'm a high school senior and I'm taking Calc II. Last week we went through integration by parts. I encountered this problem which is not on the textbook, and I couldn't solve it. I tried several different approaches but each of them led me to an…
lllluuukke
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Evaluating the limit for a point on the curve

For a point $P(a,b)$ is a point lying on the curve satisfying $$2xy^2dx + 2x^2 y dy - \tan(x^2y^2) dx =0 $$ $\lim_{a\to -\infty}b = ? $ Options are: a) $ 0$ b) $-1 $ c) $1$ d) does not exist. Attempt: If we observe carefully we get: $d(x^2…
Archer
  • 6,051
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Finding maxima and minima of a function

A couple problems are giving me trouble in finding the relative maxima/minima of the function. I think the problem stems from me possibly not finding all of the critical numbers of the function, but I don't see what I missed. Given $f(x)= 5x + 10…
Jason
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Compute the limit $\displaystyle \lim_{x\rightarrow 0}\frac{n!x^n-\sin x\sin 2x \sin 3x\cdots\sin nx}{x^{n+2}}\;\;,$

How can I calculate the given limit $$\displaystyle \lim_{x\rightarrow 0}\frac{n!x^n-\sin x\sin 2x \sin 3x\cdots\sin nx}{x^{n+2}}\,,$$ where $n\in\mathbb{N}$.
juantheron
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How does $f(x)= x \sin(\frac{\pi}{x})$ behave?

I think this function is increasing for $x>1$ but wanted to find the reason. So I thought about taking the derivative: $f(x)= x \sin(\frac{\pi}{x})$ Aplying the chain an the product rule, we get: $f'(x)= \sin(\frac{\pi}{x})-\frac{\pi}{x} \cos…
Vmimi
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$\int_{0}^{2008}x|\sin\pi x| dx$

Evaluate: $$\int_{0}^{2008}x|\sin\pi x| dx$$ That modulus sign is causing problems. How do I handle it? I am trying integration by parts I have even evaluated: $\int_0^1 {|\sin \pi x|}= \frac 2 \pi$. Not sure how to utilise it in the problem. I…
Archer
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