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.

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How to prove these integral inequalities?

a) $f(x)>0$ and $f(x)\in C[a,b]$ Prove $$\left(\int_a^bf(x)\sin x\,dx\right)^2 +\left(\int_a^bf(x)\cos x\,dx\right)^2 \le \left(\int_a^bf(x)\,dx\right)^2$$ I have tried Cauchy-Schwarz inequality but failed to prove. b) $f(x)$ is differentiable in…
Xiaolang
  • 495
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Number of real roots of polynomial and intermediate value theorem

Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a polynomial function with real coefficients satisfying \begin{align} p(x_1)<0, p(x_2)>0, p(x_3)<0,\ldots \text{(sign flips in alternating manner)} \end{align} for $x_1
Heedong Do
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Computing limit of $(1+1/n)^{n^2}$

How can I compute the limit $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n^2}$? Of course $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n} = e$, and then $\left(1+\frac{1}{n}\right)^{n^2} =…
PJ Miller
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Limit with roots

I have to evaluate the following limit: $$ \lim_{x\to 1}\dfrac{\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}} . $$ I rationalized both the numerator and the denominator two times, and still got nowhere. Also I tried…
kEoz
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Spivak Calculus chapter 7 theorem 9

I am working through Spivak Calculus chapter 7 theorem 9. There is one statement that I can't quite understand. The theorem states: If $n$ is odd, then any equation $$ \ x^n+a_{n-1}x^{n-1} +\cdots+a^0 $$ has a root. proof: we would like to prove…
ano
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Why the substitution is not working even though its bijective?

The following integral i was trying to evaluate: $$\int_0^\infty \left( \sin(1/x) - \frac{\sin(\pi/x)}{\pi} \right) \,dx.$$ ,what i did was substituting pi/x = 1/t in the second integral , which converts to first integral which tells integral is…
WizardMath
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For which $\alpha$ this sum converges? $\sum_{n=3}^{\infty} {\frac{1}{n \cdot \ln(n) \cdot \ln(\ln(n))^{\alpha}}}$

Given: $$\sum_{n=3}^{\infty} {\frac{1}{n \cdot \ln(n) \cdot \ln(\ln(n))^{\alpha}}}$$ I am asked: For what values of $\alpha$ does this summation converge? So I said, $f(n) = \frac{1}{n \cdot \ln(n) \cdot \ln(\ln(n))^{\alpha}}$. $f(n)$ is obviously…
TheNotMe
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Extreme Value Theorem Proof (Spivak)

Them: If $f$ is continuous on $[a,b]$, then there is a $y$ in $[a,b]$ such that $f(y) \geq f(x)$ for each $x \in [a,b]$ Proof. We already know that $f$ is bounded on $[a,b]$, which means that the set $$\{ f(x):x\text{ in }[a,b]\}$$ is bounded. This…
Lemon
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Is it possible for a quadratic equation to have one rational root and one irrational root.

Is it possible for a quadratic equation with rational coefficients to have one rational root and one irrational root? I have seen that there is a post about this, but I dont understand it. How can it be shown using quadratic formula?
Reader
  • 1,353
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Evaluating the limit $\lim \limits_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}$

How do you evaluate the limit $$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$$
jyoung43
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if $\sum a_{n}$ is a convergent series, what about $\sum \frac{a_{n}}{1+|a_{n}|}$?

Suppose that $\sum a_{n}$ is convergent series of real numbers. Either prove that $\sum b_{n}$ converges, or give a counter-example, when we define $b_{n}$ by $\frac{a_{n}}{1+|a_{n}|}$.
Crni
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Proving that $\lim_{x\to\infty}f(x)=\lim_{x\to 0}f(\frac{1}{x})$ for any function $f$

Prove that for any $f(x)$, $$\lim_{x\to\infty}f(x)=\lim_{x\to 0}f\left(\frac{1}{x}\right).$$ I don't know for sure if this is true, I just thought about it. It seems very intuitive for a simple $f(x)$, but for more complex ones (such as the…
Ovi
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Is it true that if $f(f(x))$ is continuous and strictly decreasing then $f$ is continuous$?$

Is it true that if $f(f(x))$ is continuous and strictly decreasing then $f$ is continuous$?$ First of all, how can $f(f(x))$ be strictly decreasing. If $f(x)$ is increasing then $f(f(x))$ is also increasing and if $f(x)$ is decreasing, then…
Mathaddict
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Half of a derivative?

How could a half derivative be computed? Not that I have found a use for what I wish I could call partial derivatives, but they are still interesting. Taking the nth derivative of the $m^{th}$ derivative of $f(x)$ is the $(n+m)^{th}$ derivative of…
Infinite Planes
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Finding solution in naturals to $a^b=b^a$ with the help of $f(x)=\frac{\ln(x)}{x}$

As the title said, I'm trying to find solution in naturals numbers to $a^b=b^a$ with the help of the function $f(x)=\large\frac{\ln(x)}{x}$. I've been reading some solutions of that problem posted in math forums, and still don't know how to…
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