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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why $\arctan(nx)$ doesn't converge uniformly?

I'd love your help with understanding what led me to conclude that $\arctan(nx)$ converges uniformly for $x \in (0, \infty)$ I wanted to check that $\lim_{n \to \infty} \sup |f_n(x)-f(x)|$ is 0 where $f(x)=\lim_{n \to \infty} f_n (x)$. $f(x)=\frac…
Jozef
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Problems with $\int_{1}^{\infty}\frac{\sin x}{x }dx$ convergence

I'd love your help with deciding whether the following integral converges or not and in what conditions: $\int_{1}^{\infty}\frac{\sin x}{x}$. 1. First, I wanted to use Dirichlet criterion: let $f,g: [a,w) \to R$ integrable function, $f$ is monotonic…
Jozef
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Show that $h(x)=x^5+3x+6$ is one to one

How do I show that $$h(x)=x^5+3x+6$$ is one to one? I set $$f(a)=f(b)$$ and try to isolate for $a$ and $b$ but I get stuck because I have a term of "$a$" that is degree $5$ and a term of "$a$" that is degree $1$ so I am unable to simplify it.
marcus
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If $f^2$ monotonically increasing in $R$ then f monotonic

Function $f$ is continuous in $R$. Prove: If $f^2$ monotonically increasing in $R$ then $f$ monotonic in $R$. Intuitively its seems pretty clear, But I don't have any idea how to "start" the prove. Any ideas? Thanks.
JaVaPG
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How can I solve this equation analytically. $\sqrt{x+\sqrt{2x+\sqrt{3x...}}}-100x\sin(x)=0$

How can I solve this equation analytically. $$\sqrt{x+\sqrt{2x+\sqrt{3x...}}}-100x\sin(x)=0$$
E.H.E
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Discriminant of quadratic formula

I noticed something that i find interesting about the quadratic formula. I hope someone can explain it to me. $f(x) = Ax^2 + Bx + c$ $f'(x) = 2Ax + B$ If I say that $f'(x) = 0$ I get that $x = -\frac{B}{2A}$. From this I get that $f(-\frac{B}{2A})$…
peanut_butter
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$\lim_{n\rightarrow \infty}n^{-\left(1+1/n\right)/2}\times \left(1^1\times 2^2\times 3^3\times\cdots\times n^n\right)^{1/{n^2}}$

Evaluate the limit $$ y=\lim_{n\rightarrow \infty}n^{-\left(1+1/n\right)/2}\times \left(1^1\times 2^2\times 3^3\times\cdots\times n^n\right)^{1/{n^2}} $$ My Attempt: When $n\rightarrow \infty$, then $n^{-\left(1+1/n\right)/2}\rightarrow…
juantheron
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Proving that $x\sin x+\cos x=x^2$ has only one positive answer

I have a homework question to prove that $$x \sin x+\cos x=x^2$$ has only one positive solution. I have easily proved that it has a positive answer by showing that $f(x)=x\operatorname{sin}x+\operatorname{cos}x-x^2$ is smaller then $0$ at…
Jason
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Area and integration question, is this area under the curve?

Find the exact area between $x$ and the graph $f(x)=(x-1)(x-2)(x-3)$. $$f(x) = x^3-6x^2+11x-6$$ I found that this is an odd shaped positive polynomial with a maxima between 1 and 2 and minima between 2 and 3. I am confused to what the question…
Ivan
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Evaluation of $\int_{0}^{\frac{\pi}{2}}\left(\frac{1+\sin 3x}{1+2\sin x}\right)dx$ and $\int_{0}^{2} \left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\right)dx$

Evaluation of Some Integrals:: $\displaystyle (a)\;\;\int_{0}^{\frac{\pi}{2}}\left(\frac{1+\sin 3x}{1+2\sin x}\right)dx\;\;\;\;\;\;(b)\;\; \int_{0}^{2} \left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\right)dx\;\;\;\;\;\;$ $\displaystyle…
juantheron
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Integrating algebraic functions

The function $y = f(x)$, restricted on the domain $ 0 < x < 1$ and satisfying $$y^{5}+y^{4} + x = 0,$$ seems to be well-defined and smooth. So how does one integrate this thing? That is, what is $\int_{0}^{1} f(x) dx$? Of course, one can use…
michaelkovarik
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Define continuity for $f(x)=\arctan(2x^3)/x^2$ at $x=0$.

$$f(x)=\dfrac{\arctan(2x^3)}{x^2}.$$ How are we allowed to define $f(x)$ at $x=0$ for it to be continuous there? Find the derivative for all $x$ real numbers. I can't see this work out since $x=0$ is not defined in the denominator. Thanks…
mathgrad93
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Proving differentiability of a function from condition of diff. of other function

The following question says: Let $$\phi(t) = \begin{cases} \dfrac{sin(t)}{t} & \text{if $t\neq 0 $} \\ 1 & \text{if $t=0$} \end{cases}$$ Show that $\phi$ is differentiable on $\mathbb R$.Let $$f(x,y) = \begin{cases}…
patang
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Showing that $\lim_{n \rightarrow \infty}\left(\frac{n - 1}{n}\right)^n = 1/e$

I would like to show $\lim_{n \rightarrow \infty}\left(\frac{n - 1}{n}\right)^n = 1/e$. I know the argument typically goes like this: Let $y = \left(\frac{n - 1}{n}\right)^n$. Then $\ln(y) = n\cdot{}\ln \left(\frac{n - 1}{n}\right)$. Taking the…
Joseph DiNatale
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Two differentiation results of $\sin^{-1}(2x\sqrt{1-x^2})$

While trying to differentiate $\sin^{-1}(2x\sqrt{1-x^2})$, if we put $x = \sin\theta$, we get, \begin{align*} y &=\sin^{-1}(2x\sqrt{1-x^2})\\ &= \sin^{-1}(2\sin\theta\sqrt{1-\sin^2\theta})\\ &= \sin^{-1}(2\sin\theta\cos\theta)\\ &=…
Masroor
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