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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Solve without using L'Hôpital's rule?

Is it possible to solve this without using L'Hôpital's rule? $$ \lim_{x\to 0} \Big(\frac{3x+1}{x}-\frac{1}{\sin x}\Big) $$ I tried to solve it but I got stuck at the $\frac{1}{\sin x}$ part. $$ =\lim_{x\to 0}\frac{3+x^{-1}}{1}-\lim_{x\to…
Derek 朕會功夫
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Evaluating definite integral of $p(x)$

Let $p(x)$ be fifth degree polynomial such that $p(x)+1$ is divisible by $(x-1)^3$ and $p(x)-1 $is divisible by $(x+1)^3 $. Then find the value of the definite integral $$\int _{-10}^{10}p(x)dx$$ Attempt: $p(x)-1 = (x+1)^3 Q(x)$ $p(x)+1 =…
Archer
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Prove that if a continuous function is injective, then it is monotonic

It is intuitive and it seems very obvious that if a function $f : X \rightarrow Y$ is continuous on whole $X$ and it's injective, then it must be monotonic, but I can't come up with any neat proof for that. Could you maybe help me?
Hagrid
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Find $\frac{\mathrm d^{100}}{\mathrm d x^{100}}\frac{x^2+1}{x^3-x}=$?

$$f(x)=\frac{x^2+1}{x^3-x}$$ $$f^{(100)}(x)=?$$ I tried differnetiating once and twice, but did not see any pattern emerging and can't guess what the 100th derivative should be. EDIT so decomposing this as…
Please Delete Account
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Inspecting the function $f(x)=-x\sqrt{1-x^2}$

We are just wrapping up the first semester calculus with drawing graphs of functions. I sometimes feel like my reasoning is a bit shady when I am doing that, so I decided to ask you people from Math.SE. I am supposed to draw a graph (and show my…
Dahn
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How to find the sum of an alternating series?

Find whether the series diverges and its sum: $$\sum_{n = 1}^\infty (-1)^{n+1} \frac{3}{5^n}.$$ I found that the series converges using the Alternating Series test because the absolute value of each $n$ decreases while the value of $n$…
Jeremy
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Problem of limit of power function

I have a problem : Find $$\lim_{x\to 0}\left(\dfrac{x^2-2x+3}{x^2-3x+2}\right)^{\dfrac{\sin x}{x}}$$ Here is my argument : $$\lim_{x\rightarrow 0}\left(\dfrac{x^2-2x+3}{x^2-3x+2}\right)^{\dfrac{\sin x}{x}}= \lim_{x\rightarrow…
knot
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Calculus Generalisation

$$ \mbox{How should I integrate}\quad \int_{0}^{\pi/2} \cos^{2011}\left(\,x\,\right)\sin\left(\,2013x\,\right)\,\mathrm{d}x\ ?. $$ I am stuck with the reduction formula which is coming out to be $\left(\,2n + 2\,\right)I_{n} = 1 + nI_{n-1}$. Also…
user1712
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Is the $n^{th}$ derivative of $\sin(x)$ just a translation of $\sin(x)$?

I noticed that $$\frac d{dx}\sin x=\cos x=\sin\left(x+\frac\pi2\right)$$ $$\frac{d^n}{dx^n}\sin x=\sin\left(x+\frac{\pi n}2\right)$$ Does this hold for any positive real value of $n$? If so, does anybody have any reasoning behind why it's just a…
Jacob Claassen
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Exercise 5.3 in Calculus Made Easy: are these answers equivalent?

Exercise 5.3 in Calculus Made Easy, by Silvanus Thompson, is to find $\mathrm d\over \mathrm dx$ when the following relationship holds ($a$ and $b$ are both constants): $$ay + bx = by - ax + (x + y)\sqrt{a^2 - b^2}$$ I tried differentiating both…
Alex D
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Show an integration equality without using $\int\frac{\mathrm dx}{x}=\ln|x|+c$

Show that $$\int_{1}^{t_1t_2}\frac{\mathrm{d} x}{x}=\int_1^{t_1}\frac{\mathrm{d} x}{x}+\int_1^{t_2}\frac{\mathrm{d} x}{x}$$ without using $$\int\frac{\mathrm{d} x}{x}=\ln|x|+c.$$
David Salinas
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Is there any continuous function that satisfies the following conditions?

Is there any continuous function $y=f(x)$ on $[0,1]$ that satisfies the following conditions? $$ f(0) = f(1) = 0, $$ and $$ f(a)^2-f(2a)f(b)+f(b)^2<0, $$ for some $0
Ana Uspekova
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Find a Continuous Function

Possible Duplicate: Universal Chord Theorem I am having a problem with this exercise. Could someone help? Suppose $a \in (0,1)$ is a real number which is not of the form $\frac{1}{n}$ for any natural number n n. Find a function f which is…
user43758
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Compute $\lim_{x \to 0^+}x\int_{x}^1 \frac{f(t)}{t^2}dt $

If $f$ is integrable on $[0,1]$ and $\displaystyle \lim_{x \to 0^+}f(x)$ exists, compute $\displaystyle \lim_{x \to 0^+}x\int_{x}^1 \dfrac{f(t)}{t^2} dt $. We can't really use L'Hospital's rule, but for $f(t) = 1, \forall t$, we get the limit to…
user19405892
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Prove whether series converges or not?

Does anyone know how to determine with proof whether the series $$\sum_{n=1}^\infty\frac{1}{n^{2+\cos(2\pi\ln(n)) }}$$ converges?
David
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