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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How do we show that the function which is its own derivative is exponential?

In my calculus class, to show that $\frac{d}{dx}e^x=e^x$ we did something like this: $$\lim_{h \to 0} \frac{a^{x+h} - a^x}{h} = a^x \lim_{h \to 0} \frac{a^h-1} h,$$ and then we defined $e$ to be the base $a$ that makes the limit $$\lim_{h \to 0}…
Jacob
  • 739
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Integration being the opposite of differentiation?

I know that if we start with an original function and take one derivative, we get another function. If we take the integral of that new function we get the original function back. So I see how they are opposite operations but I don't see how they…
King Squirrel
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1 answer

Unified notion of what "$dx$" means

The symbol $dx$ in calculus is at first introduced just as a form of notation: in differentiation, $$\frac{dy}{dx}$$ means the derivative of the function $y$ with respect to $x$. Letting $f(x) = y$, other notations are $f'(x)$, $\dot{y}$ (Newton's…
The_Sympathizer
  • 19,651
13
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2 answers

Integration by Parts implies U-substitution?

So I feel a bit strange asking a Calculus question, but this came up today while teaching. One can check that if you start with some integral, which can be see as an "obvious u-substitution problem", that you can instead use integration by parts,…
BBischof
  • 5,807
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1 answer

Does that series converge or diverge?

Does the series $$\sum \limits _{n=3}^\infty \frac{(-1)^{[\log n]}}{\sqrt{n}}$$ converge or diverge? As usually, $[x]$ denotes the integer part of $x.$
user64494
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13
votes
1 answer

Integrate $\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}$

Evaluate integral $$\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}$$ Well,I think we have $$\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}=\frac{\pi}{e}$$ and $$\int_{0}^{1}{x^{x}(1-x)^{1-x}\sin{\pi x}dx}=\frac{e\pi}{24}$$ With such nice result of…
pxchg1200
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3 answers

Finding the inverse of the arc length function

I'm just a simple high school math student, so please don't eat me =) In my calculus text, I have the formula: $$L(x) = \int_{c}^{x} \sqrt{[f'(t)]^2 + 1}\,dt$$ Where $L(x)$ is the arc length of a curve $f(x)$ from $c$ to $x$. How can I invert this…
Clark Gaebel
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Ant climbing on bush

An ant is on the ground and trying to climb on a (straight) ivy bush 10m high. It crawls up 0.1m each night, but at day, the bush grows uniformly by 0.5m (in its entire height). Will the ant ever reach the top of the ivy? If yes, in how many days?…
Khalil. Jahromi
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13
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convex function in open interval is continuous

How can I prove that a convex function ƒ defined on some open interval C is continuous on C? Thank you.
user6163
13
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1 answer

What does the second antiderivative of a function represent?

If the antiderivative represents the area under the curve from $f(0)$ to $f(x)$ what does the second antiderivative or the antiderivative of the antiderivative represent?
AJB_1070179
  • 425
13
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Solve $x^n+y^n = (x+y)^n$

Find all positive integers $n$ and real numbers $x$ and $y$ satisfying $x^n+y^n = (x+y)^n$. We first consider the case that $n$ is even. We have $x^{2k}+y^{2k} = \binom{2k}{0}x^{2k}+\binom{2k}{1}yx^{2k-1}+\cdots+\binom{2k}{2k}y^{2k}$. This can be…
Puzzled417
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3 answers

Prove that $f(x)$ be a constant

Let $f(x)$ be continuous function in R,note $$ h_{n}(x)=2^{n}\left[f\left(x+\frac{1}{2^{n}}\right)-f(x)\right]$$ with $$ |h_{n}(x)|\leq M \qquad (x\in R,n\in N)$$ and $$ h_{n}(x)\rightarrow 0 \qquad (n\rightarrow\infty)$$ Show that $f(x)$ is a…
pxchg1200
  • 2,050
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1 answer

Equality of integrals

this is q.2 of ahlfors p. 241: Show that $$\int_{-1}^{1}\frac{dt}{\sqrt {(1-t^2)(1-k^2t^2)}}=\int_{1}^{\frac{1}{k}}\frac{dt}{\sqrt {(t^2-1)(1-k^2t^2)}}$$ if and only if $k=(\sqrt{2}-1)^2$ . Thank you.
curious
  • 151
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Show that there are no functions $f: \mathbb R \to \mathbb R$ which have the intermediate value property and $f(f(x))=\cos^2(x)$

Show that there are no functions $f: \mathbb R \to \mathbb R$ which have the Darboux property (the intermediate value property) and $f(f(x))=\cos^2(x) ; \ \forall \ x\in \mathbb R$. I guess that I'd have to use the fact that $f(f(x)) \in [0,1]$,…
Lisa
  • 733
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6 answers

Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$

The question I've been given is this: Using both sides of this equation: $$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$ Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$ Then use that to find an expression…
snario
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