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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If $f(1) = 3$ and $\int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt\;\forall x,y \in \mathbb{R^{+}}\;,$ Then $f(e) =

Let $f:\mathbb{R^{+}}\rightarrow \mathbb{R}$ be a differentiable function with $f(1) = 3$ and satisfying:: $\displaystyle \int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt\;\forall x,y \in \mathbb{R^{+}}\;,$ Then $f(e) = $ $\bf{My\;…
juantheron
  • 53,015
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Differentiability implying continuity

Theorem: If $f$ is finitely differentiable at $c$, then $f$ is also continuous at $c$. But according to the definition, for a function to be differentiable at $c$, it need not even be defined at $c$, just that it should have a finite value in the…
Vikram
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How do you cancel the common factor of $\frac{\sqrt[3]{x}+2\sqrt{x}-3}{x-1}$

when faced with this kind of limit: $\lim _{x\to 1}\left(\frac{\sqrt[3]{x}+2\sqrt{x}-3}{x-1}\right)$ i know i can use l'hopital's rule to solve it because we get $\frac{0}{0}$ replacing $x$ with $1$, still i want to learn the other method which is…
Laythe Leo
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Why is $n \left(1-p^{\frac{1}{n}}\right)$ increasing in $n$?

I would need a proof that $n \left(1-p^{\frac{1}{n}}\right)$ is increasing in $n \in \mathbf{N}$ for any $p \in (0,1)$. Context I am working on a larger question and this is the last missing piece. But with this I'm a bit out of ideas (I tried the…
TomH
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Question about the chain rule and the fundamental theorem of calculus

Let $f$ be a function which satisfies the conditions of Fundamental theorem of calculus etc etc etc and $F(x) = \int_{a}^x f( \tau ) d \tau $. We know $$ F'(x) = f(x) $$ Also, by the chain rule if we have $$ F(x) = \int\limits_a^{g(x)} f ( \tau) d…
ILoveMath
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Finding the oblique asymptote of:

Given $$f(x)=\frac{x^2+1}{(x+1)^{\frac{1}{2}}}$$ how would you find the oblique asymptote of that?
John
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Find $\int \frac{x^2 - x}{x^2 +x + 1}dx$

I know the answer is $x - \ln|x^2 + x + 1|$ but I don't understand how to get it. Its in the partial fraction decomposition section of homework. The way the homework worked it is like this... $$ \int \frac{2x+1}{x^2 + x + 1} \, dx $$ I see they…
user142342
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Trigonometric Limit Question

I am beginning trigonometric limits. I believe this limit requires a substitution but I am not quite sure exactly how the substitution works. Any explanations on the process of this specific question would be wonderful. Given: $$\lim_{x \to \pi}…
nitrous2
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Area bounded by $y=e^{-x^2}\;,y=0$ and $x=0$ and $x=1$,

Let $S$ be the area of the region bounded by $y=e^{-x^2}\;,y=0$ and $x=0$ and $x=1$, Then which one is/are right Options:: $\displaystyle (a)\; S\geq \frac{1}{e}\;\;\;\;\;\;(b)\;S\geq 1-\frac{1}{e}\;\;\;\;\;\;(c)\;S\leq…
juantheron
  • 53,015
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Does the function $\frac{x}{\sqrt{1+x}}$ have an oblique asymptote?

Does the function $$\frac{x}{\sqrt{1+x}}$$ have an oblique asymptote? If so, how do we find it? I thought about long dividing, but that wouldn't work, because You can't divide square roots.
John
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How do we calculate the area of a region bounded by four different curves?

Calculate the area(express both respectively in integral with one variable) bounded by the following curves (i.e. the shape with one side corresponding to one curve): $$xy=1, \quad xy^2=3,\quad x^2-y^2=26,\quad x^2-y^3=11$$ This problem is created…
Victor
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Inverse of Torus parametrization

any tips on finding the inverse of the following map: $$(\theta,\phi)\mapsto ((2+\cos\theta)\cos(\phi),(2+\cos\theta)\sin(\phi),\sin\theta)$$ From doing it with cartesian : One of the maps is: I get $\phi^{\pm}(x_1,x_2,x_3)=(x_1,x_2)$ and…
TKM
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contradiction to chain rule.

As the chain rule states: If $f(u)$ is differentiable at point $u=g(x)$, and $g(x)$ is differentiable at $x$, then the composite function $(f\circ g)(x)=f(g(x))$ is differentiable at $x$ , and $$(f\circ g)'(x)=f'(g(x)).g'(x).$$ Then the…
spectraa
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Find an integral

$\DeclareMathOperator{\arcsinh}{arcsinh} \DeclareMathOperator{\sgn}{sgn} \newcommand{\dd}{\mathop{}\!\mathrm{d}}$ My integral follows: $$\int\limits_0^n\left(\int\limits_0^n \dfrac{1}{\sqrt{x^2+y^2}+1}\dd x\right)\dd y.$$ I attempted the…
MickG
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Evaluating an indefinite integral $\int\sqrt {x^2 + a^2} dx$

indefinite integral $$\int\sqrt {x^2 + a^2} dx$$ After some transformations and different substitution, I got stuck at this $$a^2\ln|x+(x^2+a^2)| + \int\sec\theta\tan^2\theta d\theta$$ I am not sure I am getting the first step correct. Tried…
urmish
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