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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Prove that there is $x \in [0,1]$ such that $|f''(x)| > 4$

Assume $f$ has a continuous second derivative with $f(0) = f'(0) = f'(1) = 0$ and $f(1) = 1$. Prove that there is $x \in [0,1]$ such that $|f''(x)| > 4$. We must also have that $f'$ is continuous by differentiability. Therefore, by Rolles theorem…
Puzzled417
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Is it possible to find two differentiable functions $f$ and $g$ and $g$ for which $x = f(x)g(x)$ and $f(0) = g(0) = 0$

Is it possible to find two differentiable functions $f$ and $g$ and $g$ for which $x = f(x)g(x)$ and $f(0) = g(0) = 0$? The fact that both functions have to be differentiable makes this a little more complicated, but we can say $\lim_{x \to a}…
user19405892
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Number of real solution of the equation $4\cos(e^x) = 2^x+2^{-x}$

Number of real solution of the equation $4\cos(e^x) = 2^x+2^{-x}$ $\bf{My\; Try::}$ . Now for $|x|\geq \pi>3$ No real solution exists, So we will check for $|x|<\pi$ Now i did not understand how can i calculate umber of real solution in that…
juantheron
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Incorrect integral in textbook?

I saw this integral in my textbook. Is this incorrect? Shouldn't it be $-\lambda e^{-\lambda s}$ since the integral of $e^{-\lambda s}$ is $-\frac{1}{\lambda}e^{-\lambda s}$ \begin{align*} f_S(s) &=\lambda^2\int_0^se^{-\lambda…
A user
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Find an equation of the tangent line to $y = \cos(x)+3\sin(x)$ at $x=\pi/3$

Find an equation of the tangent line to $$y = \cos(x)+3\sin(x)$$ at $x=\pi/3$. This is what I have done... Find $y$, $y= \cos(\pi/3) + 3\sin(\pi/3)$ this equals $1 + \sqrt 3/2$ Next Find $f'(x) = \sin(\pi/3) + 3\cos(\pi/3)$ this equals $3+…
pewpew
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What is $\int \log x \,\, \mathrm{d}x$?

Today one of my friends asked me what's the $$\int \log x \,\, \mathrm{d}x$$ and I was unable to answer. I think we just take the Taylor series and integrate it . Is that all or something else. Note: I am grade $11$ student so I don't know any…
Archis Welankar
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If $f$ is monotone prove it is continuous

Assume $f: [a,b] \to \mathbb{R}$ is a monotone function that satisfies the Intermediate Value Theorem. Prove that $f$ is continuous. It is sort of confusing how they say it satisfies IVT. Don't only continuous functions satisfy IVT? If instead…
user19405892
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How can a square root be defined since it has two answers?

I am aware that 1/0 is undefined for two reasons: If you would have to give an answer to this it is infinity which is not a number but a concept; The limit of 1/x for $x \to 0$ is either positive or negative infinity and since it has two limits we…
Héctor van den Boorn
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Prove $f$ attains its maximum

Assume $f$ is a function over real numbers such that $f(x)>0$ for all $x$. Suppose that $\displaystyle \lim_{x \to \infty} f(x) =\lim_{x \to -\infty} f(x) = 0.$ Prove $f$ attains its maximum. Firstly, I believe the question should…
Puzzled417
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Radius of Convergence of power series, with n! and $n^n$

During revision, I came across this problem: The set of real numbers $x$ for which the series $$\sum_{n=1}^{\infty}{\frac{n!x^{2n}}{n^n(1+x^{2n})}}$$ converges is __. I tried using the ratio test, but got stuck in the process of simplification. (The…
yoyostein
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Using Stirling's Formula to calculate ratio

I am working on this question: For equidistant points $x_j=j, 0\leqslant j\leqslant n$, $n$ even, let $$\omega(x)=(x-x_0)(x-x_1)\ldots (x-x_n)$$ Use Stirling's formula to estimate the ratio $$\frac{\omega\left(\frac…
user315747
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Chain Rule Confusion

Below i have a function that i need to use the chain rule on. My friend showed me his answer which was correct which was $-8x^7\sin(a^8+x^8)$. $$y = \cos(a^8 + x^8)$$ I am really confused as how he got that. I know that in the chain rule you bring…
soniccool
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Solution of Differential equation $\frac{xdx-ydy}{xdy-ydx} = \sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$

Solution of Differential equation $$\frac{xdx-ydy}{xdy-ydx} = \sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$$ $\bf{My\; Try::}$ Let $x=r\sec \theta$ and $y=r\tan \theta\;,$ Then $x^2-y^2=r^2$ and $xdx-ydy=rdr$ and $$\frac{y}{x} = \frac{\tan \theta }{\sec…
juantheron
  • 53,015
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$443642_{b_1}=53818_{b_2}$ in what base?

What are the bases $b_{1},b_{2}$ in $443642_{b_1}=53818_{b_2}$ What would be a general approach to this type of problems that would reduce the tedious manual calculations? (This problem is important, because we could have two unknown sources that…
user195934
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Find volume of region bound by $y=x, y=x^2$ around x-axis

Here is the problem in my textbook: Find the volume of the solid obtained by rotating the region bounded by the curves $y=x, y=x^2$ about x-axis. Here is my solution : Because equation $x = x^2$ has two roots : $0$ and $1$. we have: $$ V=…
hqt
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