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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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Show that $1/x > \log(x)^2$, for all real $x < 1$

so i'm taking a Calc 1 class right now, and I am supposed to evaluate, whether there is a constant continuation of $f(x) = x^x$, where x approaches zero and is a positive real number. I have a function $a(x) := e^x$, which is always higher than…
TeReOs13
  • 43
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Prove there exists $c,d\in(a,b)$ such that $\frac{f'(c)}{f'(d)}=\frac{e^b-e^a}{b-a}e^{-d}$

Assume that $f$ is differentiable on $(a,b)$ and continuous on $[a,b]$, $f'(x)\neq0$. Obviously it's related to the mean value theorem, so I was thinking Let $g(x)=e^xf(x)$ and applying the mean value theorem, but that didn't work out well--does the…
Lee Laindingold
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Conditions for a function to have guaranteed global extrema along $[a, b]$

Today, I took my Calculus I final. One of the questions on the final was thus (paraphrased): Which of the following must be true in order for a function to have global extrema along the interval $[a, b]$? The function must be continuous along…
Reid
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How to find y given an x of this implicit equation

This really got me thinking and I honestly have no idea. I’m guessing we don’t need the modulus sign given that is a hint but why? We were given an initial value problem: $$ \frac{\mathrm d y}{\mathrm d x} = \frac{y\cos x}{1+2y^2}, \;\; y(0) = 1…
Oceane
  • 299
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Divide a number N into two parts in such a way that three times the square of one part plus twice the square of the other part shall be a minimum.

I'm reading a book (Calculus Made Easy - Silvanus Thompson) that has the following exercise: Divide a number N into two parts in such a way that three times the square of one part plus twice the square of the other part shall be a minimum. It's…
GFlow
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If $ g\left(x\right)\geq f\left(x\right) $ and $ \intop_{a}^{b}g=\intop_{a}^{b}f $ then $ f=g $?

Let $ g:[a,b]\to\mathbb{R} $ and $ f:[a,b]\to\mathbb{R} $ be continuous functions, such that $ f\left(x\right)\leq g\left(x\right) $ for any $ x\in[a,b] $. Assume $…
FreeZe
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How does one calculate what a sum converges too?

I have little to no substantial experiences with sums at infinity beyond what the notation conveys. My question is how does one calculate what: $$\sum_{n=1}^{\infty} a(n) $$ converges to, supposing the sum to be convergent. To use a classic example,…
Euler Wannabe
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Converting equation to Cartesian coordinates

I'm having trouble figuring out how to convert this equation to Cartesian coordinates. Sorry if I didn't format my question correctly, this is my first time using this site. Any help would be appreciated! $$r =…
user1781500
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Set Of Discontinuities Of A Derivative

Prove that the set of discontinuities of a derivative of an everywhere differentiable function $f(x)$ is of 1st category. $$$$Let $f'(x)$ be a derivative of an everywhere differentiable function $f(x)$. Now as the set of discontinuities of any…
user771946
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number of solutions of $\ln|\sin x|=-x^2$

Question: Find the number of solutions of $\ln|\sin x|=-x^2$ in the interval $[-\frac{\pi}{2},\frac{3\pi}{2}]$. My try: I first looked at the interval $[-\frac{\pi}{2},0]$. In this interval as $x$ tends to $0$ the function $f(x)=\ln|\sin x|+x^2$ is…
Albus Dumbledore
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Laplacian of a generalized Whitehead potential

I'm trying to calculate the Laplacian $\triangle\Phi_W^{IJ}$ of the following "generalized Whitehead potential": $$ \Phi_W^{IJ}(\vec{x}) =…
Xenos
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Is the solution guide wrong?

Give the volume of the solid generated by revolving the region bounded by the graph of $y = \ln(x)$, the $x$-axis, the lines $x = 1$ and $x = e$, about the $y$-axis. Question 2 the solution guide gives $\frac{\pi}{2}\left(e^2+1\right)$ has the…
yiyi
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supremum of a periodic function

Let $f : \mathbb R \rightarrow \mathbb R$ be a continuous function such that $f(x+1) = f(x)$ or $f(x-1)=f(x)$ or $f(1-x)=f(x)$ . then $f$ attains its supremum? here is what I was able to do: $f(x+1)=f(x)$. let $f(x)= \sin(2*\pi*x)$. thus the…
under root
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Cartesian Product Of Open Intervals

Prove that the Cartesian product of the open intervals $$(a_1, b_1)\times(a_2, b_2)\times\cdots\times(a_{n-1}, b_{n-1})\times(a_n, b_n)$$ is an open set in $\mathbb R^n$. Let us chose any point say $x = (x_1, x_2,\ldots,x_n)$ from the set $$(a_1,…
user771946
  • 415
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Explain Why $\int\frac{f'(x)}{f(x)}\,\mathrm dx = \ln|f(x)|+C$

Explain why $$\int\frac{f'(x)}{f(x)}\,\mathrm dx = \ln|f(x)|+C.$$ Thanks!
Ofir Attia
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