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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Calculus: prove that the equation has at least 3 real roots.

$$(x + 1)^{1/2} =\frac{1}{(x − 2)^2}$$ Ok, I know how to show how an equation has at least $1$ real roots or exactly $1$ real roots, but for this equation, I know there is indeed at least $34 real roots. But I don't know, how to show it. This are my…
Joe
  • 143
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Multiplying both sides of an equation when there's a limit on one side?

Determine the value of $a\in\mathbb{R}$, such that $\displaystyle\lim_{x\to 1}\dfrac{x^2+(3-a)x+3a}{x-1}=7$ My attempt: \begin{align*} &\lim_{x\to 1} \dfrac{x^2+(3-a)x+3a}{x-1}=7\\ &\implies\lim_{x\to 1} x^2+(3-a)x+3a=7x-7\\ &\implies\lim_{x\to…
Rose
  • 515
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Analysis of the function $y=x^{\frac{1}{x}}$

The graph of the function $y=x^{\frac{1}{x}}$ for positive $x$ is as shown below: When I calculated $y$ for negative values of $x$ only some of the values between $0$ and $-1$ and only those for which $x$ is odd(whole number), were given by Excel,…
Vikram
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Find values of $a$ and $b$ that make the function continuous everywhere.

I need some help with this question: Find the values of $a$ and $b$ that make $f$ continuous everywhere. $$f(x)=\begin{cases} x^2 − 4/x-2, &\text{if }x < 2\\ ax^2-bx+1, &\text{if } 2 ≤ x ≤ 3\\ 4x - a + b, &\text{if } x ≥ 3\end{cases}$$ I started…
Michael
  • 161
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Derivative of Velocity with respect to to Distance and vice versa

Can anyone kindly explain to me that if $\frac{d}{dt}(x) = x^{'}$ then what's $\frac{d}{dx}(x^{'}) $ and $\frac{d}{dx^{'}}(x)$?
Esan
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Proving $f(x)=\frac{x^2}{1+x}$ is uniformly continuous on $[0,\infty)$

I have a homework assignment to complete and I am having trouble proving that $f(x)=\frac{x^2}{1+x}$ is uniformly continuous on $[0,\infty)$. For some reason I can't find the way to solve this one... Cans someone please help me? Thanks :) EDIT: I…
Jason
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An improper integral involving nested square roots

I have: $$I_{a,b}= \int_b^{ + \infty } \left( \sqrt {\sqrt {x + a} - \sqrt x } - \sqrt {\sqrt x - \sqrt {x - b} } \right)dx$$ with $a>0$ and $b>0$. I should determine whether this is a convergent or divergent integral. The problem is that I…
Sabaga Bachir
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Strictly convexity of a function and global minima

How to prove that a function $\mathbb{ f:R\to R}$ is strictly convex, then a critical point is a global minimum using Taylor expansion at the critical point?
Driss
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Proving uniform continuity of functions

got this questions from homework and don't have a clue how to start. I know that every real function which is continuous at a closed interval, is considered to be uniform continuous at this interval. However, my problem is how to deal with…
user3708158
  • 357
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Showing derivative of polynomial has $n$ distinct roots

Let $$f_n(x)=\frac{\mathrm d^n}{\mathrm dx^n}((1-x^2)^n)$$ Any hints on how to show that it has $n$ distinct real roots?
Kal S.
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Is the function $y=\frac{1}{1+e^{1/x}}$ continuous at $x=0$?

Here is the question which I am trying to solve: Determine if the following function is continuous at $x=0$: $$y=\frac{1}{1+e^{1/x}}$$ For continuity, we know that there are three criteria: $f(a)$ is defined limit is…
dato datuashvili
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Tangent to curve $x^3+y^3=a^3$ meets it again.

Tangent to curve $x^3+y^3=a^3$ at $(x_1,y_1)$ meets it again at $(x_2,y_2)$.How to prove that $$\frac{x_2}{x_1}+\frac{y_2}{y_1}+1=0$$ Since…
RE60K
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Prove: the function series, $\sum_{n=0}^{\infty}\frac{(-1)^{n-1}x^2}{(1+x^2)^n}$ uniformly converges

$$\sum_{n=0}^{\infty}\frac{(-1)^{n-1}x^2}{(1+x^2)^n}$$ I want to know whether this function series in uniformly converges or not. I can recognize Leibniz here, so if I'll be able to prove that $\lim_{n \to \infty} f_n(x)=f(x)=0$ the function series…
Jozef
  • 7,100
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Integrating along the wrong arc

Here's a dumb question to which I'd post my own answer if I weren't feeling too lazy right now, but maybe other points of view than mine are worth seeing too. $$ \int_{\pi/2}^{3\pi/2} \frac{1}{1-\cos\theta} \,d\theta $$ Do the tangent half-angle…
Michael Hardy
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For continuous function $f$, prove: $\int_{0}^{x} \; \left[\int_{0}^{t}f(u) \;du \right] \;dt=\int_{0}^{x} f(u)(x-u)du$

I'd love your help with proving that: For continuous function $f$, $$\int_{0}^{x} \left[\int_{0}^{t}f(u) \; du \right] \; dt = \int_{0}^{x} f(u)(x-u)du$$ I'm not quite sure what I should do with this. From Newton-Leibniz theorem I get that the…
Jozef
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