Questions tagged [calculus]

For basic questions about limits, derivatives, integrals and applications, mainly of one-variable functions.

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0answers
25 views

Where to start with an intergral involving two Heaviside step functions?

This question is a follow-up to a previous question. I am dealing with the following type of integral: $$ f(\vec x) = \int_{\mathbb R} \text{d}^3y\; (\vec x\cdot \vec y)^n H(A-|\vec y-\tfrac{1}{2}\...
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44 views

Ordinary differential equation solution ideas. [on hold]

Let $f$ be a differentiable function and $\mu>0$. Is there an explicit solution $y$ of the following first order ODE? $$ |y'(x)| = \exp\left(-\mu\sqrt{f^2(x) + (y(x)-x)^2}\right), \ \ x\in \ ]-\...
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2answers
22 views

Example of a function that is non-differentiable on an open interval

I am studying a course on single variable calculus. During a lecture, the professor mentioned in passing that there can be functions that are non-differentiable on an entire open interval. Can anyone ...
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1answer
34 views

Integrals Definite substitution from this? [on hold]

Integrals Definite substitution from $$\int^{\pi/4}_{0}\frac{8 \cos(2t)}{\sqrt {9-5 \sin t (2t)}}~dt ~?$$? u = 9 - 5 sin t(2t) du = -10(t cos (t) + sin (t)) , this correct for du = -10(t cos (t) ...
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1answer
55 views

Expressing $f(f(x))$ when $f(x)$ is a piecewise function

This is a question from MIT OpenCourseWare about calculus. I cannot find any explanation online since I have no idea what the keywords are. The question states that: $f(t)=\begin{cases} 2t & \...
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1answer
35 views

The proof for the Lebesgue differentiation theorem

Im looking at the proof of the Lebesgue differentiation theorem on wikipedia: https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem#Proof I don't see why this line is true. This looks like ...
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0answers
48 views

simplification for an challenging expression [on hold]

Can someone please help me how to simplify this $$ \left(1 - \frac{1}{\cos{\theta}}\right) a + \tan{\theta} \exp{(-i\phi)}b = 0 $$ expression to reach to the following? $$ -(\sin{\frac{\theta}{2}}\...
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0answers
26 views

Define complex antiderivative up to constant in terms of real definite integral

I know there's a complex analogue of the real antiderivative that is allowed to produce multivalued functions. For instance, the complex logarithm has some relationship to $z \mapsto \frac{1}{z}$ . ...
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1answer
21 views

How to apply steady state solution into this question

This is what my study guide defined as the steady state solution If $h(a)=0$ for some constant $a$, then the constant function $y=a$ is a solution of the DE. We sometimes called this a steady state ...
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3answers
65 views

$n$th derivative of $e^{ax}\sin(bx+c)$

How can we substitute $r \cos \alpha$ and $r \sin \alpha$ for $a$ and $b$? How, on successive differentiation, is there another $r$ multiplied?
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1answer
31 views

How to derive volume element for spherical coordinate fast (maybe informally)?

Is there a handy informal argument to derive change of variable volume element for spherical coordinate $r^2 \sin\phi$. Possibly some sort of geometric interpretation? The purpose is to not memorize ...
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2answers
37 views

the limit $\lim_{x \to 0^+} \frac{\sin(\arctan(\sin x))}{\sqrt{x} \sin 3x +x^2+ \arctan 5x}$

a hint to start to find the limit of this $\lim_{x \to 0^+} \frac{\sin(\arctan(\sin x))}{\sqrt{x} \sin 3x +x^2+ \arctan 5x}$ without using L'Hospital rule
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1answer
14 views

reparametrization function

I am trying to understand some computation, but it seems something is not going well. You will find this slide in the following link: https://www.asc.ohio-state.edu/kurtek.1/Lecture3_Srivastava.pdf ...
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1answer
33 views

From binomial theorem to differential calculus

I first posted this over at HSM, without much uptake. I'm trying to understand the development of the calculus. Does this sound plausible as one of the stages? Newton knows the binomial theorem, ...
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3answers
97 views

What is the meaning of $2^\sqrt{3}$? [duplicate]

What is the meaning of $2^\sqrt{3}$ ? one can explain $2^3=2\times2\times2$ But how can I explain $2^\sqrt{3}$ etc. ? How can I explain non-integer powers? I do not want the value. I need to know ...
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6answers
161 views

Quickly evaluating this limit: $\lim\limits_{x\to 0}\left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$

I'm reviewing for the Math GRE Subject test and came across this question in the excellent UCLA notes. $$\lim_{x\to 0}\left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right).$$ If one attacks this with naive ...
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1answer
35 views

What is the gradient of $f(x) := \langle v , F(x) v \rangle$?

Let $F: \mathbb{R}^n \to S^n$ be differentiable function at point $x \in \mathbb{R}^n$. Where $S^n$ is the space of all symmetric matrices with usual Euclidian (trace) norm. Assume $v \in \mathbb{R}^n$...
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2answers
30 views

Confront theorem [on hold]

I have to show, using the sandwich theorem that $$\lim\frac{|x|}{\sqrt{x^{4}+4x^{2}+7}}=0$$ And I have no idea how to start since I can't find any max value to $|x|$.
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6answers
200 views

What’s the difference between something that approaches infinity and something that is infinite.

I’m trying to understand what the difference between something that approaches infinity and something that is infinite is because I was told that that I cant divide infinity by infinity but for some ...
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1answer
50 views

Volume of a union of triangles

Let $t \in \mathbb{R}$ and consider the points $$A(t) = (t,t^3,t), \, B(t) = (t,t,t), \, C(t) = (0,2t,t)$$ Find the volume of $$\Omega = \bigcup_{t \in [0,1]} T(t)$$ where $T(t)$ is the triangle ...
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1answer
63 views

Is this a correct usage of triple integrals?

The problem states: Find the volume of a region bounded by $x^2+y^2=4, \quad y=z, \quad y+z=4 $ So the idea is to triple integrate over the given region. Since sketching stuff in 3D is rather ...
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1answer
34 views

$\frac{dy}{dx}$ of an n-ellipse

An n-ellipse is a generalization of a circle with $n$ foci. It can be written as $\sum\sqrt{(x-a)^2 + (y-b)^2} = R$. I am curious as to the derivative $\frac{dy}{dx}$ of this relation. Using implicit ...
2
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2answers
56 views

prove the absolute value of the integral from $a$ to $b$ of $f$ is less or equal than integral from $a$ to $b$ of the absolute value of $f$

if $f$ is integrable on $[a,b]$ , then $$\bigg\lvert\,\int_a^b{f(x) dx}\,\bigg\rvert \leq \int_a^b{\big\lvert\,f(x)\,\big\rvert\,dx}$$ I know how this works because of the area of the integrals, but ...
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0answers
40 views

I need to find the surface area between Sphere : $x^2 + y^2 + z^2 = 1$ and the Cylinder : $x^2 + y^2 = x$

I'm new to calculus and I tried to do this example but I wasn't able to. I need to find the area between the sphere : $x^2 + y^2 + z^2 = 1$ and the cylinder : $x^2 + y^2 = x$. I tried to rewrite the ...
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3answers
87 views

Is $d(xdx)=dx^2$?

There's a problem in my book that says to find the second-order derivative of $$u=f(\sqrt{x^2+y^2})$$ This is what I did $$d^2u=f''(\sqrt{}x^2+y^2)(\frac{xdx+ydy}{\sqrt{x^2+y^2}})^2+f'(\sqrt{}x^2+y^2)...
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3answers
214 views

Solve $\frac{1}{x}\cdot \cos x + \ln x \sin x = 0$

I was working on the following function: $$f(x) = \frac{\ln x}{\cos x}$$ I tried to find values of x where derivative will equal to zero. After taking derivative of $f'(x)$, I got: $$\tag 1f'(x) =...
2
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1answer
83 views

When does $\int \frac{dx}{x} = \ln|x|$ and when $\int \frac{dx}{x} = \ln(x)$?

Sorry for the provocative question but I am often see a solution where the absolute value is neglect for example: $$ \begin{cases} xuu_x+yuu_y=u^2-1, x>0\\ u(x,x^2)=x^3\\ \end{cases} $$ We ...
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2answers
69 views

Doubt in Stokes' theorem & line integral

Use Stokes' Theorem to evaluate the line integral $$ F = -y^3 dx + x^3 dy - z^3 dz$$ where C is the intersection of the cylinder$$ x^2 + y^2 = 1$$ and the plane $$x+y +z=1$$ I solved it but I have 1 ...
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3answers
77 views

How to find the limit of $\lim\limits_{n\to∞}((\frac{2}{e})^{2n}\sqrt{n})$ [on hold]

I tried L'Hospital's rule seems to be $\frac{∞}{∞}$ initially but I can't get anywhere.
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4answers
85 views

Range of $f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$

I am trying to find the range of this function: $$f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$$ So I think that means I have to find minima and maxima. Using partial derivatives gets messy, so I was ...
2
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2answers
50 views

Understand a certain step in the mean value theorem.

I'm trying to understand a certain step within the proof for the mean value theorem, which states that if $f : [a,b] \rightarrow \mathbb{R}$ is differentiable, there exists an $x_0 \in [a,b]$ with $\...
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1answer
33 views

Odd Integral converges but don't know why

The integral is $$I=\int_{0.1}^{\infty} \frac{1}{e^x*x} dx$$ Any ideas on how this converges? Been at this for a while and can't make progress.
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0answers
36 views

Determining the convergence/divergence of recursive sequence [on hold]

Is it always possible to define a sequence using both $n^{th}$ term formula and recursion formula? For example: $a_1=3$, $a_{n+1}=a_n+3$ defines the sequence $\{3,6,9,12,...\}=\{3n\}$ I am asking ...
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0answers
33 views

integral of complex function on symmetrical path

Hi everyone I have a question concerning the solution of the following exercise: Evaluate the integral of function $f(z)=\frac{Re(z)}{z^2+1}$ over the triangular path $\Gamma$ given by: $\Gamma_1$: ...
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0answers
25 views

Monotonicity of quotients of elementary symmetric polynomials

Let $n\in\mathbb{N}$. Let $\sigma_k$ ($0\leq k\leq n$) be the elementary symmetric polynomials: \begin{align} \sigma_0(x_1,\ldots,x_n):&=1 \\ \sigma_k(x_1,\ldots,x_n):&=\sum_{1\leq i_1<\...
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3answers
64 views

Is there a solution for $xy=a, xy=b$?

As per title. I have a system of equations where: $$\left\{\begin{aligned}Axy = B\\ Cxy = D \end{aligned}\right.$$ $A,B,C,D$ are constants. Is there a solution to this equation?
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1answer
69 views

finding the position function of a body moving in a medium with velocity $v(t) = -0.2t^2 + 0.7t + 0.5$

Based on an experiment lasting $2.5 \text{ minutes}$, the velocity of a body travelling in a medium may be modeled by the equation $$v(t) = -0.2t^2 + 0.7t + 0.5 \frac{\text{metres}}{\text{minute}}$$ ...
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2answers
26 views

Sketching curves with derivatives. [on hold]

The curve $y = ax^2 + bx + c$ contains the point $(0,5)$ and has a stationary point at $(2,-14)$. Calculate the values of $a, b$ and $c$. I know that stationary points have a gradient of $0$ but I am ...
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5answers
34 views

Maclaurin expansion of $\arccos(1-2x^2)$

Maclaurin expansion of $\arccos(1-2x^2)$ This is what I tried. $f'(x)=2(1-x^2)^{-1/2} \\ f''(x)=2(1-x^2)^{-3/2}+3 \cdot 2 x^2(1-x^2)^{-5/2} \\ f^{(3)}(x)=18x(1-x^2)^{-5/2}+2\cdot 3\cdot 5x^3(1-x^2)^{...
2
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1answer
36 views

Limit comparison test for series

I'm confused by one thing lct states that if you have two series, $a_n$ and $b_n$ if taking the limit of two (where $bn$ is the decent comparison where it's positive for all n meaning it's limit ...
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3answers
41 views

Find the value of $a$ for with $\sum^{\infty}_{n=0}e^{na}=2$

I was given the following problem and told to find the value of $a$: $$\sum^{\infty}_{n=0}e^{na}=2$$I understand that the answer to this question will result in 1+ a number that decreases ...
2
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1answer
60 views

An integral inequality about $f(f(x))$

Let $f:[0,1]\rightarrow \mathbb{R}$ be continuous and monotonically increasing. Assume $f(0)=0$,$f(1)=1$. Prove:$$\int_0^1{f\left( f\left( x \right) \right) dx}\le 2\int_0^1{f\left( x \right)}dx $$...
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2answers
34 views

Sequence converges means it is bounded above and below

What happens in a situation where $\{a_n\}$ has an asymptote, will there ever be a case? Because if there is one then it's no longer bounded.
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1answer
23 views

Modelling infectious diseases without Immunization not yielding intuitive answer

I am trying to predict the spread of a type of infectious disease demonstrated by the following problem. Suppose we have 1000 people and each infected person infects 10% of uninfected people per hour. ...
0
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0answers
34 views

Integration of periodic function

Let $a$ a T-periodic function. It is obvious that we have $$n\int\limits_0^T a (s) = \sum\limits_{i = 0}^{n - 1} {\int\limits_{iT}^{(i + 1)T} a (s)ds} $$ I want to estimate the following quantity$$\...
0
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2answers
71 views

Is there a function f with $ f (x) $ divergent for large x and slower than logarithmic growth rates?

Problem: Is there a function $f:[0,+\infty]\to \mathbb{R}$ with the following conditions: $f(x)\ge 0$ for $x>C$ where $C$ is constant. $\lim_{x \to \infty} f(x)=+\infty$ $\lim_{x\to +\infty} \...
2
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3answers
70 views

Find two continuous functions such that $\int_1^\infty f(x) dx$ and $\int_1^\infty g(x) dx$ converge but $\int_1^\infty f(x)g(x) dx$ diverge

Since $\int_1^\infty f(x)\,dx$ and $\int_1^\infty g(x) \,dx$ are converge, $\lim_{x\to\infty}f(x)=0$ and $\lim_{x\to\infty}g(x)=0$, so $f,g$ are fractional function, and denominator is changing more ...
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3answers
62 views

Trying to understand the following proof of $f(x) = a^x \implies f'(x) = a^x \ln a$

I was reading proof of how to obtain derivative of $a^x$ here (Click on the screenshot below to expand). Although I comprehend the proof, I have several questions about the way the proof is ...
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0answers
63 views

Textbook Error? $\big [e^{\int_{0}^{x} e^{-t^2} dt} \big ]' = \int_{0}^{x} e^{-t^2} dt \cdot e^{-x^2} $ [closed]

Textbook's Answer $ \big [e^{\int_{0}^{x} e^{-t^2} dt} \big ]' = \int_{0}^{x} e^{-t^2} dt \cdot e^{-x^2} $ My Answer $ \big [e^{\int_{0}^{x} e^{-t^2} dt} \big ]' = e^{\int_{0}^{x} e^{-t^2} dt} \...
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5answers
65 views

prove that sequence $\frac{n^2+1}{n+4}$ diverges

Can somebody help me prove this sequence? I've tried using Comparison Theorem and ended up with $a_n = \frac{n^2+1}{n+4} < \frac{n^2}n = n > N = M$. So I choose $N = M$. But I don't think ...