Questions tagged [calculus]

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

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40 views

Where this solution of $f ' (x) = g(x)$ comes from?

I would have liked to solve the following first-order linear ODE for $f(x)$: $$ f'(x) = g(x) $$ I attempted to solve it like this: $$ \int f'(x)\,dx = \int g(x)\,dx $$ $$ f(x) = \int g(x)\,dx+C $$ ...
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1answer
44 views

Show that $\int_{0}^x \frac{1}{1+t^4} dt = x - x^5 + x^9 … $ where $\lvert x\rvert$ < 1

Show that $\int_{0}^x \frac{1}{1+t^4} dt = x - x^5 + x^9 ..... $ where $\lvert x\rvert$ < 1 I tried expanding using binomial theorem, but I am unable to prove the series and the integral will be ...
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1answer
51 views

Understanding differential equations — why can't all ODEs be integrated?

I'm in the process of learning about differential equations and something keeps bothering me. I know the idea that a differential equation relates a function and its derivatives and I can do simple ...
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2answers
32 views

can radical and rational equations be linear?

I am a student and I am just started the title of linear equations. I want to learn about linear equations deeply and linear equations make me confused. I have some question to ask you that explain to ...
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0answers
15 views

The nearest point of a curve to the certain point problem

Question: Find the nearest point to point $(3,0)$ and lie on the curve of $y = x^3- 2x^2 + 3$ Honestly, it's not the question what i'm dealing with. I have another question, but the answer is ...
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1answer
17 views

Is this function, containing a cross product continuous?

Is the following function continuous ? f : $\mathbb{R^3}\times \mathbb{R^3} \rightarrow \mathbb{R^3} $ $x,y \mapsto x \times \frac{y-x}{|y-x|}$ for $x \neq y$, else $0$. Now I would have argued that ...
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0answers
15 views

Cardinality of some special sets

Consider a set $\mathcal{R} = \{\{(i,j),(k,l)\}:i,j,k,l\in\{1,2,\ldots,K\},\;\mbox{and}\; i<j, k<l\}$. The cardinality of $\mathcal{R}$ is ${K\choose 2}^2$. The set $\mathcal{R}$ can be written ...
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0answers
31 views

Optimizing a multi-parameter quadratics solution set

So, recently I was doing a physics problem and I ended up in getting this quadratic in middle of the steps: $ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$ So, this is ...
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1answer
20 views

Prove that each partial sum of a convergent series of non-negative terms cannot exceed the sum of the series by elementary calculus

Let $\sum_{n=1}^\infty a_n$ be a convergent series of non-negative terms. And its sum is denoted by $S$. Let $S_k$ be the $k$-th partial sum of the series. I would like to prove that $\forall k\in\...
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2answers
23 views

Given that $x = 5\sin(3t), t\ge 0$: Find the maximum value of $x$ and the smallest value of $t$ for which it occurs.

Given that: $$x = 5\sin(3t), t \ge 0$$ Find the maximum value of $x$ and the smallest value of $t$ for which it occurs. I have figured out the smallest value by: $$\frac{dx}{dt}=15\cos(3t)$$ when $$...
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2answers
37 views

How many maxima and minima an n-degree polynomial can have at most?

There's a statement: "Given a polynomial of degree 6, it may have up to 6 real roots, corresponding to 3 minima and 3 maxima." Is this true in general? How to get the number of maxima and ...
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2answers
58 views

Hard Differential Equation

Can anyone help me to find the solution of this ODE : $$4(y')^2-y^2+4=0.$$ I've tried to find it's solution by putting $y = e^{at}$ (for null solution) and $y = 2$ (for particular solution). My final ...
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1answer
43 views

Find the minimum of the set $A=\left\{\int_0^1(t^2 - at-b)^2 dt\, : \,a,b \in \mathbb{R}\right\}$.

Let $$A=\left\{\int_0^1(t^2 - at-b)^2 dt\, : \,a,b \in \mathbb{R}\right\}\,.$$ Find the minimum of $A$. $\textbf{My attempt:}$ Well, we have $ 0 \leq\int_0^1(t^2 - at-b)^2 dt = \frac{1}{5} - \frac{...
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0answers
21 views

find the area bounded by curve [closed]

suppose G is simple, smooth and closed curve. using line integral of a vector field calculate formula to find the area bounded by G curve.
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2answers
39 views

Length of a curve and calculas

While finding the length of a curve we assume an infinitesimal right triangle, of width $dx$ and height $dy$, so arc length is ${\sqrt{dx^2 + dy^2}}$. But my question is that actually the curve is ...
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2answers
80 views

Find the limit of the sequence $a_n=\frac{1+\sqrt{2}+\sqrt[3]{3}+\dots +\sqrt[n]{n}}{n}$ [closed]

What's the limit $a_n=\frac{1+\sqrt{2}+\sqrt[3]{3}+\dots +\sqrt[n]{n}}{n}$? I know that the limit is 1 (because $\sqrt[n]{n}\to 1$) I do not want to make it with epsilon Cauchy deffinition (I did it). ...
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1answer
31 views

Prove that for b > 0, $\iint_{\mathbb R^2}e^{-b({x^2}+{y^2})} dA = \frac{\pi}{b}$.

Using polar coordinates, prove that for $b > 0$, $$\iint_{\mathbb R^2}e^{-b({x^2}+{y^2})} dA = \frac{\pi}{b}$$ Using $\frac{\pi}{b}$, also prove that $$\int_{-\infty}^{\infty} e^{-b{x^2}} dx = \...
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1answer
67 views

Does this pattern converge anywhere?

Let $$L = \ln(2\ln(3\ln(4\ln(5\ln(6\ln(7 \cdots))))))$$ Then does $L$ converge to any finite value? If yes then how. If no then why? While fiddling around with the calculator I saw that $\ln(2\ln(3\...
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1answer
47 views

Solution of $\int_a^b f(z)g(z)dz=0$

Fix $a\in[0,1]$ and let $b$ an arbitraty value in $[a,1]$. Moreover, there is a positive and continuous function $g:[0,1]\rightarrow(0,\infty)$. I want to find a solution $f$ of $$\int_a^bf(z)g(z)dz=0$...
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1answer
26 views

A sequence-of-function question related to Newton's method

Suppose that $f(x)'>0$ and that $f(a)<0$ while $f(b)>0$, so that $f(x)=0$ has a root $r$ in the interval $(a,b)$. Newton's method of finding $r$, starting at $c$ in $(a,b)$ is as follows: let ...
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3answers
76 views

Evaluating $\lim\limits_{x\to \infty}\left(\frac{20^x-1}{19x}\right)^{\frac{1}{x}}$

Evaluate the limit $\lim\limits_{x\to \infty}\left(\dfrac{20^x-1}{19x}\right)^{\frac{1}{x}}$. My Attempt $$\lim_{x\to \infty}\left(\frac{20^x-1}{19x}\right)^{\frac{1}{x}}=\lim_{x\to \infty}\left(\frac{...
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0answers
33 views

Finding $ \frac{d^n}{dx^n} f_1(x)$ from $\frac{d}{dx} f_{k-1}(x)=f_k(x)-f_{k-1}(x) f_1(x)$

Suppose the following recursive equation holds: \begin{align} \frac{d}{dx} f_{k-1}(x)=f_k(x)-f_{k-1}(x) f_1(x) \end{align} where $f_0=1$. Question: Can we use this recursion to find \begin{align} f^{(...
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1answer
15 views

P(x)=(x-a)^2*Q(x), find Q(x) and show there exists at most one line $\ell$ that is tangent to the graph of $P(x)$ at two places. [closed]

Show that if $P(x)$ is a polynomial such that $P(a)=P'(a)=0$ then there exists a polynomial $Q(x)$ such that P(x)=(x-a)^2 * Q(x) and show that if $P(x)$ is a quartic polynomial then there exists at ...
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1answer
33 views

Is $\lim_{s \to \infty} \int f(x) g(s)dx$ equal to $\int f(x) (\lim_{s \to \infty}g(s) ) dx$?

$$\lim_{s \to \infty} \int f(x) g(s)dx = \int f(x) (\lim_{s \to \infty}g(s) ) dx$$ Is this equality true? Can you move the limit operator inside of the integral, since we're not integrating with ...
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5answers
95 views

How to integrate $\int \sqrt{1-\dfrac{1}{25x^2}}\ dx$?

How to integrate following $$\int \sqrt{1-\dfrac{1}{25x^2}}\ dx$$ What I did is here: $$\int \sqrt{1-\dfrac{1}{25x^2}}\ dx=\int \dfrac{\sqrt{25x^2-1}}{5x}\ dx$$ I substituted $5x=\sec\theta$, $dx=\...
2
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1answer
58 views

so what is the average of the sequence made from $\cos(2)$

so what is the average of the sequence made from cos(2)? $\cos(2)=0.4161468365471423869975682295007621897660...$ the first number in the sequence is the first number in the decimal expansion until it ...
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0answers
25 views

Show that the image of $\lambda$ has zero content in $\mathbb R^n$.

Let $\lambda:[0,1] \to \mathbb R^n$ be a class curve $C^1$ Part by part. Then, the imagem of $\lambda$ has null content in $\mathbb R^n$. My attempt, let $\epsilon > 0$, let's consider a ...
2
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1answer
55 views

Solving $\frac{dy}{dx}=1+(a_mx^m+a_{m-1}x^{m-1}+…+a_0)y^2$

I have a problem with the following equation, $\frac{dy}{dx}=1+P_m(x)y^2$ Where $P_m(x)$ is a polynomial function. I have solution for $P_m(x)=x$ using Mathematica, and Prof @Claude Leibovici solved ...
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1answer
50 views

Does a mean value theorem hold in this case?

Assume you have a function $f(x,y)$ such that $x$ and $y$ can take values on $[0,T]$ but we have to have that $y \ge x$. Assume also that $f(x,x)=0$. For each $x$ we have that $$\lim\limits_{\Delta t\...
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1answer
11 views

How does $\frac d{ds}(Y(s) \cdot s^3 e^{\frac{-s^2}{4}}) =s^3 e^{\frac{-s^2}{4}} (Y' - \frac{s^2-6}{2s}\cdot Y)$?

This arises when solving the differential equation Solve $Y' - \frac{s^2-6}{2s}Y=-\frac1{2x^2}$ The integrating factor turns out to be $s^3e^{\frac{-s^2}{4}}$, and then we multiply the equation above ...
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1answer
69 views

If $A = \lim_{n\to\infty} \left( {}^nC_0\cdot {}^nC_1 \cdot {}^nC_2 \cdot {}^nC_3\cdot \dotsb \cdot {}^nC_n \right)^{\frac{1}{n(n+1}}$, what is $A^2$? [closed]

If $$ A = \lim_{n\to\infty} \left( {}^nC_0\cdot {}^nC_1 \cdot {}^nC_2 \cdot {}^nC_3\cdot \dotsb \cdot {}^nC_n \right)^{\frac{1}{n(n+1}}, $$ then $A^2$ is equal to ... My Attempt: I tried taking ...
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1answer
40 views

What is the relation between the normal derivative and derivative at a point of any function?

Is the normal derivative the average of the derivative at all points of the function? If not what is the mathematical relation between them.
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1answer
41 views

When can we say that a sequence is bigger or smaller than another sequence

Let's say there are two sequences {an}, {bn}. I often see the inequality of {an} > {bn}. But I don't understand how can we make such comparison to determine if a sequence is bigger or smaller than ...
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1answer
31 views

Derivative of distributions inner product

The dot product of two distributions $u(s)$ and $v(s)$, for $s$ the parametric coordinates, is written as $u(s)^{\intercal} \, v(s)$. What would a closed form for the derivative $\frac{\partial}{\...
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4answers
94 views

Does the staircase paradox apply to areas or volume?

So there is the “staircase paradox” that is sometimes used to “show” that $\pi = 4$ (in the case of approximating a circle), or that $\sqrt{2} = 2$ (in the case of approximating the hypotenuse of a ...
6
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2answers
96 views

How do you prove that the derivative $\tan^{-1}(x)$ is equal to $\frac{1}{1+x^2}$ geometrically

How do you prove that the derivative of $\tan^{-1}(x)$ is equal to $\frac{1}{1+x^2}$ geometrically? I figured it out by working it out using implicit differentiation. I also found how to plot a semi-...
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2answers
47 views

Criteria for Pointwise Convergence of Continuous Functions

There are many criteria for uniform convergence of continuous functions, such as Stone-Weirestrass etc... However are there any known results guaranteing that a sequence of continuous functions $\{f_n\...
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0answers
30 views

Do we absolutely need laplace transforms to solve $y'' + ky = F_0 \sin(\sqrt k t)$? Can't we use a different method?

I just saw someone solve $y'' + ky = F_0 \sin(\sqrt k t)$ using Laplace transforms and it was a real slog because it involved the unwieldy transforms of $t\sin(\sqrt k t)$ and $t\cos(\sqrt k t)$ ($k, ...
1
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1answer
55 views

Problem with the parametrisation of this surface integral

I am facing troubles in understanding (read: "guessing") the correct way to parametrise this integral: $$\int_{\Sigma} \dfrac{1}{\sqrt{1 + x^2 + y^2}}\ \text{d}\sigma$$ Where $\Sigma = \{(x, ...
2
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0answers
35 views

Proving that the triple integral can be evaluated by iterated integrals

I am currently in a Calculus III class and we're learning triple integrals. While learning double integrals, I was introduced to Fubini's Theorem, stating the double integral is equal to the iterated ...
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0answers
42 views

limit of sequence proofing [closed]

I am relatively new to Calculus and I have some confusion in solving the question about the sequence below. Can anyone help me with this, please? Thanks a lot! Let $a_n = \left(3^n + 4^n\right)^{1/n}$...
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2answers
36 views

Understanding what is wrong in a limit development

I have the following limit: $$\lim_{x\to -\infty} \frac{\sqrt{4x^2-1}}{x}$$ I know that the result is $-2$ and I know how to achieve it. However on the first try I made the following development and I ...
2
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2answers
117 views

Evaluating $\int_{-\infty}^\infty\frac{\cos(2x)}{x^2+4}\:\mathrm{d}x$

As stated in the title, I want to evaluate the integral $$I=\int_{-\infty}^\infty\frac{\cos(2x)}{x^2+4}\:\mathrm{d}x$$ I'm pretty sure it evaluates to $$\frac{\pi}{2e^4}$$ But I'm not sure how to ...
1
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1answer
47 views

How do I evaluate the line integral $\int_c F\cdot dr$

How do I evaluate the line integral $$\int _c F\cdot dr$$ where $F=x^2$i$+2y^2$j and C is the curve given by r$(t)=t^2+t$ j for $t \in [0,1] $. I have started with: $\int f(t^2$i$+t$j$)$$ dr/ dt $ $...
5
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1answer
95 views

Find the area of the region bounded by $\sin(x)\sin(y)=k$ where $0 \leq x \leq \pi$, $0 \leq y \leq \pi$, and $0 \leq k \leq 1$

On the coordinate plane, the equation $\sin(x)\sin(y)=k$ where $0 \leq x \leq \pi$, $0 \leq y \leq \pi$, and $0 \leq k \leq 1$ forms a closed region. Find the area of this region in terms of $k$. I've ...
0
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1answer
33 views

When getting a function from a instant slope, can't we just put the slope as the slope itself?

I'm looking at MIT's 'The Exponential Function' video and I've seen that when we want to find a function from a instant slope, and I've realized that we need the function to equal the slope. Why can't ...
1
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1answer
32 views

Solving the differential equations as shown below

I recently came across a question in which we had to to solve the set of differential equations: $10dx/dt+x+y/2=0 $ and $6d(x-y)/dt= y$ I tried a lot to solve these equations but I was unable to do ...
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1answer
50 views

Spivak Calculus Chapter 3 Problem 10-(d)

I am currently working through Michael Spivak's „Calculus“ 3rd edition all by myself and came across this Problem which might not be that important at all, but I am still curious to find out more ...
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3answers
49 views

Find the limit of following function x tends to infinity [closed]

How to find root of this equation [2]: https://i.stack.imgur.com/P8vaa.jpg
3
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2answers
48 views

Does $x^n$ belong to $O(e^x)$ for all $n\geq 1$?

My question is essentially two-fold. I've been asked to prove that $x^5 \in$ $O(e^x)$ as $x\to \infty$, and trying to do that I decided to plot some functions of the form $x^n$ next to $e^x$, and ...

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