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

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5 views

A form for a piecewise continuous function containing only one particular piecewise constant function.

Let A be a continuous function, let B be a piecewise constant function, and let C be a multivariate continuous function. Is it true that $D(x) = C(A(x),B(x))$ could be any piecewise continuous ...
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0answers
19 views

Find an equation to the tangent line to the curve at the given point

\begin{align} x &= \cos t + \cos 2t, & y &= \sin t + \sin 2t, & \left(−1, 1\right) \end{align} Using the above information I found that $\;\frac{dy}{dx}\;$ is: \begin{align} \...
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0answers
10 views

Defining derivatives and integrals for hyperoperations > 2

Derivatives and Integrals are continuous generalizations of the Forward Difference and Summation additive operators respectively. We can do the same with multiplication and get multiplicative calculus ...
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2answers
85 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)}$$ Simple ...
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0answers
17 views

Kline calculus intuitive chapter 3 problem 14

I am starting to get frustrated as I am not able to comprehend these questions because they simply do not make sense to me. Here is the problem and answer to it. https://scienceanswers.wordpress.com/...
26
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15answers
2k views

Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later edit: . .&...
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3answers
77 views

Are there many different power series representation for a given function?

So I have to find the power series representation for $f(x) = \ln (3-x)$. I attempted the following: $$\ln(3-x) = \int {- \frac{1}{3-x} dx}$$ $$ = - \int { \frac{1}{1-(x-2)} dx}$$ $$ = - \int {\...
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8answers
42k views

Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
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3answers
28 views

Slope of a Tangent without given x value

At what point on the graph of $y=-3x^3+2x-1$ is the tangent parallel to $y=2x+10$? Now do I solve this question algebraically or do I solve it graphically since there is no specific x value given to ...
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2answers
20 views

I have to maximize this function involving absolute values

f(x) = $\frac{1}{1+|x|}$ + $\frac{1}{1+|x-2|}$ needs to be maximized. Maximizing this function means minimizing the denominators simultaneously. So I have to find the minimum value of 1+ $|x|$ and 1+...
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1answer
35 views

How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
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1answer
23 views

How can this be minimized?

I have the following function of $x_1$ and $x_2$: $$e(x_1,x_2)= (x_1^2+x_2^2)(a+n)+2a(-x_1+x_1x_2-x_2)+a^2$$ where $a$ and $n$ are real numbers. I want to find the values of $x_1$ and $x_2$ that ...
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3answers
106 views
+100

Position of Object Suspended on a String (Need Another Answer)

I'm going to try to make as few errors in typing this as possible, so please bear with me and ask me to clarify/correct whatever needed. Q: If an object is suspended on a string hung between two ...
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3answers
52 views

Does this $\sum_{n=1}^\infty(-1)^n\tan(\frac{\pi}{n+2})\sin(\frac{n\pi}{3} )$ converge and why? [on hold]

Does the series $\sum_{n=1}^\infty(-1)^n\tan(\frac{\pi}{n+2})\sin(\frac{n\pi}{3} )$ converge and why?
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1answer
13 views

Find all radiuses of convergence for this series - is my approach correct?

I'm supposed to find all radiuses of convergence for this power series: $\sum_{k=0}^{\infty} \frac{k^{2}}{3^{k}}x^{k}$ I've worked with ratio test: $\frac{{}\frac{(k+1)^{2}}{3^{k+1}}}{\frac{k^...
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0answers
14 views

continuous differentiability of a multivariable function

I have a function defined ad $Y=F(X_1,X_2,...,X_N)$, I want to prove that $F$ is continuously differentiable over $X$. Is there any theorems I can use? I tried to calculate the Jacobian matrix and ...
7
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3answers
185 views

Derivative of the magnitude of a vector. Does it exist, or not?

I have a puzzling situation involving derivatives. I want to derivate: $$ \frac{d}{dx}| \mathbf F(x)| $$ This was actually something involving physics. Lets be 2-dimensional for simplicity. Let a ...
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2answers
26 views

What's the series and what's the radius of convergence of this (power) series?

Find the convergence radiuses of this power series: $1 + n + n^{4} + n^{9} + n^{16} + n^{25} + n^{36} + ...$ First of all, I'm surprised it says $radiuses$ instead of $radius$. I know you find ...
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2answers
40 views

Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...
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0answers
27 views

Bounded convergence theorem for Riemann integrals

I will teach a analysis class to some olympiad students this month. The subject is the fundamental theorem of algebra. My approach will be to prove the following "modified" version of the bounded ...
3
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3answers
60 views

How to prove that $\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x)$

It is well-known that: $$\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x).$$ This can also be written as $$ 2\pi\delta(x)=\int^{+\infty}_{-\infty}e^{ikx}\,\mathrm dk.$$ However, I don't know how to ...
4
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3answers
204 views

Integration by parts or substitution?

$$\int_{}^{}x e^x \mathrm dx$$ One of my friends said substitution , but I can't seem to get it to work. Otherwise I also tried integration by parts but I'm not getting the same answer as wolfram. ...
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0answers
20 views

Kline Calculus intuitive approach Chapter 3 problem 12

The problem is as follows : Water drops flow out from a small opening at the rate of one drop per second and fall vertically with an acceleration of 32 ft/sec^2. Determine the distance between two ...
4
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3answers
35 views

Is $b|x|(\sin|x^2+x|)$ differentiable? $b$ can have any real value

So I get that if only $\sin|x^2+x|$ was given it is not differentiable at $x=0$, but why does it become differentiable at $0$ when a factor of $b|x|$ is introduced? And if it does, then is the ...
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1answer
132 views

help verifying equation $\int_0^ x \frac{1}{1+t^n} dt$ [on hold]

As a follow up to a previous posting addressing the integral of $1/ (t^n+1)$ for $n\in \Bbb{N}$ I found the following $$\int_0^ x \frac{1}{1+t^n}\, dt=\sum_{i=0}^{\infty}\frac{(i!)(n^i)x^{in+1}} {(x^...
5
votes
5answers
135 views

Prove that if $f(x) = \displaystyle\int_{0}^x f(t) dt$, then $f(x) = 0$

Prove that if $f(x) = \displaystyle\int_{0}^x f(t) dt$ for all $x$, then $f(x) = 0$. I first differentiated to get $f'(x) = f(x) - f(0)$. Then by the mean value theorem there exists a $c$ in $(0,x)$ ...
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1answer
21 views

simple integration artimethic error

I am trying to integrate a polynomial but I couldn't get the correct answer somehow. I feel like I'm making a mistake when evaluating the integral. $$\pi\int_{-1}^1{1-2x^2+x^4}dx=[{x-{2x^3\over3}+{x^...
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2answers
15 views

Evaluating a statement without calculating the indefinite integral

I'm cramming for a supplementary exam so you might see a ton of questions like these in the 48+ hours to come <3 The question is more of just a yes or no ; Evaluate the statement without ...
1
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1answer
25 views

A function twice differentiable exercise

We are given the function $f=f\left(u,v\right):\mathbb{R}^2\rightarrow \:\mathbb{R}$, a function twice differentiable which has the property: $$\frac{∂^2f}{∂u^2}\left(u,v\right)=\frac{∂^2f}{∂v^2}\left(...
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0answers
16 views

Fourier transform of convolution with additional dependence

There's the well-known identity $$\widehat{f*g}(\xi)=\hat f(\xi)\hat g(\xi)$$ for, say, $f,g\in\mathcal S$. Does anyone know of an extension of this to a situation like $$\mathcal F_x\{[f*g(\cdot,x)](...
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2answers
32 views

How to determine the convergence of the following series?

Good evening to everyone! I have the following series $$ \sum _{n=1}^{\infty }\left(-1\right)^{n }\left|\alpha -1\right|^n\frac{n!}{\left(n+1\right)!-n!+1} $$. I don't know from where to start to ...
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4answers
58 views

Proving $f:\Bbb R\rightarrow \Bbb R$ is onto where $f(x)=\frac{x^3}{x^2+1}$

Proving $f:\Bbb R\rightarrow \Bbb R$ is onto where $f(x)=\dfrac{x^3}{x^2+1}$ Proving it was one to one seemed easy, it seems I'm having difficult finding a way to prove it is surjective. Can I get a ...
2
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2answers
40 views

Intuition behind LHS of squeeze theorem

I am reading the solution to the following problem: Evaluate the limit of $$\lim_{x \rightarrow \infty } \left(\int^{\pi / 6}_0 (\sin t)^x dt \right)^{1/x}$$ The first step was stating that for $t \...
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0answers
26 views

Partial Derivative of Line Integral as a Potential of F

Context to the question: Say $ \{F_{k} \} \to F$ uniformly on a compact subset $K \subset T$, for $ \{F_{k} \}$ a sequence of conservative vector fields and $T$ open and connected. I've shown that ...
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2answers
30 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
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0answers
40 views

Property of a continuous function

Let $f$ - a continuous function and $$\lim_{x \to \infty} \left( \int_0^x f(t)dt + f(x) \right) = 0$$ Prove that $$\lim_{x \to \infty} f(x) = 0$$
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1answer
39 views

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$? Well, in the answer is no. it is written that $e^{x+y}$ for every $(x,y)$ has ...
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2answers
20 views

Example of continuous function with values in a closed set

Give an example of a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and an open set so that $f\left(A\right)$ is not an open set. So basically I need to find a continuous function that is ...
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1answer
48 views

Prove convergence of a sequence.

Let $a_n$ be a series of non-negative real numbers. Suppose $\sum a_n $ diverges. Prove : If $\lim(na_n)$ exists (in $\mathbb R$ or $\infty$), then $\sum \dfrac{a_n}{1+na_n}$ diverges. Thanks ...
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2answers
27 views

Show the uniform convergence of the partial derivatives

Let $f\in C_1\left(\mathbb{R}^n,\mathbb{R}\right)$, is it true that if $[a,b]$ is a closed interval and $\left(x_2,...,x_n\right)$ is fix, then for all $\varepsilon>0$ there exists $\delta$ such ...
0
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1answer
19 views

$A\subseteq R$ is an upper bounded set that contains at least two items/numbers. If $x < \sup{A}$, then $\sup{\left(A\setminus\{x\}\right)}=\sup A$

Prove: $A\subseteq \mathbb{R}$ is an upper bounded set that contains at least two items/numbers. If $x < \sup{A}$, then $\sup{\left(A\setminus\{x\}\right)}=\sup A$. My attempt: Since $A$ is upper ...
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3answers
32 views

Derivatives with different rules

I'm having trouble with this one problem that just deals with deriving. I can't seem to figure out how they got their answer. Any help would be appreciated! Thanks! $ \frac{(x+1)^2}{(x^2+1)^3} $ The ...
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1answer
46 views

Compute definite integral by hand [on hold]

How can I compute $$\int_0^1 \frac{x^3t}{(x^2+t^2)^2} \, \mathrm{dt}$$ by hand?
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1answer
22 views

Find the radius of convergence of this power series

Given: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$ I started by forming it: $\binom{2k}{k} = \frac{(2k)!}{k!*(2k-k)!} = \frac{(2k)!}{k!*k!}$ Now the problem is, I cannot write $2! * k!$ instead of $(...
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0answers
15 views

Finding a constant using limits and a given piece wise function.

The question states Find $\theta$ which makes $f(x)$ is continuous everywhere. Sorry about the notation , I don't have enough time to lookup the piece wise function syntax and I can't recall it :-) $...
5
votes
1answer
97 views

Evaluate $\int \frac {\sin(x)}{x^2 + 4x + 5}dx$

Question: Evaluate $$ \int \frac{\sin(x)}{x^2 + 4x + 5} dx=\int \frac {\sin(x)}{(x + 2)^2 + 1}dx $$ By using the change of variable $y = x + 2$ we have that $dy = dx$ then $$I = \int \frac{\...
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2answers
51 views

Given $f(x,y)$ is a continuous function, Do these integrals equal? [on hold]

Given range $\{ 0 \le x \le 1, 0 \le y \le 1\}$ Do these integrals equal? $\int_0^1(\int_0^y f(x,y)dx)dy = \int_0^1(\int_0^x f(x,y)dy)dx$ Well, the answer is no. It seems like the triangulars are ...
0
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3answers
55 views

How to solve the limit of this sequence

$\lim\limits_{n \to \infty}(\frac{1}{3\cdot 8}+\dots+\frac{1}{6(2n-1)(3n+1)})$ I have tried to split the subset into telescopic series but got no result. I also have tried to use the squeeze theorem ...
12
votes
4answers
237 views

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

I would like to obtain an equivalent form for $$ f(x)=\sum_{n=0}^{\infty} e^{-x\sqrt{n}} $$ as $x \rightarrow 0^+$. I tried without success to "remove" the $\sqrt{\cdot}$ in the summand by summing ...