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

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2answers
38 views

Use the Fourier inversion formula to compute h(x) when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$

Use the Fourier inversion formula to compute $h(x)$ when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(x)=\frac{1}{(1+y^2)^2}=\frac{1}{1+y^2} \times \frac{1}{1+y^2}=\widehat{\frac{1}{2}e^{-|x|}} \times ...
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0answers
32 views

In general, let $f \in L^2(-1,1)$ and let $g:R \to R$, $g(x)=f( \{ x \} )$. How are the Fourier transforms of f and g related?

Recall that $\{ x \}$ is a decimal part of a real number $x$. For example, if $x=3.41$, then $\{ x \}=-0.41$. Part A: Let $f(x)=2x$, with $x \in (-1,1)$ and let $g:R \to R$, $g(x)=2 \{ x \}$. Sketch ...
3
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1answer
1k views

Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts

I'm struggling with the following calculus question. Let there be two functions $f,g : [a, \infty) \to \mathbb R$ such that: $g$ is monotonic, differentiable and has a limit at zero $f$ is ...
0
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2answers
86 views

Proof that a function is continuous in R

I have some trouble with this problem, I'll write what I did Problem: $ f(x) = x^2-2x $ Prove f continuous in $\Bbb R$. My solution: I need to prove that: $$\lim_{x \to x_0} f(x) = f(x_0)$$ Or ...
4
votes
1answer
120 views

find the minimum value of $x^2-6x+9+ \dfrac{64}{x^2}$

Looking for an elegant solution. I can do by brute force, that is finding derivative and double derivative. All Ideas will be appreciated and tried by me.
4
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3answers
366 views

I believe this is a Taylor series. How do I approach it, and what formulas can I use to solve this type of problem?

Suppose that $|x| < 1$. Find the sum of the series $$2x - 4x^3 + 6x^5 - 8x^7 + \cdots$$ I'm not looking for an answer. I want to know how to appropriately solve such a question though.
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2answers
166 views

Evaluate $\int_0^1 \sqrt{2x-1} - \sqrt{x}$ $dx$

I'm trying to calculate the area between the curves $y = \sqrt{x}$ and $y= \sqrt{2x-1}$ Here's the graph: I've already tried calculating the area with respect to $y$, i.e. $\int_0^1 ...
1
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0answers
40 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
2
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3answers
117 views

Prove that $\lim \limits_{x\to \infty}e^{\frac{\ln(x)}{x}}=1$

How to prove that $\lim \limits_{x\to \infty}e^{\frac{\ln(x)}{x}}=1$? I know that $x$ grows much faster to infinity then $\ln(x)$, therefore the limit equivalent to $e^0 = 1$ but that's not a ...
2
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2answers
60 views

Partial Fractions with an irreducible quadratic factor

$\int \frac{2}{(x-4)(x^2+2x+6)} dx$. this is a partial fraction with irreducible quadratic factors. I know how to set it up and I found A, B, and C. 2 = A((x^2)+2x+6) +(x-4)(Bx+C). then I plugged ...
0
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1answer
39 views

(Dis)continuity of function in $R^2$

$$f(x,y) = \begin{cases} a+2x^{2}-b(y-c), & x^{2}>2+x\wedge y<6\\ 3+cx-y, & else \end{cases}$$ $f(x,y)$ is continuous on $R^2$ if $a=-3, b=1, c=2$ I think it's true: insert ...
1
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3answers
149 views

Is a function always a monotonically increasing function

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function? Alternatively, how does the definition of a limit guarantee that ...
2
votes
1answer
134 views

Changing a double integral into polar coordinates

I have the double integral $\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)} dydx$ And I am asked to evaluate this by changing to polar coordinates. I know the transformations are, $x=r ...
2
votes
2answers
102 views

Is $\lim_{x\to 0+} \frac{\ln(x)}{\ln(x)} = \frac{-\infty}{-\infty} = 1$

$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... = \exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$ Now, from continuity we can evaluate each term separately. ...
4
votes
5answers
94 views

Integration problem $\displaystyle \int \frac{dx}{x(x^3+8)}$

$$\int \frac{dx}{x(x^3+8)}$$ I think I'm supposed to use partial fractions, but I am unsure of how to start the problem. Any help would be appreciated.
3
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7answers
449 views

Best Math Plotting Software for Electrical Engineering

I am an electrical engineering undergrad. I would like to learn a math plotting software which would be helpful in visualizing topics in advanced calculus (my immediate need). It would also be helpful ...
0
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1answer
31 views

Using differentials, estimate the difference in the deflection between the point midway on the beam and the point 1 10 ft above it

So I've been trying to figure out the problem for about an hour and I cannot figure it out. Question: To study the effect an earthquake has on a structure, engineers look at the way a beam bends when ...
0
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4answers
85 views

How to find this limit of $\tan(n\pi)/(n-1)$?

I tried to find this limit $$\lim_{n\rightarrow1}\frac{\tan(n\pi)}{n-1}$$ but I couldn't. Here's what I tried $$ \begin{align} \lim_{n\rightarrow1}\frac{\tan(n\pi)}{n-1} &= ...
1
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2answers
54 views

Doubt in integral substitution

I am not able to figure out what substitution to use in the following integral $$ \int \frac{(x-1)e^x}{(x+1)^3}dx $$ Any help would be appreciated.
3
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2answers
140 views

I need to understand why the limit of $x\cdot \sin (1/x)$ as $x$ tends to infinity is 1

here's the question, how can I solve this: $$\lim_{x \rightarrow \infty} x\sin (1/x) $$ Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is ...
2
votes
2answers
44 views

Integral to measure error within 10^-8

If someone could give me background on HOW to solve this problem, NOT THE ANSWER, that would be appreciated. I would love to know how to approach this problem in the most efficient and universal way. ...
2
votes
1answer
131 views

Can't understand a step in the advanced calculus book by thomas P. Dence

On page 9 of the Advanced calculus book by Thomas P. Dence he defines the set $S_1 = \{x\in \Bbb Q: x\geq 0, x^2 \leq 5\}$ and he said that the supremum $U$ is less than $u=3$ and that $U$ is ...
1
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3answers
82 views

Applications of calculus

We have the following formula for area $$A = r^2(\sinθ\cosθ-\sqrt{3}\sin(θ)^2)$$ We then need to find what value θ will give maximum area, so we differentiate to get; $$ ...
0
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1answer
43 views

How to rewrite this radicand as a perfect square.

I'm not sure how to rewrite $$\sqrt{162+81e^{18t}+81e^{-18t}}$$ in the form $\sqrt{\text{something}^2}$. It says to keep in mind that $e^{z} \cdot e^{-z}=1$. Thank you.
2
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2answers
42 views

Function with a removable discontinuity

Is a function with a removable discontinuity considered continuous? Take for example $\frac{x^2-4}{x-2}$. It reduces to $x+2$.
5
votes
2answers
114 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
2
votes
2answers
108 views

Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that ...
3
votes
2answers
488 views

Maximum area of a triangle inside an ellipse.

The question says to find the maximum area of a triangle formed by joining the points $A,B$ and origin $O$, where $A$ and $B$ are points of intersection of an arbitrary line passing through $(4,5)$ ...
1
vote
1answer
43 views

What subject does this fall under (differential curves maybe?)

I found a question where I don't even know where it comes from (except, vaguely, a Calc 2 class--but in my past Calc 2 class I never saw anything like this): Find the angle of rotation needed to ...
2
votes
1answer
48 views

Check the properties of the eigenfunction corresponding to the distinct eigenvalues of an integral equation

Let $\lambda_1, \lambda_2$ be eigenvalues and $f_1 , f_2$ be the corresponding eigenfunctions for the homogeneous integral equation \begin{align} \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) ...
12
votes
1answer
225 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
1
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1answer
31 views

Property of Conditionally Convergent series

If $ \sum a_n$ be an conditionally convergent series.For any real number R, is it true that there exists a sequence$\{b_n\}$ where each $b_i=1 $ or $-1$ such that $\sum a_nb_n$ converges to R?
4
votes
3answers
123 views

Comparison test for sequences?

Let $a_n, b_n$ such that for sufficiently large $n$: $ a_n \le b_n$. Can we deduce that: $\lim_{n\to\infty}a_n = \infty \implies \lim_{n\to\infty}b_n = \infty$ $\lim_{n\to\infty}b_n = L ...
1
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3answers
110 views

Is Inverse Function Composition Commutative?

Given $f: \mathbb{R}\to(-1,1)$ is there a theorem that states $f\left(f^{-1}(x)\right) = f^{-1}\left(f(x)\right)$. In example, is $\tanh{\left(\tanh^{-1}{(x^2)}\right)} ...
1
vote
2answers
111 views

How to find $\sum_{r=1}^{n} r^2\cos {(r\theta)}$

How do you find the sum $$S(\theta)=\displaystyle\sum_{r=1}^{n} r^2\cos {(r\theta)}$$? I observed that if $f(\theta)$ is $\sum\cos {(r\theta)}$, then $S(\theta)=-f''(\theta)$. Help will be ...
2
votes
5answers
207 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
0
votes
0answers
21 views

Calculus the mass of fluid flowing through the hemisphere $x^2 + y^2 + z^2 = 1, z \geq 0$

Part a) The flux density vector is $\vec F = x \hat i -(2x + y) \hat j + z \hat k$ and the unit normal vector points outward. Part b) Solve the exercise if the planar base of the hemisphere is ...
1
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3answers
113 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
0
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2answers
35 views

Associative property for series

Are those equation always valid: $$\sum a_n + b_n = \sum a_n + \sum b_n$$ $$\sum_{k=1}^n(a_{k+1}+a_k)-\sum_{k=1}^na_k=\sum_{k=1}^na_{k+1}$$
0
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2answers
31 views

How to get a partial sum formula

Let S denote sum from 1 to n of (k-1)/k! . I tried obtaining a partial sum formula, but I couldn't get too far. WolframAlpha comes with quite a simple form, but I fail to see how they got there . Can ...
1
vote
2answers
103 views

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$. So if I create a number line marking a-b and a+b. If that the integral above has 5 different answers depending on where (-t,t) is located on the number ...
3
votes
1answer
60 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
2
votes
1answer
126 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
1
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1answer
41 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
0
votes
2answers
20 views

Problem about moving sides of triangle

Imagine a triangle XOY which sides lie on x-axis and y-axis with hypotenuse XY of length 5 m. Suppose the point X moves away from the (0,0) along x-axis with speed = 1 m per second. What speed the ...
2
votes
1answer
44 views

$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$

How to solve this: $$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$$ The answer is supposed to be 39. My attempt: ...
6
votes
8answers
350 views

Evaluate: $\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$

Find the following limit $$\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$$ How do I find this limit? If I had to guess I'd say it converges to $-2$ but the usual things like L'Hôpital or clever ...
0
votes
1answer
50 views

simple vector multiplication

can anyone explain why n1 x n2 is (-1,7,26) I thought it is n1= (5,-3,1) and n2=(2,4,-1) (5,3,1) * (2,4,-1) = (10,12,-1) what am i missing here? here is the example I can't figure out
2
votes
0answers
947 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
3
votes
1answer
1k views

sin(x) infinite product formula: how did Euler prove it?

I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by ...