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

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7
votes
0answers
118 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
0
votes
0answers
19 views

Taking partial of a function with two arguments inside the integral

I have a function $ f(y,w) $ which is jointly concave and twice differentiable. Let $ f^{(2)}(\cdot, \cdot) $ denote the partial of $ f $ with respect to its second argument i.e. $ f^{(2)}(y,w) = ...
0
votes
0answers
40 views

Proving a function constant

The problem is to prove f is a constant function given that: $$f:R \to Q $$ is a continuous function. I was hoping to show two cases given a is rational and b is irrational and the equate the two. ...
2
votes
6answers
67 views

Evaluating $\lim_{x\to\infty}\left(\frac{2+x}{-3+x}\right)^{x}=e^5$

Show That$$\lim_{x\to\infty}\left(\frac{2+x}{-3+x}\right)^{x}=e^5$$ How does one reach this result? I keep getting $+3$ in the power and not only $5$.
2
votes
2answers
45 views

Proving and Finding a limit

I need to find the following limit and prove using the definition of limits. $$\lim_{x\to1} {x \over x+1} = \frac 1 2$$. Following the definition: $$\forall \epsilon \exists \delta : \lvert x - c ...
0
votes
2answers
40 views

How to prove the infimum of this subset

consider $A\:=\:\left\{\frac{n-m}{n+m}\:;\:m<n\:\:n,m\:\in \mathbb{N}\right\}$ How to prove that $1$ is the supremum and $0$ is the infimum? I go to the definition of infimum, and see that by ...
4
votes
1answer
64 views

Evaluation of $\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} dt$

I need to get the value of the following definite integral $v\in \mathbb R^+$ $$\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} ...
0
votes
1answer
15 views

surface area of a spherical balloon is increasing at a rate of 100 cm²/s

The surface area of a spherical balloon is increasing at a rate of 100 cm²/s when the balloon has a volume of 100 cm³. Determine the rate at which the volume is increasing at that point.
0
votes
1answer
17 views

How does cos theta equals to dot product of OP and OQ/ [mod(OP)*mod(OQ)]

O is the origin OP and OQ are vectors, for simplicity lets denote P as vector with magnitude 1 facing angle of 45 degrees from positive x axis. Let's denote Q as vector with magnitude 2 facing ...
5
votes
6answers
95 views

How do I prove this limit does not exist: $\lim_{x\rightarrow 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} $

How do I prove that this limit does not exist? $$\lim_{x\rightarrow 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} $$ My attempt: When you approach from from left towards zero , say i take -0.00000000000001 . ...
8
votes
4answers
312 views

Asymptotic behaviour of a sequence with finite sum

I'm working on$$ u_n=\sum_{k=0}^{n}\frac{1}{k^2+(n-k)^2} $$ to find out an asymptotic behaviour as $n \rightarrow +\infty.$ I've already seen that $u_n$ tends to $0$. Thanks for your help.
0
votes
1answer
53 views

How do i prove this limit does not exist : $\displaystyle \lim_{x\rightarrow 0}\frac{1}{x}\sin\left(\frac{1}{x}\right)$

How do i prove this limit does not exist: $$\lim_{x\rightarrow 0}\frac{1}{x}\sin\left(\frac{1}{x}\right)$$ In book they have done using Cauchy criteria for finite limits ,which i dont seem to ...
0
votes
1answer
46 views

How to integrate these two?

For A>0, $$\int^\infty_0 \frac{t*E^{-At}}{t^2+1}dt$$ $$\int^\infty_0 \frac{E^{-At}}{t^2+1}dt$$ My purpose is to derive $$\lim_{A\rightarrow \infty}SinA\int^\infty_0 ...
1
vote
1answer
20 views

Find zeroes of trigonometric polynomial

I know this is a rudimentary question but I'm not really sure how to do this. For my homework problem I have to verify some error term of trapazoidal quadrature. I end up with $$f^{(3)} = -8\sin ...
0
votes
1answer
47 views

Show that the sequence of products $\prod_{k=1}^n (1+1/k^3)$ converges

$$ a_{n} = 1 + \frac{1}{n^3} $$ Show that the sequence is converges $$ \lim_{n \rightarrow \infty} \left(1 + \frac{1}{1^3}\right)\left(1 + \frac{1}{2^3}\right)\left(1 + \frac{1}{3^3}\right) \ldots ...
3
votes
1answer
51 views

$f(x)$ is a differentiable function for $x\in[a,b]$, and $f'(a)=f'(b)$, prove: there is a $\theta$ such that…

$f(x)$ is a differentiable function for $x\in[a,b]$ ($f'(x)$ may not continuously), and $f'(a)=f'(b)$, prove: there is a $\theta$ such that $$f'(\theta)=\frac{f(a)-f(\theta)}{a-\theta}$$ I think we ...
1
vote
0answers
14 views

Proofs regarding about all Second Derivative Test cases (Inconclusive & Single Variable)

This is how I would prove f''(c) > 0 that f(c) has local min and I would easily flip the inequalities and state a conclusion for f''(c) < 0 that f(c) has local max. Quick Proof for f''(c) > 0 ...
1
vote
1answer
30 views

the Fourier transform of a constant

How to calculate the Fourier transform of a constant without the aid of duality property? In other words, how do I calculate $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$
-1
votes
0answers
22 views

Linear approximation of f(g(x)) [closed]

Suppose that the functions f(x) and g(x) are differentiable everywhere. Some values for these functions and their derivatives are given in the table. x f (x) f ′(x) ...
1
vote
1answer
40 views

Solving for C when we have $C\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2=1$

Solving for C when we have $C\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2=1$ $$\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2$$ ...
1
vote
2answers
20 views

Finding all tangent lines that pass through a specific point (not the origin)

I was given the function $y = x^3-x$ and told to find all tangent lines that pass through $(-2,2)$. Not sure what steps to take past finding the derivative.
-2
votes
1answer
42 views

maths- econA reasonably realistic model of a firm’s costs is given by the short-run Cobb- Douglas cost curve C=Tq^1/a+F w [closed]

A reasonably realistic model of a firm’s costs is given by the short-run Cobb- Douglas cost curve $$C=Tq^{1/\alpha}+F$$ where $C$ is total cost, $q$ is output, $\alpha$ is a positive parametric ...
4
votes
1answer
65 views

Then is $f_a$ continious?

Excuse me for the bad title, here's the question Given a differentiable function defined on R. For a given number $a$, $\forall x\in \mathbb R, x\neq a$, by mean value theorem, there exists a ...
0
votes
0answers
13 views

Finding limits of double integrals to contain shapes

I need to find double integral expressions to contain the region in the picture below. In order to do this, my plan was to split the region into two triangles, as indicated. I cannot seem to find ...
0
votes
0answers
34 views

Minimizing average cost through optimization [closed]

A reasonably realistic model of a Firms cost is given by the short-run Cobb-Douglas cost curve: C=T(q^1/a)+F where C is total cost, q is output, a is positive parametric constant, F is the fixed cost, ...
1
vote
0answers
20 views

Triple integral, finding the volume between two planes and a surface in 3D

So I have tried to solve this problem, but I'm running into a problem, because the top circle (intersection of the function with z=1) when you project it onto the xy plane is smaller than the circle ...
1
vote
0answers
34 views

Laplace equation on infinite strip

I'm trying to solve the following problem using the Fourier transform: $$u_{xx}+u_{yy}=0$$ on the domain $\;0\lt y\lt b$ , $-\infty\lt x \lt \infty \;$ with the following conditions: $$ u(x,0)= ...
0
votes
1answer
30 views

Let $a_n \rightarrow a$. Show that $\liminf(a_n-b_n)=a-\limsup(b_n)$.

Assignment: Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences of real numbers with $a_n \rightarrow a \in \mathbb{R}$. Show that: $$\liminf(a_n-b_n)=a-\limsup(b_n)$$ ...
2
votes
2answers
30 views

Every concave function that is nonnegative on its domain is log-concave?

This is a statment from Wiki. I'm not sure why this is true: If: $f(\theta x+(1-\theta y) \geq \theta f( x) + (1-\theta)f(y)$ And $f(\cdot) \geq 0$ then: $$f(\theta x+(1-\theta y) \geq f( ...
1
vote
8answers
130 views

Proof $x=\sin(x+1)$ has one solution in $\mathbb{R}$

I have this problem : Proof $x=\sin(x+1)$ has one solution in $\mathbb{R}$. I got stuck and I don't go how to "move" on. My proof $f(x)=\sin(x+1)-x$ If $f$ is injective function then ...
2
votes
2answers
43 views

Proving $\lim _{n\to \infty }a_{n+1}=\lim _{n\to \infty }b_{n+1}$ where $a_{n+1}=\frac{a_n+b_n}{2}\:$, $b_{n+1}=\sqrt{a_n\cdot \:b_n}$

$a_1,\:b_1>0$ $a_{n+1}=\frac{a_n+b_n}{2},\:b_{n+1}=\sqrt{a_n\cdot b_n}$ The question asks to prove that: $\lim _{n\to \infty }\left(a_n\right)=\lim \:_{n\to \:\infty \:}\left(b_n\right)$. ...
1
vote
0answers
27 views

Show that the derivative of a fraction is always negative.

I'm looking for a neat way to show that the derivative with respect to n of: $$\frac{0.5^n}{\sum_{x\in G}{x^n+(1-x)^n}}$$ is always negative, when G is a finite set such that $x \in G \Rightarrow ...
0
votes
3answers
37 views

Derivative of Equation

So i have this problem with the function: $$U(x)=\frac{A^2}{x^2} - \frac{A}{x}$$ I need to find the derivative of $U$ to find the min and max values. It says in the problem that $A$ is a positive ...
3
votes
2answers
62 views

Can the choice of epsilon be arbitrary in epsilon-delta proofs?

I've been reading Spivak's chapter on limits and something that I don't feel I understand entirely is how the epsilon is decided upon. It makes sense to me in the context of $\,|f(x)-L|<\epsilon$ ...
4
votes
6answers
89 views

Find the $n$th derivative of $f(x) =\frac{x^n}{1-x}$

Question Find the $n$th derivative of $f(x) =\frac{x^n}{1-x}$ What I've managed thus far First I thought that I might be able to discern a pattern by calculating the first few derivatives of ...
2
votes
0answers
37 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
0
votes
0answers
32 views

Derivative norm(d^T*x)

Is the following derivative correct? $$ \nabla ||d^Tx||_2=\frac{1}{||d^Tx||_2}dd^Tx, $$ where $d \in R^n$ and $x \in R^n$. Thanks, Tom
5
votes
2answers
126 views

Evaluation of $-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $

How would I integrate this: $$-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $$ I have tried several methods but can't seem to work this out.
0
votes
1answer
19 views

Properties of exponents with logarithims

Does $$e^{x(\ln(\ln(1/x^2)))}$$ equal $$ e^{\ln(x(\ln(1/x^2)))}$$ ? I'm having trouble taking the limit of $\ln(1/x^2)^x$ as $x$ approaches $0$. I know that I should put the function in terms of ...
1
vote
2answers
87 views

Invent transformation mapping ellipsoid to unit sphere

Invent a transformation that maps the ellipsoid $ x^2+8y^2+6z^2+4xy-2xz+4yz=9$ onto the unit sphere. I don't even know where to begin with this question, any help would be appreciated.
-3
votes
2answers
26 views

Show that $a c\equiv b c\pmod m,\;a. b, c, m \in \mathbb Z$ and $m \geq 2 $ does not imply $a\equiv b \pmod m$ [closed]

Show that $a c\equiv b c\pmod m$ with $a. b, c, m \in \mathbb Z$ and $m \geq 2 $ does not imply $a\equiv b \pmod m.$
-2
votes
3answers
43 views

Evaluating $ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $ [closed]

I'm trying to evaluate this integral: $$ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $$ where $k$ is a real parameter.
-3
votes
0answers
8 views

Approximate length of curve with trapezoidal rule [closed]

This question is from my math teacher: "Approximate the length of of h(x) = $\sqrt x$ between x =1 and x=4 using the trapezoidal rule with 6 sub intervals. Show all work and estimate to3 decimal ...
0
votes
1answer
34 views

Formula for Taylor Polynomials

I'm studying from Calculus: Early Transcendentals, by Briggs & Cochran, and the authors give the definition of Taylor Polynomials as: $p_n(x) = \sum \limits_{k=0}^n c_k(x-a)^k$, where the ...
1
vote
4answers
102 views

Indefinite integral of $\frac{\sqrt{x}}{\sqrt{x}+1}$

For this I tried using the substitution technique, but it got me nowhere near the right answer. What my notepad looks like: $$f(x) = \dfrac{\sqrt{x}}{\sqrt{x}+1}$$ and $$F(x) = \int f(x) = ...
0
votes
1answer
43 views

Optimization to minimize cost using the function C=Tq^(1/a)+F

I was given the function of $C=Tq^{1/a}+F$ where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is the fixed cost, and $T$ measures the technology available (also a ...
4
votes
1answer
38 views

Spivak tough limit proof-verification

Suppose there is a $\delta > 0$ such that $f(x) = g(x)$ when $0 < |x - a| < \delta$. Prove that $\displaystyle \lim_{x\to a} f(x) = \lim_{x \to a} g(x)$. $|f(x) - L| < \epsilon$ for $|x - ...
11
votes
2answers
112 views

How to prove $\large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$

I was given a challenge of sorting the following numbers. $\Large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$. After some work I was able to figure out the order. How can one ...
2
votes
0answers
19 views

Equalities of Integrals with simple expressions

I'm trying to understand why the following equalities are correct: The integral of ($sinx - {\frac{2}{\pi}})^2 $ from $0$ to $\pi$ is exactly the same as the integral $(sinx - {\frac{2}{\pi}}) ...
0
votes
1answer
14 views

Finding radius when performing shell method

Find the volume of the region generated by revolving $y = -x^3$ and $y = -\sqrt x$ around the $x$-axis. I don't understand how the radius component is $-y$; why not $+y$?