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

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2
votes
0answers
13 views

Supremum of a functional on $C^1([0,1])$

I've tried to solve the following problem (1.1.30 from Berkeley Problems in Mathematics, no solution is provided therein), but I'm not sure if my solution is correct. I wanted to ask you for your ...
0
votes
0answers
14 views

Find a formula for $f(x, y)$ given the following assumptions…?

I've been going through some examples in my textbook ready for a uni exam in a few days, and I am having difficulty with a few of the questions, in particular this one: A gene is a sequence of ...
0
votes
1answer
15 views

For the line $y = mx$, let $m = tan(\theta)$. Write $f(x, mx)$ as a function of $\theta$..?

I have a problem, and I am not sure how to solve it. This is the problem from my book: let $f(x, y)$ be given by the function: $$ f(x, y) = \begin{cases} \frac{2xy}{x^2 + y^2}, & (x, ...
0
votes
0answers
12 views

Hooke's Law - Modelling a 'Bungee Jump'

A question in my textbook asked me to find a general equation for depth fallen by an object of mass $75kg$ thrown from a bridge whilst tied to an elastic rope. Below the bridge there is a stream of ...
0
votes
1answer
13 views

Find $\varphi_{1}$ from $q_{1}$=$A_{1}$sin($\omega$t+$\varphi_{1}$)

I have $q_{1}$=$A_{1}$sin($\omega$t+$\varphi_{1}$) where $q_{1}$=0 and $\dot q_{1}$=$v_{0}$ and I must find $\varphi_{1}$? I know that $\varphi_{1}$ must be zero but i must demonstrate it first. Any ...
0
votes
0answers
14 views

Distance from point to line through Archimedean solid?

I have a problem I am trying to solve, but I am not having very much luck. This is the question: The Archimedean solids are 13 semi-regular polyhedrons whose faces are regular polygons of two ...
2
votes
1answer
43 views

Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
0
votes
2answers
29 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
0
votes
1answer
15 views

solving the area using both axis

Consider a region between the 2 following equations... $x= (y-3)^2 + 3$ and $y= -x^2+5$ bounded by the horizontal lines $y=5$ and $y= -1$. Set up the integral using the y-axis and then set up an ...
0
votes
1answer
28 views

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem.

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem. For what range of values of $x$ will this ...
0
votes
1answer
52 views

Finding an integral. [on hold]

Evaluate $$\!\int (x^5\sqrt{x} + x\sqrt[4]{x})\ \mathrm{d}x$$ My attempt: I tried to factor out a $\sqrt{x}$ and I got $$\sqrt{x}\int\! x^5+x\sqrt[3]{x} \ \mathrm{d}x$$ But here I cannot factor a ...
-2
votes
3answers
33 views

write an expression [on hold]

A word processor determines the width of the body of text on a page. The page is 11 inches wide and has two equal size margins of x inches on each side of the text. Write a formula that gives the ...
1
vote
1answer
24 views

Prove the convergence of sequences

Let $x_{n} = 0 $ if $n < 100 $ and $x_{n} = 1$ if $n \geq 100$ prove $x_n$ converges and find its limit. I started by letting $\epsilon > 0$, as per normal, and choosing $n \geq 100$ as well as ...
0
votes
1answer
18 views

Interpreting a 3 dimensional Graph?

The points P(a, b, c) and Q(2, 3, 5) are symmetric in the sense given. Find a, b, c. About the xy-plane. How does one find the answer to this without graphing. I just do not understand how one would ...
7
votes
2answers
121 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
3
votes
2answers
67 views

Problem with two variable limit where $\lim\limits_{(x,y) \to (-1, 8)} xy = -8$ using only definition

So we just started with two-variable limits. The definition is quite straight forward though my head is still giving it a few spins. I thought doing a couple of examples would help me. Come the second ...
0
votes
1answer
14 views

Using Distance Formula to Graph and Find Equation

My Question : Write an equation for the plane which passes through the point P(3, 1,−2) and satisfies the given condition. Parallel to the xy-plane. How does one use the equation : $$(x − a)^2 + (y − ...
2
votes
0answers
49 views

How to evaluate the integral $\int\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ [on hold]

Please help me in doing this integration. $\int_{0}^{m}\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ where m is a positive number.
3
votes
0answers
38 views

Subdifferential of integral

I am currently trying to extend my knowledge about subdifferentials. Now I am stuck at a particular property of the subdifferential. In this "paper" ...
0
votes
2answers
39 views

Physical interpretation for the curl of a field

I was supposed to compute the curl of a field for a fairly simple assignment and got the following : $$\nabla \times F = (0,0,y-e^{x+y}) \text{ ; } (x,y) \in [0,1]\times [0,1]$$ However, I'm unable ...
1
vote
2answers
62 views

Wolfram Alpha “x = derivative x”

Asking Wolfram Alpha $x = \text{derivative } x$, I was expecting $e^x$, being that the derivative of $e^x$ is $e^x$, Wolfram Alpha however yields $x = 1$. Is this stating that the derivative of a ...
1
vote
1answer
38 views

limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$

What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$ I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
1
vote
1answer
31 views

Analytically Understanding The Definite Integral As A Limit Of Sums

With naive intuition one can obviously see that the definite integral as infinite subdivisions of an area under a curve, within the finite interval "a to b", from which the function of integration ...
0
votes
2answers
29 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
2
votes
2answers
517 views

Very basic calculus question - do you think there's a typo?

I'm helping someone with their homework before they go back to A-Levels, and I came across the following question which I think is miswritten: Find the gradient at $x=1$ of the equation ...
3
votes
2answers
65 views

Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I ...
3
votes
3answers
100 views

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$? It is well-known that $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}$, so ...
0
votes
0answers
12 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
3
votes
4answers
77 views

Show $\frac{\sin(x)}{x}>\cos(x)$ for $0<x<\pi$ using the Mean Value Theorem

I'm trying to show the inequality $$\frac{\sin(x)}{x}>\cos(x)$$ by for $0<x<\pi$ using the Mean Value Theorem, but I don't know how to start. I can show that $\sin(x)<x$, but I can't see ...
1
vote
2answers
44 views

Calculate double integral $\iint_A \sin (x+y) dxdy$

Calculate double integral $$\iint_A \sin (x+y) dxdy$$ where: $$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$$ How to calculate that? $x+y$ in sin is confusing as i do not ...
3
votes
2answers
39 views

Double integral $\int\int_A y dx dy$

Calculate Double integral $$\iint_A y dxdy$$ where: $$A=\{(x,y)\in\mathbb{R}^2 : x^2+y^2\le4, y \ge 0 \}$$ I do not know what would be the limit of integration if i change this to polar coordinates. ...
0
votes
1answer
74 views

Why is it incorrect to integrate by $d(2x)$?

I tried to prove the volume of a cone. If you let the radius be $r$ and let the height be equal to the radius, then all you need to do is integrate the area of a circle with radius $r$ by $dr$. ...
0
votes
0answers
26 views

Bungy Jump Model

Let's say there is a rope that has been designed so that it's modulus of elasticity is known. I have been given the information that the rope is stretched to twice it's natural length when there is a ...
1
vote
2answers
122 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
-1
votes
2answers
30 views

A Crucial Observation On Li's Criterion for the Riemann Hypothesis?

In 1997, Xian Jin Li formulated an interesting criterion whose validity is completely equivalent to the Riemann Hypothesis, namely: Define the real number $(n-1)!\lambda_n$ to be the $n-th$ ...
0
votes
1answer
18 views

Limit of a Monotonic Increasing and Non-Bounded Function

I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution? Let $f$ be a function. The ...
0
votes
0answers
27 views

How does one integrate a function where the numerator is a polynomial of a degree n, and the denominator is a polynomial under root of degree m<n?

How does one integrate a function where the numerator is a polynomial of degree $n$, and the denominator is a polynomial under root of degree $m$ $(m<n)$? A random example being ...
1
vote
0answers
24 views

ramanujan type sum about functional equation

Could you prove the following series numericaly i could not verify the computer take a lot of time $$\sum _{k=1}^{\infty } -\frac{16 x^2 \left(\pi \coth \left(\frac{\pi ^2 (2 k-1)}{2 x}\right)-\pi ...
0
votes
1answer
21 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
0
votes
3answers
64 views

How to prove that this function is integrable on $[0,1]$

Here I tried to find two step functions, one of them is less than $f$ on $[0,1]$ whereas one of them is greater than $f$ on the same closed interval, to prove this function is Riemann-integrable on ...
0
votes
1answer
40 views

Number of solutions of the differential equation ${dy}\over {dx}$=$y^{1/3}$ $y(0)=0$

The given differential equation is ${dy}\over {dx}$=$y^{1/3}$, $y(0)=0$ I got the solution $$y^{2/3}={{2}\over {3}}x$$ $$i.e. y^{2}={{8}\over {27}} x^{3}$$ $$i.e. y= \pm \sqrt{{{8}\over ...
2
votes
1answer
67 views

Is there a way of solving integrals where the numerator is an integral of the denominator?

Is there a way of solving integrals where the numerator is an integral of the denominator? I was evaluating the integral $$\int \frac{x-\sin x}{1-\cos x}\mathrm{d}x$$. I separated the numerator into ...
-2
votes
0answers
9 views

sturm liouville Problem finding function eigenvalues given [on hold]

find a problem whose eigenvalues are 1, cosx, cos2x effort done: calculated ao, a1, a2.... using Fourier series formula
-3
votes
0answers
27 views

Verification of an indefinite integral with trigonometric functions [on hold]

I was making this integral $\int \frac{dx}{\sin(x) + \cos(2x)}$ and i end up with this result: $\frac {2}{\sqrt3}\ln({\frac{\tan(x/2) + 2 -\sqrt3}{\tan(x/2) + 2 +\sqrt3}})\ - \frac ...
0
votes
3answers
27 views

Solving ODE y'(x)=2 x y(x), using power expansion

I have this equation: $$y'(x)=2 x y(x)$$, I want to solve this ODE with differential equation with power expansion. I get a problem cause I do not get how to equate the coefficients. $$2 x ...
2
votes
1answer
28 views

Circle Packing, Estimate only of number of smaller circles in a circle.

Given x number of circles of radius r what is a good approximate size Radius for a bigger circle which they fit in. To explain in actual problem terms. I want to move units in a video games which ...
0
votes
3answers
46 views

Solving for a Limit Given a Limit

$$ \text{Given}\; \lim_{x \to 1} \frac{f(x)-4}{x-1} = 10, \;\text{evaluate}\; \lim_{x \to 1} f(x) $$ I'm wondering if anyone can give me some tips on how to approach this problem. I ...
11
votes
3answers
150 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
1
vote
2answers
35 views

Find Taylor series for $f(x)=e^x$ at $c=3$. Then simplify the series and show how it could have been obtained directly from the series $f$ at $c=0$.

Find the Taylor series for $f(x)=e^x$ about the point $c=3$. Then simplify the series and show how it could have been obtained directly from the series for $f$ about $c=0$. Taylor's Theorem: ...
1
vote
0answers
31 views

does the equivalence class of an element in a set is the set itself?

what does equivalence class mean? I am trying to understand I think that I am a little confused so let's take this example: Suppose that we have the relation $2x+3y$ is a number less than or equal ...