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

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-1
votes
2answers
60 views

Find the value of the integral $\int_{-\infty}^{\infty} \frac{1}{1+(a-y)^2} \frac{1}{1+y^2} dy$

Can someone help me with the evaluation of the integral \begin{equation*} \int_{-\infty}^{\infty} \frac{1}{1+(a-y)^2} \frac{1}{1+y^2} dy? \end{equation*}
0
votes
0answers
26 views

what would the equation of a torus be by making the circunference $(y-2)^2+ z^2 = 0$ and $x=0$ turn along the $z$ axis

What I understand of the question is that I have to, somehow, give the equation of the torus that results of spinning the circumference $$(y-2)^2 + z^2 = 0$$ and $$x=0$$ which as far as I know is just ...
27
votes
2answers
978 views

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral: $$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$, and lookups ...
1
vote
3answers
76 views

prove that $\lim_{x \rightarrow 0^+}\frac{1}{x} \int_0^x\sin(\frac{\pi}{t})dt =0$ [closed]

I want to show that \begin{equation*} \lim_{x \rightarrow 0^+}\frac{1}{x} \int_0^x\sin(\frac{\pi}{t})dt =0. \end{equation*} Any idea?
5
votes
2answers
191k views

Calculus question taking derivative to find horizontal tangent line

How would I solve this problem? Find the point where the tangent line is horizontal in the following function: $$f(x)=(x-2)(x^2-x-11)$$ I computed the derivative: $\quad ...
-2
votes
1answer
67 views

How to integrate a function with a nested absolute value: $|x^2 - 2|x||$? [closed]

I need help with this problem, $$\int_0^4|x^2 - 2|x||dx$$ what should I do with $2|x|$ ?
3
votes
7answers
296 views

Find the absolute maximum and minimum value of $f$

I have to find the absolute maximum and minimum value of $$f(x, y)=\sin x+\cos y$$ on the rectangle $[0, 2\pi] \times [0, 2\pi]$. I have done the following: $$\nabla f=(\cos x, -\sin y)$$ ...
1
vote
2answers
36 views

Second order differential equations where rhs $= 6e^2\cos(3x)$

Solve the differrential equation $$y'' - 4y' + 13y' = 6e^{2x}\cos(3x)$$ where $y(0)=3$ and $y'(0)=-8$ I think we start like... For the homogenous case $$\lambda^2 -4\lambda + 13 = 0 $$ ...
3
votes
3answers
106 views

evaluate the sum $\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$

I'm trying to evaluate this sum $$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$ I have no idea how to deal with it. With one sum I can, with partial-fraction decomposition, ...
-1
votes
1answer
61 views

Theorem of Lagrange multipliers - Extremas of $f$ [duplicate]

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are ...
0
votes
1answer
522 views

Differential of the greatest integer function

So I know that the derivative of the greatest integer function is zero. That is if $f(x) = [x]$ then $df/dx = 0$. Then, a friend asked me for the differential , $df$ of $f(x)$. My answer was zero. He ...
-1
votes
1answer
22 views

continuous functions and limit existance

Let, $C\in \mathbb R$ and let $f(x)= Cx^2+1$ if $x \geq 2$ , $f(x)= 10-x$ if $x<2$ for what value of $C$ is $f(x)$ a continuous function.
0
votes
4answers
51 views

First order differential equation: did i solve this equation right

So i'm trying to solve: $$x^2\frac{dy}{dx} + 2xy = y^3$$ I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ I think i've solved it and ...
0
votes
0answers
23 views

Optimal Space-Travel Departure Time (Issues deriving and solving complex expressions).

Problem This problem aims to determine the optimal time to depart for an intergalactic destination, taking into account the fact that in a number of years technology back on the planet you left may ...
3
votes
4answers
709 views

Prove that limit doesn’t exist anywhere? [closed]

I'm doing some practice problems and am having trouble answering these problems: Consider the following function $$f(x)=\begin{cases}1, & \text{if } x\in \Bbb Q\\ -1, & \text{if } x\in \Bbb ...
1
vote
1answer
39 views

Apply chain rule to $u = y^{1 - n}$ in order to find $\frac{du}{dx}$

Let $u = y^{1 - n}$. I know that, by using the chain rule: $$\frac{du}{dx} = \frac{du}{dy} \cdot \frac{dy}{dx}$$ Also, I know that $\frac{du}{dy} = (1 - n)y^{-n} = \frac{1 - n}{y^{n}}$ Now, for ...
6
votes
2answers
105 views

A simple way to find $\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$

I was reading an exam paper used to identify gifted high-school students, and I encountered the following problem: $$\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$$ Using ...
5
votes
3answers
317 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
1
vote
1answer
64 views

Why is the chain rule applied to derivatives of trigonometric functions?

I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives? ...
0
votes
0answers
32 views

continue on some strange summation formulas ..by william Gosper

could you show if is it true the following expressions? $$\sum _{z=1}^{\infty } \frac{(-1)^z \cos \left(\sqrt{\pi ^2 a^2+b z^2+c}\right)}{z^2}=\frac{b \sin \left(\sqrt{\pi ^2 a^2+c}\right)}{4 ...
1
vote
1answer
44 views

First order differential equation: how do I prove that $u$ satisfies the differential equation

So I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ now it says: Show that if $y$ is the solution of the above Bernoulli differential ...
1
vote
2answers
40 views

Deriving energy equation (Kinetic)

A particle of mass $m$ moves on the $x$-axis under a force $$F(x)=-2x+2\epsilon x^2$$ Use newton's second law, $F=m\ddot x$ to derive the energy equation $$\frac{1}{2}m\dot x^2+V(x)=E_0$$ where ...
3
votes
2answers
50 views

How to precisely define $C^\infty$ in $f(x) \in C^\infty$

In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$ Is there a more rigorous way to define ...
0
votes
2answers
309 views

Determine if z is a function of x and y. $6x-4y+2z=10$

"Determine if z is a function of x and y. $6x-4y+2z=10$. Find the formula" All i did was equate for z $$z = 5-3x+2y$$ That is the formula. And It's pretty obvious that the answers are unique but i ...
0
votes
0answers
101 views

How fast is the distance between two points changing.

I am having a difficulty with the following question from my calculus unit. Bus station A is located 100km west of bus station B. At 12pm a bus leaves station A driving south at 70km/h and a bus ...
1
vote
1answer
34 views

Taylor Polynomial - intuition

How do adding higher derivatives of the function on the same point gives a better approximation?
3
votes
3answers
310 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
3
votes
2answers
107 views

Why are radians used in calculus. [duplicate]

Ok, please ignore my silliness. So, why do we use radians in calculus and why is it considered more scientific than degrees. And how did mathematicians know or prove that radians would work for all ...
2
votes
1answer
30 views

Shortest Path with Constraint

What is the length of the shortest path that goes from $(0,2)$ to $(12,1)$ that touches the $x$-axis? I tried using calculus to solve this problem (i.e.: distance is: $$ \sqrt{(x-0)^2 + (0-2)^2} + ...
-2
votes
1answer
119 views

What radius circle to remove from unit circle to make golden earring?

A circular lamina of radius $x$ is removed from a circular lamina of radius $1$. If the center of gravity is at the edge of the smaller circle (along the line connecting the two centers) what is $x$? ...
0
votes
1answer
47 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
0
votes
4answers
69 views

Big-O notation — is it mainly used to classify rate of growth or rate of decay to zero?

For example, $e^{x} = 1 + x + x^2/2 + O(x^3)$, and we interpret $O(x^3)$ as the remainder term that goes to zero like $x^3$. What's the primary usage of Big-O notation? (strictly in math classes, ...
3
votes
1answer
66 views

Show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is definable in $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f)$

For the structure $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f), n_f=1 $ show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is a definable set. My issue here is how to ...
12
votes
4answers
289 views

Prove that $\sinh(\cosh(x)) \geq \cosh(\sinh(x))$

Prove that $$\sinh(\cosh(x)) \geq \cosh(\sinh(x))$$ I tried to tackle this problem by integrating both lhs and rhs, in order to get two functions who show clearly that inequality holds. I've ...
1
vote
0answers
66 views

How to compute the unit outer normal at the point in a curve?

Given a smooth closed curve $f(x,y)=0$, How to compute the unit outer normal at each point $(x_{0},y_{0})$ in the curve?
1
vote
1answer
105 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
2
votes
4answers
113 views

Why do we need $\sup$ and $\inf$ when we have $\max$ and $\min$. [duplicate]

In my analysis text, it seems that $\max$ and $\min$ are replaced by $\sup$ and $\inf$ for 1D single variable function, why is this the case?
0
votes
2answers
179 views

Why is $f'(c) = \text{does not exist}$ a critical point?

In my lecture the prof wrote that when the derivative does not exist at a point it is also a critical point I can understand that $f'(c) = 0$ indicates that we have a flat place on our curve, so ...
2
votes
3answers
79 views

Real Methods to Evaluate $2 \int_{-1}^{1}x^2 \sqrt{1-x^2}dx$

I was recently contacted by a friend to find the values of the two following integrals by any means. $$ I=2\int_{-1}^{1}x^2 \sqrt{1-x^2}dx$$ $$ J=\int_{-1}^{1}(1-x^2) \sqrt{1-x^2}dx$$ The first ...
0
votes
2answers
271 views

volumes solving for dx or dy

The only problem I have with this is knowing when you are solving for dx or dy. For example, this question which says find the volume of the solid created by rotating the region bounded by y = 2x-4, ...
1
vote
1answer
43 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
2
votes
4answers
234 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{n=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
0
votes
2answers
49 views

What is the derivative of $\arcsin(x/4)$?

I tried it and got $\frac{1}{4\sqrt{1-\frac{x^2}{16}}}$ But WolframAlpha is saying that the correct answer is $\frac{1}{\sqrt{16-x^2}}$ What did I do wrong, and what is the correct way of solving ...
1
vote
1answer
43 views

Series convergence and Big O

I am trying to prove that if there exists $\theta \in \mathbb{R}$ such that $f(n) = \mathcal{O}(n^{\theta})$, then $\sum\limits_{n=1}^\infty \frac{f(n)}{n^s}$ converges. Intuitively it makes sense ...
5
votes
0answers
81 views

Exponential integral equation

I want to find the function $q(x)$ such that the following integral equation is satisfied for all $s\ge 0$: $$ \frac{p}{as + 1}\int_0^\infty \exp \left( -\frac{xs}{as + 1} \right)q(x)dx = ...
0
votes
2answers
798 views

Cube Root function not differentiable

Why is the cube root function not differentiable at x=0? I graphed it and the curve looks a bit vertical at that point, is that why? Can someone give a good explanation please.
2
votes
1answer
225 views

Application Stokes's Theorem

I am a bit unsure the way Stoke's theorem is applied in this case. Evaluate $\oint\limits_C {xydx + yzdy + zxdz} $ around the triangle with vertices $(1,0,0), (0,1,0), and (0,0,1)$, oriented ...
6
votes
7answers
2k views

The box has minimum surface area [duplicate]

Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube. Could you give me some hints what we are supposed to do?? $$$$ EDIT: Having found that for ...
1
vote
1answer
28 views

Finding Recursive Function

Let $f(x)=e^\frac{-1}{x}$ Prove in induction that the general form of the n-th derive is: $$f^{(n)}(x)=P_n(\frac{1}{x})\cdot e^\frac{-1}{x}$$ For $n=0$: $P_0(x)=1$ Assume for n: ...
3
votes
3answers
165 views

Finding $\lim_{x \to 0^+} x^{\sin x}$

Find $\displaystyle \lim_{x \to 0^+} x^{\sin x}$ This is how I started but I get to a dead end fast: $\displaystyle\lim_{x \to 0^+} e^{\ln x^{\sin x}}=\lim_{x \to 0^+} e^{\sin x \ln x}$ I ...