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

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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 ...
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4answers
105 views

Prove that limit doesn’t exist anywhere?

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 ...
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2answers
51 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 ...
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13 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 ...
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1answer
51 views

How to find $\frac{0}{0}$ limit without L'Hôpital's rule

I am having trouble solving this limit. I tried applying L'Hôpital's rule but I got $\frac{0}{0}$. $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + ...
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1answer
36 views

How to find bounds of this integral $\int_0^{10} \frac{x}{\sinh \frac{x}{2}}dx$

How to find bounds of this integral: $$\int_0^{10} \frac{x}{\sinh \frac{x}{2}}dx$$ I try but I get that integral not converges. Thank you.
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1answer
32 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 ...
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2answers
19 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 ...
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1answer
34 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 ...
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152 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 ...
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0answers
34 views

Is $lim$ an operator? [duplicate]

In my calculus I lecture notes the prof said that $lim$ satisfies the properties of linearity as well as multiplicity. This looks like what an operator might do. Can we characterize $lim$ as an ...
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2answers
30 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 ...
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29 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 ...
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1answer
22 views

Taylor Polynomial - intuition

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

What is the volume and surface area of the 1-Sphere?

I am reading a post on here that mentioned something about the 1-sphere. I know that a 2-sphere is a circle, and 3-sphere is a volume, but what is this 1-sphere and how do you calculate the volume and ...
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2answers
68 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 ...
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1answer
13 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} + ...
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0answers
25 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?
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18 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 $$(y-2)^2 + x^2 = 0$$ and $$x=0$$ which as far as I know is just ...
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4answers
39 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, ...
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2answers
39 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 ...
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1answer
31 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 ...
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2answers
46 views

How should I plan to study/prepare for Calculus One this summer (I know this has been asked before, but my situation is a bit unique)? [on hold]

I took Calculus 1 in the fall semester last year believing that I was going to ace it because of how good I was at math in high school (got an A in every math class up to and including PreCalculus ...
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4answers
69 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?
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2answers
55 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 ...
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3answers
62 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 ...
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3answers
46 views

Improper Integral: $\int_{-\infty}^\infty\frac{e^{-t}}{1+e^{-2t}}\ dt$

$$\int_{-\infty}^\infty\frac{e^{-t}}{1+e^{-2t}}\ dt$$ I have the antiderivative as $$-\arctan e^{-t}$$ but when I do it out, I end up getting $$-\frac\pi4 + 0 - \frac\pi2+\frac\pi4$$ However, I ...
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2answers
34 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, ...
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1answer
38 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? ...
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57 views
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5answers
78 views

Convergence of $\sum_{n=1}^{\infty}(1-n\sin\frac{1}{n})$ [on hold]

Can someone help me to understand how to find out if this series absolutely convergent and regular converges: $$\sum_{n=1}^{\infty}(1-n\sin\tfrac{1}{n})$$
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3answers
46 views

Simple calculus question (limits)

So I have to calculate the following limit $$\lim_{u\downarrow 1}\frac{\frac{2u}{3}-\frac{2}{3u^2}}{2\sqrt{\frac{u^2}{3}-1+\frac{2}{3u}}}.$$ I tried to use L'Hopitals rule, but it doesnt work it ...
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2answers
43 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 ...
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1answer
41 views

Approximate the value of the integral with an error less than $ 10^{-3}$

Approximate the value of the integral with an error less than $ 10^{-3}$ [Do not add the numbers in the sum!] $$\large \int_0^1 \sin (x^2)dx$$ So this is what i have tried and am stuck from there. ...
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1answer
24 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 ...
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1answer
35 views

Find the nth derivative of $x/(x^2 +1)(x+2)$ [on hold]

Find the nth derivative of $\dfrac{x}{(x^2 +1)(x+2)}$, Pls show me the step by step solution. I got the partial fraction decomposition as $\dfrac{2x+1}{5(x^2 +1)} + \dfrac{2}{5(x+2)}$. Can't figure ...
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1answer
47 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 ...
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0answers
48 views

Problem 6 of calculus [on hold]

I am having a hard time on problem 6 in the calculus book. How do you arrive at this result?
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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: ...
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3answers
272 views

solution to differential equation from deriving power series

Find the solution of the differential equation $$y'= 2xy$$ statisfying $y(0)=1$, by assuming that it can be written as a power series of the form $$ y(x)=\sum_{n=0}^\infty a_nx^n.$$ Im advised to ...
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1answer
46 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 ...
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1answer
17 views

Concavity and quasiconcavity… [on hold]

How do you explain the difference between concavity and quasi concavity? or convexity and quasi convexity?
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1answer
32 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
3
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3answers
71 views

Solving with integration by parts: $\int \frac 1 {x\ln^2x}dx$

Solving: $$\int \frac 1 {x\ln^2x}dx$$ with parts. $$\int \frac 1 {x\ln^2x}dx= \int \frac {(\ln x)'} {\ln^2x}dx \overset{parts} = \frac {1} {\ln x}-\int \frac {(\ln x)} {(\ln^2x)'}dx$$ $$\int ...
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1answer
22 views

Proving Two Taylor Polynomials Are Equal

I am trying to prove the Following: Let there two polynomials: $p(x),q(x)$ at a degree on $n$ at most, and $$f(x)=p(x)+o(x-x_0)^n=q(x)+o(x-x_0)^n$$ therefore $p(x)=q(x)$ I have come to the ...
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4answers
34 views

Limit of a rational function with radicals [on hold]

How do I solve this limit: $$\lim_{x\to0}\frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}$$
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1answer
19 views

How we can find $A_{(\Gamma_f)}$?

We have $f,g:[-4,4]\rightarrow\mathbb{R}$, $f(x)=x^2+2$ and $g(x)=x+4$. We need to find the crowd area between the graphs f and g. I know that $A_{(\Gamma_f)}=\int_a^b|f(x)|dx$ but in this case how ...
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0answers
53 views

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ using elementary, high school techniques [duplicate]

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ $$$$ I was given this integral by a friend who saw this here on MSE. He asked me if I could solve it using the very ...
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1answer
28 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
2
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2answers
35 views

nonnegative Riemann-integrable function, infimum

$f$ is a nonnegative Riemann-integrable function on $(0,1)$ and $f(x)\ge\sqrt{\int_0^xf(t)dt}$ for $x\in(0,1)$. Find $\inf\frac{f(x)}{x}$ I have no idea how to work out the assumption, for equality ...