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

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0answers
8 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 ...
0
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1answer
18 views

Taylor Polynomial - intuition

How do adding higher derivatives of the function on the same point gives a better approximation?
0
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3answers
31 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 ...
3
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2answers
66 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 ...
1
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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} + ...
1
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0answers
22 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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0answers
17 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 ...
0
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4answers
37 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, ...
0
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1answer
31 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
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1answer
30 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
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2answers
43 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 ...
2
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4answers
67 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
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2answers
52 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
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3answers
59 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 ...
2
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3answers
43 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 ...
0
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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, ...
0
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1answer
27 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: $$\frac{dy}{dx}cos 2x=[2x]'*[cos 2x]'=-2 sin 2x$$ Why isn't it like in other variable derivatives? Like ...
5
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0answers
44 views
1
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5answers
70 views

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

Can someone help me to understand how to find out if this series absolutely convergent and regular converges: $$\sum_{n=1}^{\infty}(1-n\sin\tfrac{1}{n})$$
0
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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 ...
0
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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 ...
1
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1answer
40 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. ...
1
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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 ...
-2
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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 ...
0
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0answers
16 views

Calculating the surface integral of a vector field.

We have a vector field, $\vec{F}$, defined by $F=\nabla \wedge A$ where A is $$A = \left(\begin{matrix} yz^2\\ -3xy \\ x^3y^3 \end{matrix}\right)$$ I get $F$ to be $$F= \left(\begin{matrix} 3x^3y^2\\ ...
2
votes
1answer
44 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?
1
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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: ...
7
votes
3answers
264 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 ...
3
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1answer
42 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 ...
-1
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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?
2
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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
votes
3answers
70 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 ...
1
vote
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: $$\lim_{x\to0}\frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}$$
1
vote
1answer
18 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 ...
0
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0answers
52 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 ...
1
vote
1answer
27 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
votes
2answers
34 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 ...
1
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2answers
34 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
0
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0answers
10 views

proof with value functions

I have a following setting ; $$V_{1}\left(S\right)=V_{1}^{-}\left(S\right)+V_{1}^{+}\left(S\right)$$ where $V_{1}^{-}\left(S\right)$ is the value function before time $T_{1}^{*}$ and ...
1
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0answers
16 views

does this yield convergence to smooth limit function?

let $u: M \times [0,T) \mapsto \mathbb{R}$ with $M$ compact Riemannian manifold (but would also be helpful first to just consider compact $K \subset \mathbb{R}^n$), and $u$ smooth as a function of $M$ ...
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votes
3answers
45 views

Maximum & minimum area of rectangle outside another.

Find the maximum & minimum area of an outer rotated rectangle when the inner rectangle has the side lengths $a$ and $b$. Here's an image: I have already tried to relate the side of ...
3
votes
1answer
50 views

If a polynomial maps a region onto a neighborhood of zero, does it follow that it has a zero in some “robust” sense?

Let $B^n\subseteq\Bbb R^n$ be a unit ball, $P: B^n\to\Bbb R^m$ is polynomial in each component, and assume that the image of $P$ contains $0$ in its interior. Does it follow that for some $\epsilon$, ...
0
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0answers
20 views

follow-up question to Hake's theorem in Bartle's book

My question is based in here. Why is it that $b$ forces to be a tag of $[x_{m-1},b]$? I can't get the right trick. Can you please give me some hints? Thanks
0
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4answers
38 views

Values of $a$ for which $ f(x)=8 a x-a \sin6x -7x - \sin 5x $ increases

Please help me in this question: Find all the values of the parameter $a$ for which the function $f(x)= 8 a x-a \sin6 x -7 x - \sin 5 x $ increases and has no critical points for all real $x$. I ...
2
votes
3answers
81 views

Calculate an integral depending on n

Is there a way (simple or not) to calculate the following integral? $$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$ Thanks
4
votes
1answer
26 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, an example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
1
vote
3answers
50 views

How to solve this particular indetermination: $0*\infty$

The limit in question is: $$\lim_{x\to\infty} 2n\left(\sqrt{n^6+5n^2}-n^3\right)$$ By looking it up on wolfram alpha I found out the answer is 5 but I am not so sure how to arrive to it. I tried to ...
1
vote
3answers
58 views

taking the limit $\lim\limits_{n\rightarrow \infty} {\frac{(3^{n+1} + 4)(7^n-47)}{(7^{n+1}-47)(3^n +4)} }$

I need help with a guide on how i will deal with this kind of problem.. This a part of my solution in series convergence. I find it hard taking the limit of this: $$\lim_{n\rightarrow \infty} ...