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

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1answer
97 views

Does there exist such a pentagon that can be covered by a circle?

Does there exist a pentagon in which every two nonadjacent vertices is connected by a diagonal and the minimal height of every triangle formed by the sides and diagonals of the pentagon whose two ...
1
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1answer
260 views

Electrostatic Potential Energy

How is the boxed step , physically as well as mathematically justified and correct ? Source:Wiki http://en.wikipedia.org/wiki/Electric_potential_energy As work done = $- \Delta U $. for Conservative ...
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1answer
54 views

Creating multivariable functions

Getting ready to go into multivariable calculus and I have a problem given to me by one of the teachers that will be teaching it next year. So here it is! Write a multivariable function (i.e ...
2
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2answers
92 views

Darboux's sums of $f(x)=\frac{1}{x}$

Consider the function $f(x)=\frac{1}{x}$ on $[1,2]$. How to find $\epsilon>0$, s.t for all partition $P$ of $[1,2]$ with $\lambda(P)<\epsilon$, we will have that $$|U_{f,P}-L_{f,P}|<0.01$$ ...
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3answers
132 views

Radius of convergence for $\sum_{n=1}^{\infty} { \frac{1}{n!} \cdot x^{n!}} $

How do we find the radius of convergence $R$ of this power sum? $$\sum_{n=1}^{\infty} { \frac{1}{n!} \cdot x^{n!}} $$ How do we handle the $n!$ as the power of $x$?
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2answers
115 views

Finding the radius of convergence and what does it mean

We have just started learning this and I do not really understand it fully. Given: $$ \sum_{n=1}^{\infty} {(\pi^n + n + 1) \cdot x^n}$$ We should check what is the radius of convergence $R$ and what ...
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2answers
380 views

Finding a relative minimum. (I don't understand how its possible for the answer sheet to be correct)

The function $f$ is given by $f\left(x\right) = 9x^{2/3}+3x-6$ has a relative minimum at $x =$ (A) $-8\;\;$ (B) $-\sqrt[3]{2}\;\;\;$ (C) $-1\;\;\;\;$ (D)$-1/8\;\;\;$ (E) $0$ The answer ...
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2answers
320 views

Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
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2answers
66 views

Turning points on $2\sin x - x$

I'm self teaching and doing a book exercise which asks: "Considering only positive values of x, locate the first two turning points on the curve $2\sin x - x$ and determine whether they are maximum or ...
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1answer
285 views

Fourier Transform on Infinite Strip Poisson Equation

Im trying to solve the following Poisson equation: $$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$ $$u(x,0) = 0,\ u(x,1) = 0$$ $$u(x,y) \to 0\ ...
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3answers
584 views

how to find the value of a function from the first and second derivative.

The function $f$ is twice differentiable, and the graph of $f$ has no points of inflection. if $f\left(6\right)=3,\, f^{\prime}\left(6\right) = -1/2,$ and $f^{\prime\prime}\left(6\right) = -2$ ...
5
votes
1answer
395 views

The Constant Function Theorem first of all $\,$?

I quote Thomas W.Tucker $\,$ "... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
3
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2answers
115 views

What's the value of $\displaystyle y^{(n)}$when $\displaystyle y=\frac{x^n}{(x+1)^2(x+2)^2}$

What's the value of $\displaystyle y^{(n)}$when $\displaystyle y=\frac{x^n}{(x+1)^2(x+2)^2}$? My Try:Let $\displaystyle y_n=\frac{x^n}{(x+1)^2(x+2)^2}$,so $\displaystyle y_n=xy_{n-1}$.According to ...
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2answers
88 views

Integrate $e^{f(x)}$

Just wondering how I can integrate $\displaystyle xe^{ \large {-x^2/(2\sigma^2)}}$ Tried using substitution where $U(x) = x^2$ but I kept getting a $x^2$ at the denominator which is incorrect. I ...
4
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1answer
181 views

Checking $f_n(x) = \frac{nx}{n+1}$ for uniform convergence

I need to check this for uniform convergence $$ f_n(x) = \frac{nx}{n+1} , \quad (x \in \mathbb R)$$ Here's what I did so far. (edited due to a comment telling me the way) $$ \sup_{x \in \mathbb R} ...
3
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2answers
66 views

Derivative of vectors

I know very little about vector calculus. What is the derivative of $\langle\alpha,\alpha\rangle$ (dot product) and $\alpha^TK\alpha$ and $\langle\alpha,y\rangle$. All these derivatives are by the ...
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2answers
104 views

Bound for erf function

For small $\epsilon \geq 0$ Is $erf(\epsilon) \leq \epsilon$ Can somebody give me the hint
1
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1answer
101 views

Calculating Limits Using Polar Coordinates-Phylosophical question!

I'm having difficult times understanding the rules regarding polar coordinates, in the context of calculating limits. On the one hand, I understand that when we take $\theta$ constant, then the path ...
1
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1answer
106 views

position question from velocity and given point.

A particle moves along the $x$-axis so that at any time $t\geq 0$, its velocity is given by $v\left(t\right)=\sin\left(2t\right)$. If the position of the particle at time $t = \frac{\pi}{2}$ is $x = ...
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2answers
130 views

Find a function $F(x)$ whose derivative is $f(x) = \sin(3x)$, without using integrals

Find a function $F(x)$ whose derivative is $f(x) = \sin(3x)$. I'm not supposed to use integrals. How to do this?
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1answer
1k views

$f$ not differentiable at $(0,0)$ but all directional derivatives exist

Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto \begin{cases} 0 & \text{for } (x,y)=(0,0) \\ \frac{x^3}{x^2+y^2} & \text{for } (x,y) \neq ...
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2answers
54 views

Why $\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$

can someone explain me why $\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$ Many thanks
2
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2answers
93 views

Quadratic surface maximization and Hessians

If we have that the contours of a response surface are elliptical and the response is given by the following function: $$\large \exp\left(-\left(w^2 + \frac{1}{4}l^2 -\frac{1}{4} \cdot w \cdot ...
1
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1answer
109 views

An example of differentiability in $\mathbb{R}^n$ everywhere but not at origin.

I came across such problem: $g:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^1$-function with $g(\theta+\pi)=-g(\theta)$ for all $x$. Define a function $f: \mathbb{R}^2\rightarrow \mathbb{R}$ as ...
0
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1answer
75 views

Application of the Multivariate Chain Rule

I'm working on a problem about the application of the multivariate chain rule, and I think I've got the right answer, but I would appreciate if someone could verify what I've done. Problem: $F(x,y) = ...
2
votes
2answers
742 views

May I use the triangle inequality for infinite series?

I have to prove the following statement: Let $\lim_{n\to \infty}r_n=0$. Show that $\forall\varepsilon>0 \ \ \ \exists \, n_0 \in \mathbb N \ \ \ \forall x \in(-1,1):$ $$\left\lvert ...
2
votes
1answer
39 views

Open Mapping In the Reals

This problem is throwing me for a loop. Any help would be appreciated. If $f: \mathbb{R} \rightarrow \mathbb{R}$ and $f(x) = x^3 + 3x^2 + cx + 3$, f is an open map for which values of $c$?
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2answers
336 views

Optimizing the area of a rectangle

A rectangular field is bounded on one side by a river and on the other three sides by a fence. Additional fencing is used to divide the field into three smaller rectangles, each of equal area. 1080 ...
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1answer
93 views

Boundedness of a function

I consider this function of $z=x+iy$ with $y>0$: $$f(z)=\bigg|\frac{1}{\alpha-i{z}}\bigg|$$ with $\alpha> 0$ ($\alpha\in\mathbb{R}$). Is it bounded? making calculation we have ...
0
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1answer
75 views

Finding the continuity of function

A function f is defined by: $$f(x) = \begin{cases} x^4 + 1, \ \text{ if }x < 0,\\ \\ 0, \ \ \ \ \ \ \ \ \ \text{ if } x = 0,\\ \\ x^2 + 1, \ \text{ if } x > 0. ...
2
votes
2answers
92 views

limit of trigonometric function

How to find the limit of this question $$\lim_{x \rightarrow a} \left( \frac{\sin x}{\sin a} \right)^{\frac{1}{x-a}}$$ where $a \neq k\pi$ with k an integer. We can write this as : ...
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1answer
164 views

Testing $\sum_{n=1}^{\infty} {e^{nx} \cos(nx)} \; , \; x \in (-\infty,-1]$ for uniform convergence

Is $$\sum_{n=1}^{\infty} {e^{nx} \cos(nx)} \; , \; x \in (-\infty,-1]$$ uniformly convergent? I said the following: $e^{nx}\cos(nx) \leq e^{nx} \sim \frac{1}{e^{n|x|}} $ because $\cos(nx) \leq 1$ ...
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0answers
34 views

Identity for reducing the power in the parameters of Hypergeometric functions

Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function $ _2F_1(a,b;c;z^d)$ (d is a real) let's say to z? Is there also any identity for ...
1
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1answer
184 views

Rate of Change problem?

The rate of change (in thousands of people per year) of the population of a town between 2000 and 2012 can be modeled by $$R(t) = 1.5e^{0.03t},$$ where $t$ is the number of years after 2000. Assume ...
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2answers
117 views

Testing if $f_n(x) = \sqrt[n]{x}$ converges uniformly.

Given: $$f_n(x) = \sqrt[n]{x} \; , x \in [0,1]$$ Is the above series of functions converge uniformly? and how do we check it?
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0answers
216 views

The volume in cubic meters of water in an aquarium is given by the polynomial $V(x)=x^3-16x^2+79x-120$

The volume in cubic meters of water in an aquarium is given by the polynomial $$V(x)=x^3-16x^2+79x-120$$ If the depth in feet can be represented by y, what are the possible dimensions of the ...
3
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1answer
48 views

Proof for nonexistance of a partial limit

I would like to verify the correctness of my proof for part b of the question (assume part a was proved). Thanks Question: $a_n $is a sequence s.t. $0,2 \in P(a_n)$ and $\forall n \in \Bbb N: ...
2
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0answers
287 views

These unknown uniformly differentiable functions

Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point). Given $\epsilon>0$, choose a partition $P \, : \, ...
1
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1answer
229 views

Calc question on radioactive decay

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12C (a stable isotope), and carbon-14, denoted 14C (a radioactive ...
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3answers
224 views

Differential calculus - Reviewing and drawing graph

I have missed math class for a few weeks and I'm quite behind with the new stuff learned by the others, so I'm stuck with a problem here. The main problem is, I'm going to have hard time explaining ...
3
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3answers
190 views

Show that $(x_n-y_n)$ converges to $x-y$.

Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$. If it's asking about $(x_n+y_n)$. I know that I can ...
2
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1answer
113 views

Line integral of $F = r \times k$ on hemisphere

Exam revision - Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere $r=c$, with $z \geq 0$, where $F = r \times k$ and $k$ is the unit vector ...
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3answers
616 views

infinity times infinitesimal - what happens?

So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration. Also, can we divide infinite number by ...
2
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2answers
113 views

Need help solving $\int x \sqrt{\frac {a^2 - x^2} { a^2 + x^2 }} dx$

I have a complicated integral to solve. Can someone provide a better way to solve it than what i did - dividing by a inside the root, and then putting $ t = x / a $, and then putting $t^2 = \cos ...
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4answers
54 views

Is it always true that $\lim_{x\to\infty} [f(x)+c_1]/[g(x)+c_2]= \lim_{x\to\infty}f(x)/g(x)$?

Is it true that $$\lim\limits_{x\to\infty} \frac{f(x)+c_1}{g(x)+c_2}= \lim\limits_{x\to\infty} \frac{f(x)}{g(x)}?$$ If so, can you prove it? Thanks!
4
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1answer
107 views

Triple Integrals in Spherical Coordinates

Use spherical coordinates to to find the volume of a solid bounded above by $x^2 + y^2 + z^2 = z$ and below by $z$ $=$ $\sqrt{x^2 + y^2}$
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1answer
597 views

Related rates & derivatives/integration

A ladder 10 feet long leans against a vertical wall. If the bottom if the ladder slides away from the base of the wall at a speed of 2ft/sec, how fast is the angle between the ladder and the wall ...
0
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1answer
98 views

Double/Multiple Integrals

Consider a circular lawn with a radius of 10 ft. Assumethat a sprinkler distributes water in a radial fashion according to the formula $$f(r) = \frac{r}{16} - \frac{r^2}{160}$$ (measured in cubic ...
2
votes
4answers
130 views

Finding the limit of function - irrational function

How can I find the following limit: $$ \lim_{x \rightarrow -1 }\left(\frac{1+\sqrt[5]{x}}{1+\sqrt[7]{x}}\right)$$
1
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1answer
84 views

Sine Fourier Series? How do I get to this answer?

Calculate the Sine Fourier series expansion for $\displaystyle f(t) = t^2 $ in $\displaystyle 0 < t < 2.$ I know I need to use $\displaystyle ∑ B_n \sin\left(\frac{nπt}{2}\right)$ I know the ...