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

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-3
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0answers
13 views

Elasticity Calculus 1

The demand function for a product is given by p = (−0.05x + 120)/(0.01x + 4) where p is the price per unit when x units are demanded. (a) Determine the intervals on which the demand is elastic or ...
0
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1answer
15 views

Optimization problem: Calculus 1

A company manufactures and sells $x$ units of a product per week. The weekly average cost in dollars per unit is $C =\frac13 x^2 + 9x + 17 + \frac{1552}{x}$ and the selling price in dollars per unit ...
0
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1answer
24 views

if $\lim a_n = \infty$ and $\lim b_n = B$, then $\lim (a_n+b_n) = \infty$

I'm having trouble starting the proof not sure exactly how to go about it. So far I know that for a sequence to go to infinity it means that for all $n >0$ there exists $n_0$ for all $n$ greater ...
1
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1answer
38 views

What is a pullback in simple calculus context?

The definition of a pullback provided by my text is quite accessible Let $\phi : M \to N$, $f:N \to \mathbb{R}$, then $f\circ \phi: M \to \mathbb{R}$, where $\phi^*f = f\circ\phi$ and $\phi^*$ is ...
0
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0answers
41 views

Polynomial: Number of solutions

Functions of polynomials often have more than one solution. For example, $x^2 = b$ with positive $b$ has two solutions for $x$. How does that work for higher polynomials? Say, I have for positive ...
0
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1answer
24 views

show that the sequence $b_n$ is monotone and find its limit

let $b_1 >0$, $b_{n+1} = 3(1+b_n)/(3+b_n)$. show that $\{b_n\}$ is monotone, $0<b_n<3$, and deduce that $\lim b_n = \sqrt 3$. im having trouble showing the monotone part.
0
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0answers
18 views

functions (on intervals) in vector spaces

I'm studying mathematical methods for solving physics and engineering problems. I've looked in a few books and I'm curious about functions being manipulated like vectors. How could I find out more ...
1
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0answers
30 views

prove Gaussian integral using polar cordinates

The proof method is to equate expression$\mathrm{\iint_{-\infty}^\infty\,e^{-(x^2+y^2)}}$ (Cartesian)with $\mathrm{\int_0^{2\pi}\int_0^{\infty}e^{-r^2}drd\theta}$(polar) however, the answer goes ...
7
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2answers
102 views

Solution to $y'=y^2-4$

I recognize this as a separable differential equation and receive the expression: $\frac{dy}{y^2-4}=dx$ The issue comes about when evaluating the left hand side integral: $\frac{dy}{y^2-4}$ I ...
1
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1answer
21 views

What are the basic rules for manipulating diverging infinite series?

This is something that I played around with in Calc II, and it really confuses me: $s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots = \infty$ $s - s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots $ $ \ \ \ \ ...
0
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0answers
12 views

Exercise about max and min of a 2D function with absolute value

I haven't done an exercise like this so, please, tell me if the proceeding is wrong and any kind of observations that you think can help me. Find global max and min of $$f(x,y)=|x^2-y|$$ in ...
2
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2answers
44 views

Example of a limit question requiring infinite applications of L'Hospital's rule to get a result

I'm looking for a limit of the form $\lim_{x \to ?}\frac{f(x)}{g(x)}$ such that any arbitrary number of iterations of L'Hospital's rule results in an indeterminate form and the limit that could (most ...
1
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1answer
34 views

Geometric proof of the Cross Product magnitude

Most proofs of the magnitude of the cross product are algebraic in nature, I find I learn best visually / geometrically. Is there a breakdown of the proof of the magnitude of the cross product using ...
0
votes
2answers
53 views

recurrence relations Solving for $b_n$

Define a sequence by $b_1=\sqrt{2}, b_2=\sqrt{2+\sqrt{2}}$ and in general $b_{n+1}=\sqrt{2+b_n}$ I'm having a hard time solving what $b_n$ is using recurrence relations.
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2answers
41 views

True or False - Convergence

can someone give me some hints about this question - True or False: For all $0<a<1$: $\displaystyle\sum_{n=1}^{\infty}\frac{a}{a^2+n^2}<\frac{\pi}{4}+\frac{1}{2}$
1
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1answer
19 views

Show a Function is strictly monotone Increasing, and what does it say about its inverse?

For example: $$g(x)=x^3-3x^2-1 \quad, \quad x\in [2,+\infty]$$ What I have tried to do was to take the first Derivative. I get $$ g'(x)=3x^2-6x$$ I then check the sign of Derivative of g(x) at ...
1
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1answer
18 views

Existence of a smooth function with given derivative roots

Is there a smooth function $f$ that for all $n\in\mathbb{Z}_+$, $f^{(n)}(n)=0$ i.e. $n$th derivative at the point $n$ is zero and $f^{(n)}(x)\ne 0$ for all $x\in\mathbb R\setminus \{n\}$? If there is ...
1
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1answer
10 views

Function of Jointly Distributed and Convolution

Looking into the continuous case of the sum of jointly distributed RVs in an example in my textbook and there are a few steps missing that I can't seem to wrap my head around. If $X$ and $Y$ are ...
2
votes
2answers
31 views

Determine the set of values of $x$ such that this series converge

Determine the set of values of $x$ such that this series converge: $$\sum^{\infty}_{n=1} \frac{e^n+1}{e^{2n}+n} x^n$$ My work: If $x\geq e$, we have $$\frac{e^n+1}{e^{2n}+n} x^n \geq ...
1
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1answer
17 views

Heat Equation on $[0,l]$ with Neumann boundary conditions

I was reading the following pdf about the heat equation on an interval $[0,l]$ with Neumann conditions, http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf i.e. ...
0
votes
1answer
10 views

Improper integral confusing step

The following passage is in my textbook: $$A(S) = \int_0^{\infty} f(E) \max(S-E,0)dE$$ This simplifies to $$A(S) = \int_0^{S} f(E)(S-E) dE$$ Now this is from a finance textbook so it might ...
5
votes
4answers
73 views

What is an exact characterization for the functions $f$ such that $xf'(x) \leq 2f(x)$?

What is an exact characterization for the functions $f$ such that $xf'(x) \leq 2f(x)$? I know, for instance, that the inequality holds for all functions $f(x) = c_0 + c_1x + c_2x^2$, with $c_0, c_1, ...
3
votes
1answer
31 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
0
votes
2answers
43 views

One point following another moving in a straight line?

There is a plane with two points on it, let's say A and B. A starts at an arbitrary constant point, let's say $(0, 0)$, and $B$ at a point that needs to be tested, which we'll call $(c, d)$. A moves ...
0
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2answers
30 views

Compute the maximum of $|f(z)|$ when $|z| \leq 1$ and $f(z)=\sin (z)$

Compute the maximum of $|f(z)|$ when $|z| \leq 1$ and $f(z)=\sin (z)$ So since $f$ is holomorphic on $|z| \leq 1$, we know we'll find the max of $|f(z)|$ on $|z|=1$. So: ...
0
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1answer
30 views

Spivak Ch1 Proof Critiques

I've started working through Spivak's Calculus. I'm going into senior year after this summer, took the AP Calculus BC test last year, and wanted to get a firmer foundation in calculus before I take ...
0
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1answer
10 views

calculating center of mass of the semicircle which the density at any point is proportional the distance from the center

Assuming the radius is r, and the origin is put on the center of the semicircle. Using polar coordinates. first, because symmetry, the $\bar{x}$ is 0, now trying to find $\bar{y}$: the mass of the ...
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1answer
32 views

Center of circle when three points in $3$-space are given

How do we find center of a circle passing through three points: $ A(x_1,y_2,z_3),B(x_1,y_2,z_3),C(x_1,y_2,z_3) $? Can we minimize $ (d_{OA}+...+... ) $ with condition $ d_{OA}=...=... ,$ ...
0
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3answers
49 views

The maclaurin series $ f(x) =\frac {x^3} {2+ x^2}$

I know we have exams today and I am doing practise since our lecture; said we need to review our Maclaurin series and I found this question and I wanted to know how one would approach it. Find the ...
0
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0answers
36 views

Doubt on the comparison test: can I still evaluate $\lim \limits_{n \to \infty} \frac{a_{n}}{b_{n}}$ if $b_{n}$ might be $0$ for some $n$?

Suppose that $(b_{n})_{n \in \mathbb{N}}$ is a sequence which is not identically equal to $0$ (but which may have elements equal to $0$). Suppose also that I know that $\sum b_{n}$ converges ...
2
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2answers
30 views

Differentiating both sides of an inequality with monotonic functions

If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$? (Note: I've seen several questions asking the same thing without ...
6
votes
2answers
136 views

Why $\lim$ of $\cos(f)$ equals to $\cos$ of $\lim(f)$?

Let $$\lim_{n\rightarrow \infty}\left(\cos\left(\frac{n\pi}{n+1}\right) \right) = \cos\left(\lim_{n\rightarrow \infty}\left(\frac{n\pi}{n+1} \right)\right)$$ Why the $\cos(x)$ function can be ...
2
votes
3answers
46 views

Is the argument I used to evaluate the convergence of the series $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$ right?

If $a,b,c$ be real constants, analyze the convergence of $$\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$$ What I tried to: I compared the general term of my series to $\frac{1}{n}$: ...
1
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1answer
18 views

General theorem about all inflection points

Let $f$ be a function. I know that, if $c\in dom(f)$ and either $f''(c)=0$ or $f''(c)$ is undefined, then $c$ may be an inflection point. Can there be inflection points such that $c\in dom(f)$ and ...
0
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1answer
74 views

How long will it take me to learn calculus? [on hold]

First and foremost I want to point out that I am 14 years old currently and love mathematics. Summer is coming and that means I'll have 2 months to try to learn calculus to as far as I can get ...
0
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0answers
37 views

Suggestion for modern reference on calculus

I need a book for reference in conference paper. Actually, i use Green's theorem: If functions $P(x,y)$ and $Q(x,y)$ satisfy $$\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y}$$ in ...
0
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1answer
18 views

Taylor Polynomial- Choosing A Point

How does the point we choose to develop the Taylor Polynomial has effect on the approximation? I came across Runge's phenomenon, so roughly speaking we can say we should not develop near the ends of ...
0
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0answers
32 views

integral inequalities and continuous functions [on hold]

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
0
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1answer
16 views

maximum of function in bounded area

How can i calculate maximum of $ \frac{-1}{(x+y+3)^{2}} $ in [-1 1]x[-1 1] with non numeric method. I know that -0.2 is maximum of this function with numeric method and The Hesian matrix is zero . ...
1
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1answer
33 views

Wrong derivation of limit of Cesàro mean

It's known that $$\lim_{n\rightarrow\infty}x_{n}=a\Rightarrow\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}x_{i}}{n}=a$$ Consider the following derivation: ...
3
votes
1answer
39 views

Asymptotic ratio of two series

Assume $\{a_n\}$ and $\{b_n\}$ are two positive series such that $$\sum_{n}a_n=\sum_n b_n=1.$$ Assume also for all $n$, $\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k$ and $$\lim_{n\rightarrow ...
0
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1answer
27 views

Non-STEM applications of Calculus? [on hold]

I was talking with a friend, who is a liberal arts major, about the everyday applications of math. We agreed about how Algebra directly applied to the average person's life, but the only examples of ...
0
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1answer
44 views

What situations should $\oint$ be used? [on hold]

Can someone clarify when we should use the contour integration symbol?
0
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2answers
25 views

Yes or No: The following $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ holds

Can someone verify whether $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ is mathematically rigorous?
0
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1answer
25 views

Visualizing Integration

Imagine you have the equation for the circumference of a circle which is $2*\pi*r$ and you want to integrate the perimeter of a circle to get its area. We could do this using an indefinite integral ...
3
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3answers
68 views

simplifying $\int{\sqrt{1-4x^2}}\ dx$

i used the substitution $$x=\frac{\sin{u}}{2}$$ and I got to $$\frac{1}{4}(\frac{1}{2}\sin{(2\arcsin(2x))}+\arcsin(2x))+c$$ and $$2x=\sin(u)$$ and drew a triangle now im stuck... the answer is
0
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2answers
58 views

Integral of $e^{x^3}$

How do I find the integral of $e^{x^3}$. I have to do find the following integral and when I try to do integration by parts, I cannot find the integral of $e^{x^3}$. $$\int x^2 e^{x^3} ...
0
votes
1answer
27 views

Calculating volume by disc integration

What is the volume $V$ of the object created when the area formed by the lines $$y=x$$ $$y = 2-x^2$$ $$0 \le y \le 2$$ is rotated around the $y$-axis? It says that the answer is $\dfrac{5\pi}{6}$. ...
1
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0answers
15 views

Verify the divergence theorem on a ball?

When I verify the divergence theorem on the ball. I got $div(F)=1$, so $\int_{B_R(0)}div(F)dx$ is the volume of the ball, which is $\frac{4 \pi R^3}{3}$. And $n=(x,y,z)/R$, so $\int_{\partial ...
1
vote
2answers
52 views

Antiderivative of $\frac {dy}{dx}$

This is probably a very simple question, but I think its interesting. What I would think, based on my intuition (which I think is correct in this case) is that $$\int \frac {dy}{dx}=y$$ However, ...