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

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1answer
57 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 $a,...
2
votes
1answer
183 views

prove Gaussian integral using polar coordinates

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 into ...
7
votes
2answers
138 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
40 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 $ $ \ \ \ \ \...
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0answers
101 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 ...
3
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2answers
85 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 ...
0
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1answer
159 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 ...
1
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2answers
62 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
77 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
25 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 $x=...
2
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1answer
30 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
14 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
49 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 \frac{e^n+1}{...
1
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1answer
65 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
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1answer
19 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
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4answers
87 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
124 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 ...
2
votes
3answers
152 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 ...
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2answers
61 views

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

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: $|f(z)|=|\sin(z)|=|\frac{e^...
0
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1answer
66 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
317 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
48 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}=...=... ,$ ...
-1
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3answers
91 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 ...
2
votes
2answers
114 views

Integral of $\frac{\sin^2(nx/2)}{\sin^2(x/2)}$ over $[-\pi,\pi]$.

I would like to show that $$\frac{1}{n\pi}\int_{-\pi}^\pi \frac{\sin^2(nx/2)}{2\sin^2(x/2)} dx = 1$$ My attempt is very similar to the accepted answer to this question. $$\int_{-\pi}^\pi \frac{\...
0
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0answers
37 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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3answers
70 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 ...
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2answers
176 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
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3answers
61 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 $\frac{1}{n}$: $$\lim \...
1
vote
1answer
26 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 $f''...
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0answers
57 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
24 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 ...
1
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1answer
104 views

How to integrate $\int dx \frac{1}{\cosh^2 x +a^2}$

How to integrate that function $$\int \frac{1}{\cosh^2 x +a^2}dx$$, What I did was rewrite $$\cosh^2 x = ({\frac {{e}^x+{e}^{-x}}{2}})^2 $$ then $$\int \frac{1}{({\frac {{e}^x+{e}^{-x}}{2}})^2 +a^...
0
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1answer
24 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
51 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: $$\lim_{n\rightarrow\infty}\frac{\sum_{i=...
4
votes
1answer
92 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 +\infty}\frac{...
1
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1answer
78 views

What situations should $\oint$ be used?

Sometimes people put a circle through the integral symbol: $\oint$ What does this mean, and when should we use this integration symbol?
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2answers
42 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?
3
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3answers
76 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
99 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} \;\mathrm{d}x$...
0
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1answer
57 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 $\dfrac{5\pi}{6}$. ...
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0answers
28 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 B_R(0)}...
1
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1answer
58 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, ...
0
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1answer
30 views

Expansion for Partial Fractions for $(3-2x)/(x^2+6x+9)$

I'm trying to expand $(3-2x)/(x^2+6x+9)$ into partial fractions to integrate. I'm doing $$(3-2x)/((x+3)^2)=A/(x+3)+B(x+3)^2$$ $$(A(x+3)+B)/((x+3)^2)=3-2x$$ for x=0:$$(3A+B)/9=3$$ for x=1: $$(4A+B)/...
3
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2answers
140 views

from Carathéodory Derivative definition to the derivative of $\sin(x)$

A function $f$ is Carathéodory differentiable at $a$ if there exists a function $\phi$ which is continuous at a such that $$f(x)-f(a)=\phi(x)(x-a).$$ For $f(x) = x^n$, $\phi(x) = x^{n-1} + ax^{n-2}...
0
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2answers
127 views

Let $f(x)=e^{2x}$. The line L is the tangent to the curve of $f$ at $(1,e^2)$. Find the equation of $L$ in the form $y=ax+b$ [closed]

please help ! calculus ! really need to do this for my final exam. HELP its tomorrow
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0answers
60 views

Partial derivatives - Chain rule

Let $f(x, y, z)=e^{xz}\tan (yz)$ and $x=g(s, t)$, $y=h(s, t)$, $z=k(s, t)$. We set $m(s, t)=f(g(s, t), h(s, t), k(s, t))$. Find a formula for $m_{st}$ using the chain rule and verify that the result ...
2
votes
2answers
55 views

Find the maximum and minimum of the function $f$

Find the maximum and minimum of $f(x, y)=xy-y+x-1$ at the set $x^2+y^2\leq 2$. I have done the following: Since the region $x^2+y^2\leq 2$ is closed, $f$ has a maximum and a minimum, which is ...
1
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4answers
64 views

Use the definition of a limit to prove that the limit is equal to zero?

All I can think of to start is to state that: $$|n-∞| < \delta \Rightarrow |(c/n^2)-0| < \epsilon$$ But I don't know where to go from there
0
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2answers
71 views

In complex variables, why is |z-1| < 5 an open disk centered at +1, where the boundary is a circle of radius 5?

How can I justify this basic concept? Use the definition of the modulus? Write z = $e^{i\theta}$? ...and why is |z+1| < 5 ...centered at -1 and not +1? Thanks, Edit: it is always the basic ...
0
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1answer
45 views

Solve the system of differential equations

I plan on adding more into later just a bit stuck, researching it at the moment. Solve the system of differential equations $$\begin{bmatrix} x'\\y' \end{bmatrix} - \begin{bmatrix} -11&15\\ -...