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

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6 views

If $f:\mathbb{R}\to\mathbb{R}^2$ is of class $C^1$, show that $f$ does not carry $\mathbb{R}$ onto $\mathbb{R}^2$

If $f:\mathbb{R}\to\mathbb{R}^2$ is of class $C^1$, show that $f$ does not carry $\mathbb{R}$ onto $\mathbb{R}^2$. In few words I have to show that $f(\mathbb{R})$ contains no open set of ...
3
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0answers
27 views

Solve trigonometric integral

Please help me to solve the following integral: $$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx.$$ I have tried a lot, but no results. I only transformed this integral to the ...
2
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1answer
10 views

Validating inexistence of a Max in a continuous function

In my assignment I have to prove the following statement: Let $f$ be a continuous function which satisfies the following: For every $x \in \Bbb R $ there is $y>x$ such that $f(y)>f(x)$ ...
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3answers
40 views

Find $\lim_{x\rightarrow 0} \dfrac{36^x-9^x-4^x+1}{\sqrt 2 -\sqrt{1+\cos(x)}}$

What is the limit of this function as $x \rightarrow 0$ ? $$\lim_{x\rightarrow 0} \frac{36^x-9^x-4^x+1}{\sqrt 2 -\sqrt{1+\cos(x)}}$$ I have tried simplifying the expression in different ways. ...
2
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2answers
19 views

Using spherical coordinates to find volume of a region

Use spherical coordinates to find the volume of the region lying above $z = \sqrt{3x^2+3y^2}$ and within the $x^2+y^2+z^2=2az$, $a>0$. So far I know that the first graph is a cone and the second ...
3
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3answers
42 views

Wave equation $u_{xx}+u_{xt}- u_{tt}=0$

Does anybody know how we can solve the equation $u_{xx}+ u_{xt}- u_{tt}=0$ with $u(x,0):=g(x)$ and $u_t(x,0):=h(x)?$ I mean it is known how to do this for the wave equation see here but I don't know ...
2
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1answer
27 views

limit question - $\lim_{x\rightarrow a}g(f(x))=c$

Can i say that if $\lim_{x\rightarrow a}f(x)=b$ and $\lim_{x\rightarrow b}g(x)=c$ then $\lim_{x\rightarrow a}g(f(x))=c$ ? I don't think so but don't know how to prove it. Thanks.
2
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1answer
32 views

Circular definition of tangent line and derivative

I'm trying to understand the deep relations between the tangent line to the graph of a function $f$ at a given point $P$, and the derivative of $f$ at the same point. Indeed, in many books the ...
6
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1answer
45 views

Bound on first derivative $\max \left(\frac{|f'(x)|^2}{f(x)} \right) \le 2 \max |f''(x)|$ [duplicate]

I want to show that for a function $f \in C_c^2((a,b))$ strictly positive the inequality $$\max \left(\frac{|f'(x)|^2}{f(x)} \right) \le 2 \max |f''(x)|$$ holds. I noticed that the left term is ...
3
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0answers
15 views

An infinite sum of polygammas

Please help me with the proof of what follows. $$\sum _{m=0}^{\infty } (z+1)_{-m} \psi (z-m) \prod _{k=0}^m \frac{1}{\psi (z-k+1)}=1$$ The real part of z is not an integer.
3
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2answers
28 views

An infinite sum

Can someone help me prove the below? Thanks. $$\sum _{k=1}^{\infty } \frac{\Gamma (k)^2}{\prod _{m=1}^k (x \Gamma (m)+m)}=\frac{1}{x}$$
2
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4answers
49 views

Proof that a polynomial has a minimum in $\Bbb R$

I have to prove to following statement and I am having a really hard time here. There it is: Prove that the following polynomial has a minimum in $\Bbb R$ $$p(x)=x^4 + a_3x^3 + a_2x^2 + a_1x + ...
1
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1answer
12 views

What does it mean “Laplace transformable functions”

I am reading about the The convolution operation, and the notion Laplace transformable functions is mentioned there. Doe anyone know what is the definition of Laplace transformable functions? Thank ...
1
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1answer
16 views

Implications of zero divergence ($\nabla \cdot F$) when finding the flux

Say we are given a vector field $$F=(-x^2/2+xy,xy+y^2,-3yz-3)$$ with the property $\nabla\cdot F=0$. If we would like to find the flux through the part of the surface $x^2+y^2+2z^2=3$ that lies ...
2
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3answers
27 views

Find all the values of x, for which the series converges.

$\sum\limits_{n=1}^∞ (x^2/(x^2+4))^n$ I did try to use the ratio test and I ended up with $| x^2/(x^2+4)|<1$ I don't have any idea what to do after this, how do I solve for x?
5
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0answers
32 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
2
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1answer
29 views

Can a function which is periodically undefined have a limit as x goes to infinity?

I'm currently preparing for a calculus test. I was trying to solve the exercises of the test of last year, and one of the questions was: Give a full limit research of this function: ...
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3answers
35 views

Limit Problem-To find the value of a and b,when the value of the limit is given.

Here,is a limit problem: $\lim \limits _{x \to 0} {x^3 \over {\sqrt {a+x}} (bx - \sin x)} = 1$. Here, $a \in \mathbb R _+$. The question is to find the values of $a$ and $b$. Here is my workout. ...
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0answers
29 views

How can I resolve: $ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $ via numerical solution?

How can I resolve a second-order ODE via Euler method? By example in the next ODE: $$ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $$ I know Euler method: $x_{i+1} = x_{i} + ...
2
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2answers
75 views

What is a good reference that connects calculus with differential geometry?

It seems that most texts on differential geometry books tend to take a quantum leap from calculus without refering the latter. Differentials suddenly becomes forms, functions suddenly becomes ...
1
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1answer
46 views

Why is the following number always positive?

Consider two points in the Euclidean plane: $A=(A_1,A_2),B=(B_1,B_2)\in\mathbb{R}^2$, and some fixed real number $\lambda\in(0,1)$. The claim is that the following expression is always a positive ...
3
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2answers
69 views

Prove this limit $\lim \limits_{x\to\infty}f(x)=0$

I have this problem in real analysis. I think it needs integral factor or knowledge of ODE to prove, but not sure how to it. Here is the question: Let $f$ be a real valued continuous function on ...
-4
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0answers
26 views

Elasticity Calculus 1 [on hold]

The demand function for a product is given by: $$ p =\dfrac{ −0.05\, x + 120}{0.01\,x + 4 } $$ where p is the price per unit when x units are demanded. (a) Determine the intervals on which the ...
0
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2answers
31 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
32 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 ...
2
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1answer
49 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
44 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
27 views

show that the sequence $b_n$ is monotone and find its limit [on hold]

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.
1
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0answers
31 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
122 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 $ $ \ \ \ \ ...
1
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0answers
22 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
47 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
40 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
57 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
51 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
20 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
20 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
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2answers
33 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
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1answer
11 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
79 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
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1answer
32 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
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2answers
55 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
31 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
34 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 ...
1
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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
52 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 ...