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

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0
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0answers
18 views

Imaginary part tends to zero

Does anybody have an idea how to show that for $|x|< \pi$ the imaginary part of the following sequence of functions $f_m$ tends to zero for $m \rightarrow \infty.$ $$f_m(x):=\left( ...
1
vote
1answer
31 views

For $0≤a≤b≤c$, show that $\lim\sqrt[n]{a^n+b^n+c^n}=c$

For $0≤a≤b≤c$, show that $\lim\sqrt[n]{a^n+b^n+c^n}=c$ I think I am making some silly mistake with my "proof". If it is indeed correct, another question emerges. So, my attempt is: Since $0≤a≤b≤c$, ...
5
votes
0answers
17 views

Evaluating $\int_0^1 \frac{z \log ^2\left(\sqrt{z^2+1}-1\right)}{\sqrt{1-z^2}} \, dz$

What real analysis tools would you employ for this kind of integral? $$\int_0^1 \frac{z \log ^2\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$
1
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1answer
25 views

On the (short) proof of integration by parts

Consider the proof of integration by parts given here: http://mathworld.wolfram.com/IntegrationbyParts.html . I do not understand this equality: $\int d(uv)=uv$. Of course I am missing something, ...
1
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1answer
25 views

Showing that the gradient is orthogonal to level surface

It is well known that the gradient of a function (which is sufficiently well behaving) $g(x)$ is orthogonal to its level surface, for example $g(x)=0$. I have seen the following derivation of this ...
0
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3answers
32 views

limits and infinity

I'm having trouble wrapping my head around some of the 'rules' of limits. For example, $$ \lim_{x\to \infty} \sqrt{x^2 -2} - \sqrt{x^2 + 1} $$ becomes $$ \sqrt{\lim_{x\to \infty} (x^2) -2} - ...
0
votes
1answer
12 views

Work done by F (vector field) on C (curve)

Let $C$ be the rectangle, with vertices at points $(0,0)$, $(0,1)$, $(2,0)$, $(2,1)$ and an anticlockwise orientation. Let $$F=(P(x,y),Q(x,y))$$ be a vector field with $$P=y^2+e^{x^2}+ye^{xy}$$ and ...
0
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0answers
18 views

Recommendation about studying calculus.

I will be first year undergrad in Physics next year. I've studied calculus from stewart so far and finished it until the integral part. Now, I am studying calculus from Apostol, because it starts with ...
1
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1answer
39 views

Show equality of a given function with a series in $ℝ$

Show that: $$2x\cos x-\sin x=4\sum_{n=2}^\infty \frac{(-1)^n}{n^2-1}\sin(nx)$$ Supposedly, this can be proved by using Fourier series, by choosing the right function but I have been thus far unable ...
4
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1answer
50 views

Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.

$f(x)$ is a differentiable function and $g(x)$ is a double differentiable function such that $|f(x)|\leqslant 1$ and $f'(x)=g(x)$. If $$f(0)^2+g(0)^2=9$$ then prove that there exists some ...
0
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1answer
22 views

Compactly supported functions; continuous continuation?

While I was trying to prove something else, I stumbled over this question. Let $f \in C^2_c(\mathbb{R}^n; \mathbb{R})$ be non-negative everywhere does this mean that $$g(x):=\frac{||\nabla ...
-2
votes
1answer
26 views

show that $f$ is concave iff $-f$ is convex.

let $f : S \subset \Bbb R^n \to \Bbb R$ be a function. then, I want to show that $f$ is concave iff $-f$ is convex. definition of convexity: $x,y\in S$ and $\alpha \in (0,1)$ $f(\alpha ...
1
vote
1answer
30 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 ...
9
votes
4answers
204 views

Solve trigonometric integral $\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx $

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
votes
0answers
17 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 $f(y)>f(x)$ ...
1
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3answers
57 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. ...
3
votes
2answers
20 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 ...
5
votes
3answers
59 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
votes
1answer
30 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
votes
1answer
33 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
votes
1answer
62 views

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

I want to show that for a function $f \in C_c^2((a,b))$ non-negative, the inequality $$\sup \left(\frac{|f'(x)|^2}{f(x)} \right) \le 2 \sup |f''(x)|$$ holds. I noticed that the left term is equal ...
3
votes
0answers
19 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
votes
2answers
34 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
votes
4answers
51 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
vote
1answer
13 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
votes
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
votes
0answers
38 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
votes
1answer
30 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: ...
1
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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
31 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
votes
2answers
79 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
vote
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
votes
2answers
71 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
votes
0answers
33 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
votes
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
votes
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
50 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.
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0answers
32 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
votes
2answers
123 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
votes
2answers
48 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
vote
1answer
42 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
vote
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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vote
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
vote
1answer
21 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
vote
1answer
23 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 ...