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

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1answer
13 views

Understanding proof of The Ratio Root test

Now this is how I reason. I first try to identify which method that is used to give the proof. I am however so bad at identifying if there are any "hidden" quantifiers in the text. (if there are ...
0
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1answer
11 views

Integral evaluation involving trignometric functions

How to explain the following equality? (Part of an integral calculation): $$\frac{2}{2\pi}\int_{-\pi}^\pi \left| \sin x \right| (\cos nx + i\sin nx) dx = \frac{4}{2\pi}\int_0^{2\pi} \sin x \cos nx ...
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0answers
25 views

A book like Michael Spivaks Calculus, for multivariate Calculus.

Is there a book like Michael Spivaks Calculus, that is for Multivariate Calculus? That is a "real analysis" multivariate calculus book?
-1
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1answer
16 views

Limit of function - non existence

Show that the following limit does not exist: $$\lim_{x \to 1}\frac{3x^4-8x^3+5}{x^3-x^2-x+1}$$
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2answers
34 views

An example of a function for which the equality $M_1 = 2 \sqrt{M_0M_2}$ holds.

Let $f$ be twice differentiable on $(a,\infty),a\in \Bbb R$ and let $$M_k = \sup \{|f^k(x)|\mid x \in (a, \infty) \} < \infty, k=0,1,2.$$ $a)$ Prove that $M_1 \leq 2 \sqrt{M_0M_2}$. $b)$ Give an ...
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0answers
20 views

Find maximum area of triangle defined by tangent line to $y=e^{-x}$

Take a point $P(a,e^{-a})$ $(a>-1)$ on the curve $C:y=e^{-x}$. Let $S(a)$ be the area of the triangle surrounded by the tangent line to $C$ at $P$, the $x$-axis and the $y$-axis. (1) Find the ...
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1answer
20 views

Finding derivative of a split function using the definition of derivative

I have this function: $ f(x) = \begin{cases} \frac{sin^2(3x)}{x}, & \text{if $x\ne0$} \\ 0, & \text{if $x=0$} \end{cases} $ How would I find the derivative of it using the definition of the ...
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0answers
15 views

finding an equation of a plane containing two lines

I just wonder how I can find an equation of a plane given two lines like this question: find an equation of a plane containing the lines $$ L1=\left\{ \begin{array}{c} x+y+z=2 \\ 3x+4y-z=3 ...
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0answers
5 views

Does $[n*\hat f(n) - \frac{i}{2\pi}(f(2\pi) - f(0))] \rightarrow 0$, when $f(2\pi) \not= f(0)$

One more thing to note, $f:[0,2\pi] \rightarrow \mathbb{C}$ is continuously differentiable on $[0,2\pi]$. I tried doing a bunch of tricks, integration by parts, moving stuff in and out of integrals, ...
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1answer
36 views

Proving that $f(x)=\frac{1000x^{14}-7x^{11}+12x+7}{(x^7-1)^2+1}$ is a bounded function

I want to prove that function $f: \Bbb{R} \rightarrow \Bbb{R}$ such that: $$f(x)=\frac{1000x^{14}-7x^{11}+12x+7}{(x^7-1)^2+1}$$ is bounded. A good place to start would be to check limits as x goes ...
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0answers
7 views

Prove that exists $\epsilon >0$ such that $S\cap C\cap B((0,0,0),\epsilon)=\{(0,0,0)\}$

I can't find the way to do this exercise. We consider $S=\{(x,y,z) \in \mathbb{R^3}: f(x,y,z)=0 \}$, where $f$ is a $C^1$function on $\mathbb{R^3}$ such that $f(0,0,0)=0$ and $\nabla ...
4
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3answers
60 views

Is my proof that $\lim_{x\rightarrow 0} x\sin\frac{1}{x}=0$ correct?

I tried to solve this limit: $$\lim_{x\rightarrow 0} x\sin\frac{1}{x}$$ And I arrived at the answer that $\lim_{x\rightarrow 0} x\sin\frac{1}{x}=0$. Is my solution correct? $\lim_{x\rightarrow 0} ...
0
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0answers
17 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
1
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1answer
31 views

Splitting polygon in half. [on hold]

Let $P$ be a convex polygon in the plane. Prove that there is a vertical line which splits P onto two polygons of equal area. I tried to use intermediate value theorem with no luck.
3
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5answers
94 views

Evaluating $\lim_{x\rightarrow\pi}\frac{\sin x}{x^2-\pi ^2}$ without L'Hopital

I need to calculate the following limit (without using L'Hopital - I haven't gotten to derivatives yet): $$\lim_{x\rightarrow\pi}\frac{\sin x}{x^2-\pi ^2}$$ We have $\sin$ function in the numerator ...
1
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3answers
45 views

What do we know about $\sin^{2} n$?

We all know that $-1 < \sin(n) < 1$. What about $\sin^2(n)$? What can we say about it? The main question is find the limit of $$\lim_{n\to\infty }\frac{\sin^2 n}{2^n}.$$
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4answers
41 views

Evaluating $\lim_\limits{x \to 0 }(\frac{\tan x}{x})^{\frac{1}{x^2}}$

Any ideas on how to tackle this limit? $$\lim_{x \to 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}$$ I tried many ways but only got more complex stages, not easier ones...
0
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1answer
14 views

Show that a complex map is onto

I consider $\mathbb{C}$ as a real vector space. For $(a,b) \in \mathbb{C}^{2}$, consider the map : $F_{a,b} \, ; \, \mathbb{C} \, \rightarrow \, \mathbb{C}^{\ast}$ such that : $$ \forall z \in ...
4
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1answer
27 views

Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$.

Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$. First of all I need to prove that $f(x)$ is well-defined. I'm not so sure what does it mean. Basically I ...
0
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1answer
16 views

differential of inner product of functions from $R^n \to R^n$

I'm trying to find the differential of an inner product. Let $f:R^n \to R^n$ be $C^1(R^n) $ and let $x\in R^n,0 \neq v\in R^n$ . What is the derivative of $<f(x),v>$ ? If f was $R \to R^n $ ...
3
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2answers
59 views

Evaluating $\int\frac{\mathrm dx}{\sqrt{\lfloor 1+ \sqrt{1+x}\rfloor}}$

How can I solve this integral? $$\int\frac{\mathrm dx}{\sqrt{\lfloor 1+ \sqrt{1+x}\rfloor}}$$
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0answers
26 views

How do I verify that $\int_0^1 (1-t) \, f''(t) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$ [on hold]

How do I verify that: $$\int_0^1 (1-t) \, f''(ht+x) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$$
0
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1answer
25 views

Limit of a sequence of a supremum.

Problem: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a\leq c < d \leq b$ such that $f(c)=f(a)$ and $f(d)=f(b)$ and ...
0
votes
1answer
26 views

Little o(h) limit about h=0

I understand that generally if a function $f(h)$ is described as $o(h)$ that $f(h)$ has a smaller rate of growth than $h$ (like it would have to be $\sqrt{h}$). i.e. $\sqrt{h} = o(h)$, just like (for ...
2
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1answer
33 views

Proof that the equation $x^2=\sin x $ has only one real solution different than $0$

I started doing it as following: Let $f(x) = \sin x - x^2$ Using the fact that $\sin x> x-\frac{x^3}{3!}$, I got that $f(\frac{1}{2})>0$ and, as $0<\sin 1< 1 , f(1)<0$. So, as ...
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1answer
17 views

If $∇f(a)\cdot y ≤ 0$ for every vector $y$, why does $\nabla f(a)$ have to be zero?

If $f$ is differentiable at every point in $B(a)$ and $f(x)≤f(a)$ for all $x$ in $B(a)$, prove that $∇f(a)=0$. I actually did some work and found out that $∇f(a)\cdot y ≤ 0$ for every vector $y$. ...
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2answers
38 views

What are the standard defintions of “counterclockwise” and “clockwise” in 3d space?

I'm in Calc III right now, and I'm a little confused as to what constitutes "clockwise", and "counterclockwise" rotations when dealing with the various planes in 3d-space. Of course, it's obvious in ...
2
votes
2answers
59 views

How to solve such an integration analytically?

$\displaystyle\int^{2\pi}_{0} e^{ia \cos{\theta}}d\theta$ where $a$ is some constant. Can it be solved with some substitution? I tried it by expanding the exponential series but that was not proper ...
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0answers
17 views

Let $f$ be a scalar field such that $f ' (a ;-y)$ exsits [on hold]

Let $f$ be a scalar field where derivative of $f$ at point a with respect to vector $-y$ exists, $f '(a;-y)$ exists. Is it always true for any nonzero vector $y , f '(a;-y) = - f '(a;y)$?
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2answers
44 views

What is $\int \frac{1}{\sqrt{25y^2-10y-3}}dy$

$= \int \dfrac{1}{\sqrt{(5y-1)^2-4}}dy$ $=\int \dfrac{1}{\sqrt{u^2-4}}\dfrac{du}{5}, \quad U$ substitution $=\int \dfrac{1}{10\cos(\theta)} 2\cos(\theta) d\theta, \quad$ Trig substitution $= ...
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2answers
24 views

What does it really mean by a derivative in a sense of something per unit.

Suppose we are given the differential equation $\frac{dP(t)}{dt}=kP$ where $P(t)$ is a function of population with variable time measured in years. And say $k>0$ is the relative growth rate of the ...
1
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2answers
23 views

Finding a good comparison/bound for determining the convergence of a series

The series is defined as follows: $b_0=1,b_1=-7,b_k=2b_{k-1}+b_{k-2}$. I need to find a good comparison sequence to determine whether $\sum_{k\geq0}1/b_k$ converges. I considered using $1/k^2$, which ...
0
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1answer
78 views

Do every math operation derive from sum?

I've been told sometimes, that every math operation (sum, subtraction, exponentiation, square rooting, so on) can be transformed to a sum of operands. For example, subtraction can be made as ...
1
vote
1answer
58 views

Evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$

I Have to evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$ with an error with no more than $10^{-10}$ using taylor approximation $ p_{2n-1}(x) \approx\arctan(x)$ . So, After manipulation, I get: ...
1
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1answer
20 views

nullity and rank of the linear transformation $T: T [ p (x)]= p(x+1)$

Let $V$ be the linear space of all polynomials $p(x)$ of degree $\le n$. if $p$ belongs to $V$ and $q = T(p)$, means that $q(x) = p(x+1)$ for all real $x$. find nullity and rank of the linear ...
1
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1answer
29 views

Nullity and rank of the linear transformation $T[f(t)] = \int_a^b f(t) \sin (x-t) dt ~\forall~x \in [a,b]$

Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f\in V, g=T(f)$ means that $$g(x)=\int_a^b f(t)\sin(x-t)\,dt\hspace{1 cm} for\ a\le x\le b$$ Then, the nullity and rank ...
2
votes
2answers
55 views

Show that total energy is conserved

Question is as followed $\textbf{F} = f(r) \textbf{r}$ where $r = |\textbf{r}|$ $$U(r) = -\int rf(r)dr$$ $$K=\frac{1}{2} m|\textbf{v}|^2$$ Show that $E=K+U$ is constant by deriving ...
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1answer
34 views

Finding $a$ and $b$ so that the function is continuous

$$f(x) = \begin{cases} \displaystyle\frac{x^2-4}{x-2}&\quad x<2\\[0.4em] ax^2-bx+3&\quad 2 \leq x <3\\[0.3em] 2x-a+b&\quad x \geq 3 \end{cases}$$ I can't make the right limit of ...
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0answers
37 views

Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ .

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
1
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1answer
19 views

Linear Motion and derivatives

A particle moving along a line has position $s(t)=t^4-18t^2m$ at time $t$ seconds. At which times does the particle pass through the origin? At which times is the particle instantaneously motionless ...
3
votes
3answers
55 views

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. I tried few things but it wouldn't work out. I would appreciate your help.
1
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2answers
56 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
1
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1answer
18 views

Showing $\limsup_{h \to {0}}\frac{O(h^2)}{h^2}<\infty$

Let $$y(h)=1-2\sin^{2}(2\pi h) , f(y)=\frac{2}{1+\sqrt(1-y^2)} $$ Justify the statement $$f(y(h))=2-4\sqrt{2}\pi+O(h^2)$$ where $$\limsup_{h \to {0}}\frac{O(h^2)}{h^2}<\infty$$
0
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1answer
24 views

Find the equation of the tangent line to the curve $3x^3 + 3y^2-11=4xy-x$ at the point $(1,-1)$.

The answer choices are given below: a) $5x + 7y = -2$ b) $-7x+5y = -12$ c) $-5x + 7y = -12$ d) $7x+5y = - 12$ e) $-7x + 5y = 2$
3
votes
0answers
37 views

Non-analytic smooth function

The Wikipedia page (http://en.wikipedia.org/wiki/Non-analytic_smooth_function) proves that $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is a ...
2
votes
1answer
34 views

Confusion in understanding a proof in Apostol's Calculus I

I'm using Apostol's Calculus I as my introductory book to the subject and I'm stuck trying to understand the proof of Theorem 1.13 (page 79) which, as I understand, provides a way to compute the value ...
1
vote
2answers
34 views

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$ I've seen a proof which goes like this. $$ \lim_{x\to\infty} \frac{\frac{\arctan x}{x^2}}{\frac{1}{x^2}} = \frac{\pi}{2} > ...
0
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0answers
19 views

Are 'similar' differentials equal? Specifics enclosed.

I'm trying to show that two infinitesimally small changes in an angle are actually equal, i.e. I want to say that $d\phi = d\phi '$, where the change in $\phi '$ is caused by a change in $\phi$. Here ...
1
vote
0answers
39 views

is the sequence $[ne]$ convergence?

Is the sequence $a_n=[ne]$ convergence or partially convergence? ($e$ is the Euler's number and the bracket mean the integer part function.)
2
votes
4answers
116 views

Convergence of $\int_0^{\infty} x \cos (x^6)\,dx$

I feel that $\int_0^{\infty} x \cos (x^6) dx$ is convergent using the regular first year definition of an integral. I have been trying to convince a university professor of this, but according to him ...