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

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0
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0answers
4 views

Find the lengths of the given curves

So I have a problem where I need to find the length of a given curve using integration. I've probably put about 2 hours into this question but I'm completely stumped as to solving it. Here is the ...
0
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1answer
17 views

Integration Convergence/Divergence Questtion

$$ \int\limits_0^{\pi} \frac{ dt}{\sqrt{t} + \sin t }$$ How can one tell if this integral converges or diverges? Integral of $1/(\sqrt{t}+\sin(t))$ from $0$ to $\pi$. I can't even find the ...
5
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0answers
43 views

if $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$

I have a question $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$ in [0,$\infty$)? can we say that the $f(\sqrt{x})$ also can be infinitely differentiable in ...
0
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1answer
20 views

How to factorize this.

We just started calculus and busy with limits. we were told that use a limit as long as it does not make the equation undefined. So the question is: $\displaystyle \lim_{x\to 0} \dfrac{2x}{x^2+x}$ ...
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1answer
25 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
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1answer
9 views

Trajectory of a particle for $t\ge1$ given $r(t)$ for $0\le t \le 1$.

I have a question on the process for which to solving this question. It is a homework question, and I already have the answer, but I am not sure on the correct process to attaining that answer. The ...
0
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1answer
27 views

how to proof that this function is zero

given $f$ continuous and diferentiable into $\mathbb{R}$ such that $\forall x\in\mathbb{R},|f'(x)|\le|f(x)|$ and $f(0)=0$ then proof that $f(x)=0$ atempt: taking $x>0$, since $f$ is ...
3
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1answer
34 views

How to use U substitution for the integral $\int\frac{8}{49+x^2}\,dx$?

So, the following is my problem. $$\int\frac{8}{49+x^2}\,dx$$ I understand this. I should first take out the constant which is 8 so it'll be $$8\int\frac{1}{49+x^2}\,dx$$ Then I should factor out the ...
3
votes
1answer
24 views

Why does $\int_b^{b+\Delta b}f(x)\;dx=f(b)\Delta b+\mathcal{O}(\Delta b^2)$

On this page it is shown that: $$\frac{\partial}{\partial b}\left(\int_a^bf(x)\;dx\right)=\lim_{\Delta b\rightarrow 0}\frac{1}{\Delta b}\int_b^{b+\Delta b}f(x)\;dx=\lim_{\Delta b\rightarrow ...
3
votes
1answer
45 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
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2answers
46 views

Proving that a polynomial has a positive root

So I want to prove that a polynomial $ P(x)=a_nx^n+a_{n−1}x^{n−1}+.....+a_1x+a_0 $ has a positive root. I'm given that $ a_n $ is positive and $ a_0 $ is negative. I want to know how to apply the ...
3
votes
3answers
47 views

Computing integrals in terms of $\pi$

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 17. We have defined $\pi$ to be the area of a unit circular disk. In ...
0
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1answer
50 views

Tricky differentiation

$$\begin{aligned}f(x) = 1 - \left ( \frac{u-x}{u-l} \right )^{B}\end{aligned}$$ After differentiating, the answer is: $$\begin{aligned}f'(x) = \left ( \frac{B}{u-l} \right )\left ( \frac{u-x}{u-l} ...
1
vote
1answer
32 views

Understanding graphically the convergence of alternating harmonic and divergence of harmonic

I understand the rules of convergence of a series so that I know that $\sum \frac 1 n$ (the harmonic series) diverges and $\sum \frac 1 {n^2}$ squared converges. It doesn't make sense graphically to ...
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2answers
89 views

Second Fundamental theorem of calculus

I need to use the second Fundamental theorem of calculus to work out: $$\int_{0}^\frac{\pi}{8}\tan(2x)\mathrm dx$$ Firstly it is clear that $\tan(2x)$ is continuous on ...
1
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1answer
18 views

Difference Operators

Let $K$ be a field. Given a map $f\colon K\longrightarrow K$, and $h\not=0$ define $\Delta_h f$ to be the map $x\longmapsto\dfrac{f(x+h)-f(x)}{h}$. Then $\Delta_h^j f$ is defined for $j=0,1,2,\dots$. ...
1
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1answer
25 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
18 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
42 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?
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1answer
20 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
43 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
24 views

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

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 ...
0
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1answer
21 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 ...
0
votes
1answer
31 views

finding an equation of a plane containing two lines [on hold]

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 ...
0
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0answers
10 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, ...
0
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1answer
38 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
20 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
votes
3answers
62 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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1answer
46 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
vote
1answer
35 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.
4
votes
5answers
124 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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2answers
53 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 $$\lim_{n\to\infty }\frac{\sin^2 n}{2^n}.$$
1
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4answers
46 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
17 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
votes
1answer
28 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
votes
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
60 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
27 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: $$\int_0^1 (1-t) \, f''(ht+x) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$$
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1answer
27 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
28 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
votes
1answer
35 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 $y$. ...
0
votes
2answers
39 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
68 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 ...
1
vote
0answers
18 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 $y , f '(a;-y) = - f '(a;y)$?
1
vote
2answers
46 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 $= ...
0
votes
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
vote
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
votes
1answer
80 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
74 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: ...