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

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0
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1answer
20 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$cos^2 x + sin^2 x=1$$ Express the number $2$, ...
0
votes
0answers
14 views

$\|P_n\| \to 0$ and $\|Q_n\| \to 0$, but such that $\lim_n S(f; P_n) \neq \lim_n S(f; Q_n)$ · TO show that $f$ is not Riemann Integrable.

Suppose that $f$ is bounded on $[a, b]$ and that there exists two sequences of tagged partitions of $[a, b]$ such that $\|P_n\| \to 0$ and $\|Q_n\| \to 0$, but such that $\lim_n S(f; P_n) \neq \lim_n ...
0
votes
0answers
8 views

Zeros of an analytic function

How to prove zeros of an analytical function (non-zero function) is always countable?
0
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0answers
7 views

Infinite sub-sequences that make up a sequence

A sequence $\{a_n\}$ can be broken into sub-sequences, $\{a_n\}^1_{k_1}, \{a_n\}^2_{k_2}, \dots,\{a_n\}^m_{k_m}$, if every element in $\{a_n\}$ belongs to at least one of the sub-sequences. I had to ...
0
votes
4answers
33 views

Find $y'$ and $y''$ : $ y=x^2\ln(2x)$

for $x> 0$ : $ y=x^2\ln(2x)$ Product rule: $$(x^2)\cdot[\ln (2x)]'+ (\ln (2x))\cdot[x^2]' $$ $$y'= x^2\frac{1}{2x}\cdot (2)+\ln(2x)\cdot(2x) =2x\ln(2x)$$ ...
0
votes
1answer
35 views

Why does the following limit give two answers?

I want to calculate $$ \lim_{t \to 0} \frac{t^2}{sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2sin(t)cos(t)} \implies \lim_{t \to 0} \frac{2t}{sin(2t)}$$ and ...
1
vote
1answer
30 views

Linear subspace

Let $$E=\mathcal{F}(\mathbb{R},\mathbb{R})$$ $$F=\{ f\in E\mid f(x)= e^{3x}(a\cos(2x)+b\sin(2x)),\quad x\in \mathbb{R},a,b\in\mathbb{R} ) \}$$ Show that : $F$ is linear subspace of $E$ My ...
5
votes
3answers
128 views

Closed form for a binomial series

I am wondering if any knows how to compute a closed form for the following two series. $$\sum_{m=1}^{n}\frac{(-1)^m}{m^2}\binom{2n}{n+m}$$ $$\sum_{m=1}^{n}\frac{(-1)^m}{m^4}\binom{2n}{n+m}$$ ...
0
votes
2answers
12 views

How to prove this claim about this function?

I thnk first I need to show the that the function is monotonic. Or maybe use it for the proof of this claim : for all $$c > 0 % MathType!MTEF!2!1!+- % ...
0
votes
0answers
12 views

Derivative of indicator function and summation

I would like to take the derivative of $G = \sum_{i=1}^{n}\Big(\mathbb{1}\{i \geq x + k\} v(x) + \mathbb{1}\{i < x + k\} v(i)\Big)$ with respect to $x$, where $\mathbb{1}\{\cdot\}$ is the indicator ...
2
votes
3answers
62 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
7
votes
7answers
171 views

Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$

I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on ...
2
votes
1answer
27 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
3
votes
3answers
58 views

How does parametrization of the intersection of two surfaces induce a space curve?

Given a two surfaces say: $z=1-y$ and $ x^2+y^2+z^2=1$, we find that they intersect at: $$x^2-2yz=0$$ How is the above a space curve? Is it not just another surface? And why do we need to introduce ...
1
vote
5answers
54 views

Differentiate the Function: y=$\ ln\ tan^2x$

y=$(ln\ tan^2x)$ = $2(ln\ tan\ x)\cdot (ln\ tan\ x)'$ =$2(ln\ tan\ x)\cdot \frac{1}{(tan\ x)'}$ =$\frac{2(ln\ tan\ x)}{sec^2x}$ Is this right if not what am I doing wrong?
1
vote
2answers
51 views

Differentiate the Function $ h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$

Differentiate the function $$h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$$ My try: $$h(z) = \frac{1}{2}\ln\left(a^2-z^2\right)-\frac{1}{2}\ln\left(a^2+z^2\right)$$ so $$h'(z) = ...
1
vote
2answers
43 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
0
votes
2answers
36 views

Deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$, by calculating the areas of regular twelve-sided polygons.

Calculate the areas of regular dodecagons (twelve-sided polygons) inscribed and circumscribed about a unit circular disk and thereby deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$. This is a ...
3
votes
1answer
48 views

Finding $\sum\limits_{k=0}^n (f(k)g(k))$ (calculus of finite difference)

So, I'm working though Smoryński's Logical Number Theory and I'm stuck on the following exercise. Define $\Delta f(x) = f(x+1) - f(x)$. Given this, it's not difficult to show that $\Delta (f(x) g(x)) ...
0
votes
2answers
68 views

Alternate formulation of Calculus

Calculus is almost always made rigorous by one of two approaches: Riemann-Sums or Infinitesimals. Students seem to have a lot of trouble with Riemann Sums. So the following approach occurred to me ...
0
votes
1answer
49 views

For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge?

For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge? I have seen a duplicate of this question but the answer there, though very good and creative, isn't clear about negative values. When ...
-3
votes
0answers
38 views

What comes after Spivak/Apostol/Courant?

Calculus texts by these authors are the essential archetypal references for undergraduate students. Subsequent courses: linear algebra, multivariable calculus, differential equations. Any in the ...
0
votes
2answers
36 views

Differentiate the Function: $g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$

$$g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$$ $$=\ln\ (\ln\ u)-\ln(1+\ln\ (2u))$$ This is the part where I get a little confused. Keep in mind I am using this formula $$\frac{d}{dx}[\ln ...
3
votes
0answers
43 views

Limit of a sum.

While fixing my answer to this question I noticed that (actually the question is equivalent to this modulo some algebra) $$\frac{1}{2}=\lim_{x\to\infty}\sum_{i=0}^\infty ...
0
votes
0answers
19 views

Does the Trichotomy Property apply?

Problem: If a, b >= 0 and a^2 < b^2, then a < b. I believe I may show this to be true by analyzing the cases where a or b are equal to zero, a = b and, a > b and subsequently using the ...
1
vote
1answer
56 views

How to master the Calculus?

I believe its a broad topic to ask a question about but the strange fact is I'm not confident at it. I do know the formulas and I've also done quite a few sums but I'm never confident that I can do a ...
0
votes
1answer
563 views

The normal line intersects a curve at two points. What is the other point?

The line that is normal to the curve $\displaystyle x^2 + xy - 2y^2 = 0 $ at $\displaystyle (4,4)$ intersects the curve at what other point? I can not find an example of how to do this equation. Can ...
1
vote
2answers
22 views

What is the area of the part of the surface $z=yx$ bounded by $x^2+y^2=1$?

A parametrization of the part of the surface $z=yx$ bounded by $x^2+y^2=1$ is \begin{align} x &= u \cos v \\ y &= u \sin v \\ z &= \frac12 u^2 \sin 2v, \end{align} or $$r(u,v)=u \cos v \, ...
0
votes
3answers
46 views

find a basis of F

This question is related to that one Linear subspace Let $$E=\mathcal{F}(\mathbb{R},\mathbb{R})$$ $$F=\{ f\in E\mid f(x)= e^{3x}(a\cos(2x)+b\sin(2x)),\quad x\in \mathbb{R},a,b\in\mathbb{R} ) ...
-3
votes
3answers
77 views

How to calculate the following limit?

Calculate the following limit where $n \in \mathbb{Z}$ and log is to the base $e$ $$\lim_{x\to\infty} \log \prod_{n=2}^{x} \Bigg(1+\frac{1}{n}\Bigg)^{1/n}$$
1
vote
2answers
30 views

Differentiate the Function: $g(y)=ln\frac{(2y+1)^5}{\sqrt{y^2+1}}$

$g(y)=ln\frac{(2y+1)^5}{\sqrt{y^2+1}}$ $g(y)= ln(2y+1)^5-ln\sqrt{y^2-1}$ g'(y)=$\frac{5(2y+1)^4\cdot (2)}{(2y+1)^5}-\frac{2(y^2+1)(2y)}{(y^2+1)^2}$ At this point I can cancel the (2y+1) from the ...
0
votes
3answers
34 views

Differentiate the function: $f(u)=\frac{u}{1+\ln u}$

$f(u)=\frac{u}{1+\ln u}$ $=\ln u-\ln 1+\ln u$ =$\frac{1}{u}-\frac{1}{1}+\frac{1}{u}$ =$\frac{1-u+1}{u^2}\ =-\frac{u}{u^2}-\frac{1}{u}$ Is my answer correct?
0
votes
4answers
119 views

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to ...
4
votes
3answers
74 views

How do I evaluate $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$?

How do i evaluate this limit : $$\displaystyle \lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+............+\frac{1}{\sqrt{n}}\right)$$ ? Thank you for any ...
1
vote
2answers
51 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...
5
votes
4answers
17k views

What does smooth curve mean?

In this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?
1
vote
2answers
43 views

How to show the integral $\int_e^\infty \left(\frac{e}{t}\right)^t dt$ converges?

Let $$I=\int_e^\infty \left(\frac{e}{t}\right)^t dt$$ How to show it converges? I tried to find some inequality to compare with.
1
vote
3answers
35 views

Differentiating the exponent power series

We know that $$ e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} $$ We know that the series is uniformly convergent everywhere, and therefore we can differentiate term by term, i.e $$ ...
-1
votes
4answers
34 views

Differentiation question (quotient rule)

Find $f'(x)$ if: $$f(x)=\displaystyle {6x \over \sqrt{ 1+x^2}}$$ Ans: $\displaystyle {6 \over (1+x^2) \sqrt {1+x^2}} $ My problem is that after applying the quotient rule, i can't simplify it to ...
1
vote
1answer
12 views

Line integral of vector field/Why doesn't my solution work?

The question in its entirety: Determine for which constants A & B the vector field $$\mathbb{F} = (Axln(z))\mathbb{i} + (By^2z)\mathbb{j} + ((\frac{x^2}{z})+y^3)\mathbb{j}$$ is conservative. If ...
2
votes
3answers
98 views

$\int_1^\infty 1/\sqrt{1+e^x}dx$

Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or ...
-3
votes
0answers
57 views
0
votes
1answer
19 views

Line integral - direction of traversal

I've always just accepted that the value of the line integral $$ \int\limits_C f(x,y)\ \mathrm{d}s $$ is independent of the direction in which $C$ is traversed, but when I actually try to satisfy ...
1
vote
2answers
50 views

Show there exist a constant $c\in \Bbb{C}$ such that $\int_{0}^{1}|{f-c}|^2<{1\over 36}$

Let $f:\Bbb{R}\to \Bbb{C}$ be a $1$-periodic function, $f\in C^1$ and $\int_{0}^{1}|f'|^2\le 1$. a. Show $\sum_{k\ne 0}|{\hat{f}(n)}|^2\le {1\over 4\pi^2}$ (I did it already, and that question is ...
2
votes
2answers
62 views

why $\lim_{x \to \frac{\pi}{4}} \frac{\cos 2x}{\sin x-\cos x}=-\sqrt{2}$?

I have this very simple limit to find $$\lim_{x \to \frac{\pi}{4}} \frac{\cos (2x)}{\sin x-\cos x}$$ which is equal to $-\sqrt{2}$. However I can get the outcome as mentioned, or $\sqrt{2}$ in the ...
1
vote
2answers
41 views

What is the general definition of a discriminant? (Not just the definition for polynomials)

For example, in regards to the second derivative test for a function of two variables, $D=f_{xx}f_{yy}-(f_{xy})^2$ is refered to as the "second derivative test discriminant." I know that D is the ...
1
vote
2answers
44 views

Does the Fourier series converge at $x=0$?

Let $f(x)$, a $2\pi$ periodic funciton such that $f(0) = 1$ and for every $0\ne x\in[-\pi,\pi]$: $f(x) = 1 + \sin \frac{\pi^2}{x}$. Is the Fourier series of $f(x)$ converges at $x=0$? If so, what ...
-2
votes
5answers
144 views

Help finding the limit of this series $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$

How can I go about finding the limit of $$\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots = \sum_{k = 1}^{\infty} \frac{1}{2^{k+1}}?$$ Could I use the absolute value theorem? I have a ...
0
votes
1answer
41 views

Help understanding the result of a formula

I need some help understand the middle section of this formula. $$OA^2 = (100-40)^2 + 50^2=10^2(61)\to OA = r = 10\sqrt{61} $$ and $$\sin(\angle OCB ) = \frac{30}{r} = \frac{3}{\sqrt{61}}, ...
2
votes
2answers
54 views

Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?

Problem 3. Show that for a sequence $(x_n)$ the following are true: (i) $\lim x_n=0$ if and only if $\lim |x_n|=0$. (ii) $\lim x_n=L$ implies $\lim |x_n|=|L|$. Is the converse true? Prove or ...