Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.

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0
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2answers
16 views

Finding values that make a series convergent

So I was given the following prompt: When $x=−2$, for what values of p does the series converge? $$\sum_{n=1}^\infty\left(\frac{(-1)^{n+1}(x-3)^n}{5^n\cdot n^p}\right)$$ I ended up working out this ...
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1answer
19 views

Taylor Series, find the sum of a given convergent series

By recognizing the series $$-(\frac{1}{2})-\frac{(\frac{1}{2})^2}{2}-\frac{(\frac{1}{2})^3}{3}-⋯\frac{(\frac{1}{2})^n}{n}-⋯$$ as a Taylor Series evaluated at a particular value of x, find the sum of ...
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2answers
24 views

The function $f$ is defined as $f(x) = x^{2} - 1$. What is the $x$-coordinate of the point on the function's graph that is closest to the origin?

The function $f$ is defined as $f(x) = x^{2} - 1$. What is the $x$-coordinate of the point on the function's graph that is closest to the origin? A. $-\sqrt{3}/3$ B. $-\sqrt{2}/2$ C. $0$ D. $\sqrt{3}/...
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0answers
27 views

Finding a limit of integral involving $n$

Let for $n \geq 0$, $$I_n = \frac{1}{\pi^n} \int^{\pi}_{0} x^n e^x \sin x dx, \quad J_n = \frac{1}{\pi^n} \int^{\pi}_{0} x^n e^x \cos x dx,$$ then you can get: $$\displaystyle \lim_{n \to \infty} I_n ...
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1answer
17 views

Inverse functions when one is given functions $f(x)$ but then integrating with respect to $y$

I know that when finding area between curves, and they're both given as functions $f\left(x\right)$ then the integral becomes something like $\int ( f_1 - f_2 ) dx$ where $f_1$ is the top function ...
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0answers
29 views

Prove that the sequence (1 + (−1)^n) is divergent [closed]

Use the definition to show that the sequence (1 + (−1)^n) is divergent.
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1answer
28 views

Finding Taylor Series of Indefinite Integral $\cos(x^2)$ around $x = 0$

How would I find the Taylor series around $x = 0$ for this integral? $$\int \cos(x^2)$$ My first point of confusion is if it is around $x = 0$, doesn't that make it a Maclaurin series? Would I go ...
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1answer
5 views

For each set of conditions below give a formula for a function that satisfies the conditions or explain why one cannot exist.

(a) A function with exactly two horizontal asymptotes and exactly two vertical asymptotes, but is defined everywhere else on $\mathbb{R}$. (b) A continuous function on $\mathbb{R}$ with $f(2)=−3$, $f(−...
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2answers
48 views

Calculate the following limit: $\lim\limits_{x\to\infty}\frac{\left(1+\cos\left(2^{-x}\right)\right)^x}{2^x}$ [closed]

Hi I've been stuck trying to calculate this limit, I'd appreciate the help. $$\lim\limits_{x\to\infty}\frac{\left(1+\cos\left(2^{-x}\right)\right)^x}{2^x}$$
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2answers
56 views

$\lim_{n \to \infty} \sqrt[k]{\prod_{k=1}^n \left(1+ \frac{k}{n}\right)}$

I was studying for my Calculus test and I've got stuck in the following activity: Compute $$\lim_{n \to \infty} \sqrt[k]{\prod_{k=1}^n \left(1+ \frac{k}{n}\right)}$$ I've tried to solve it using ...
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3answers
22 views

Confirmation of polar coordinate calculation

$$\lim_{x,y \to 0,0} \frac{x^2y + xy^2}{ x^2+y^2}$$ Polar coordinates $$x= r \cdot \cos \theta$$ $$y = r \cdot \sin \theta$$ $$\lim_{r \to 0} \frac{(r \cos \theta)(r \sin \theta) + (r \cos \theta) (r \...
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1answer
25 views

If $f_n$ converges uniformly and that exists a sequence $\{t_n\}$ such that $f_n(t_n)\to x$, does this imply that $t_n$ converges?

I have a problem that essentially boils down to the above question. Another important fact is that $f$ is continuous. My approach has basically been a proof by contradiction. We assume that the ...
5
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1answer
39 views

Two form surface integral over sphere

I'm trying to compute $\int_M \omega$ with \begin{align*} \omega &= x^4 dy \wedge dz + y^4 dz \wedge dx + z^4 dx \wedge dy, \\ M &: x^2 + y^2 + z^2 = R^2. \end{align*} I have done this in two ...
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0answers
23 views

The slopes of the graph of $ \cos cx $ belong to the interval $ [- c, \: c] $. Find the missing numbers:

The slopes of the graph of $ \cos cx $ belong to the interval $ [- c, \: c] $. Find the missing numbers, If $ | x-a | < ? $ Then $ | \cos 2x- \cos 2a | <\varepsilon $ If $ | y-b | <\delta $ ...
2
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1answer
75 views

A question on an integral inequality

I happened to have learned the following question: Let $K \subset \mathbb{R}^2$ be a convex domain with $0\in K$. Prove that if $$\frac{1}{2\pi}\int_{K}e^{-\frac{|x|^2}{2}}\, dx=\frac{1}{2},$$ then $$\...
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0answers
23 views

Statistical efficiency in context of neural networks?

According to https://en.wikipedia.org/wiki/Efficiency_(statistics), an estimator with statistical efficiency is one that "... needs fewer observations than a less efficient one to achieve a given ...
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2answers
44 views

Derivative of $\int_0^x(f(u) \cdot u)\,\mathrm{d}u$

I have two integrals that I want to calculate its derivative: $$\int_0^x(f(u) \cdot u)du ~~~~~ , ~~~~~ \int_0^x(f(u))du$$ So from what I understand: $[\int_0^x(f(u))du]'=f(x) \cdot 1 \cdot x = x \cdot ...
2
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1answer
55 views

Is there a specific reason why the majority of textbooks say “positive integer” instead of “natural number”?

This question is derived from just curiosity, but is there a specific reason for the situation I stated in the title? I am pretty sure that "positive integer" = {1, 2, 3, 4, ...}, which is ...
1
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0answers
22 views

Integration of a shifted Gaussian multiplied by a Bessel function

I am trying to solve an integral like $$ F(q, a, b) = \int_0^{+\infty}dx e^{-(x-a)^2/b^2}J_0(qx)x \quad where\quad a, b > 0; q > 0 $$ For a similar result, it is well known that $\int_0^{+\infty}...
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1answer
29 views

Trying to solve an equation involving the floor function for an algorithm

I have a c++ function which performs the following operation: ...
4
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2answers
92 views

how to solve $\int\frac{x^2}{x^2+1}dx$ by using series

I want to solve $\int\frac{x^2}{x^2+1}dx$ by using series, but I did something wrong. Correct solution: \begin{align*} \int\frac{x^2}{x^2+1}dx &= \int\frac{(x^2+1)-1}{x^2+1}dx \\ &= \int\left(...
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0answers
28 views

Monotonicity of this function

Let $a,b,c$ be positive real numbers and $f(x)= \frac{a^x}{b^x+c^x}+\frac{b^x}{a^x+c^x}+\frac{c^x}{a^x+b^x}$. Prove that $f$ is increasing on $[0,\infty)$ and decreasing on $(-\infty,0]$. Before ...
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1answer
20 views

Stokes Formula with wedge product

$$\alpha=\sum^m_{i=1} x^idx^1\land...\widehat {dx^i} ...\land dx^m$$ Compute ${\int_S}_{m-1} j^*\alpha$ where $j$ is the inclusion map of $S_{m-1}$ in $\mathbb R^m$. I'm mostly an intuitionistic proof ...
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1answer
23 views

Intuitive error in finding Volume of sphere using single integration.

To calculate the volume of a sphere of radius $\mathbf R$, I considered a thin disc of radius $ r= \mathbf Rsin\theta$ ,where $\theta$ is the angle radius vector on the circumference makes with the ...
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1answer
20 views

Calculate impulse response of a dynamic system

I am given an exercise where I need to find the impulse response of the system $$ y_n = 1.02(y_{n-1} + x_n). $$ I have looked online but haven't been able to find an example with a system with memory. ...
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2answers
38 views

Estimate sum over squares $f(r)=\sum_{k_1,k_2 \in \mathbb Z; \vert k_1 \vert,\vert k_2\vert \ge r} \frac{1}{(k_1^2+k_2^2-k_1k_2)^2}.$

I want to upper-bound the following sum $$f(r)=\sum_{k_1,k_2 \in \mathbb Z; \vert k_1 \vert,\vert k_2\vert \ge r} \frac{1}{(k_1^2+k_2^2-k_1k_2)^2}.$$ One simple way I could think of was to use polar ...
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0answers
11 views

a fully manageable but not fully observable second order system

I am currently trying to understand the concepts of manageability and observability. Can you give me an example of a fully manageable but not fully observable second order system?
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0answers
28 views

Show that for all integers 𝑛: b) If 𝑛 | 58, then 𝑛+7 and 𝑛^2+9 are coprime.

Part a) stated : If 𝑑 is an integer such that 𝑑|𝑛+7 and 𝑑|𝑛2+9, then 𝑑|58. The next part (part b) is what i need help with. I thought I'd state this here for context. Working out: In order to ...
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0answers
15 views

Find a Hardy-Littlewood Maximal Function for which the following holds

I want to find a Hardy-Littlewood maximal function for $f:\mathbb{R}\rightarrow \mathbb{R}$ for which we have $|f(x)| \le 1$ for every $x\in \mathbb{R}$ and also $f(0)=0$. But then we would also have $...
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0answers
42 views

Is this manipulation of the Taylor series valid?

The Taylor series of a function $f(x)$, (assuming differentiability) is: $$\tag{1}f(x)=f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2+...$$ If I take $x=x+h$ I get: $$\tag{2}f(x+h)=f(a)+f'(a)(x+h-a)+\frac{1}...
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0answers
36 views

Show that for all integers 𝑛: If 𝑑 is an integer such that 𝑑|𝑛+7 and 𝑑|𝑛2+9, then 𝑑|58.

This is the working out I have so far. Any checks to see if it is sufficient would be highly appreciated! Since $𝑑|𝑛+7$, $𝑑| (n+7)^2= n^2+14n+49.$ By adding $(n^2+9)$, we get: $n^2+14n+49 +(n^2+9)$ ...
6
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0answers
192 views

Number of distinct real roots of $f(f(f(f(f(f(f(f(f(f(f(x^{2020} f(x)))))))))))) = 2$

Given the $4th$ degree polynomial $f(x)$ with the graph below. How many distinct real roots does $$f(f(f(f(f(f(f(f(f(f(f(x^{2020} f(x)))))))))))) = 2$$ have? $(12, 18, 20$ or $22)$ Here is the ...
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0answers
22 views

Integral for the boundary of a polygon

I was reading a paper on the generalized Buffon's needle problem and I am a little confused about this statement. Could anyone please explain why this is true? Thank you in advance. Consider a regular ...
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0answers
6 views

Riemann invariants of a special hyperbolic system

How can one compute the Riemann invariants of the following one dimensional hyperbolic system? $$\begin{pmatrix} u \\ v \end{pmatrix}_t + \begin{pmatrix} -v & -u \\ |v|-k & \mathrm{sgn}(v)u \...
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1answer
30 views

Finding a planes equation containing a specific curve

What is the algorithm if I wanna fond a planes equation that contains a specific curve in space (a continuous curve $\gamma(t)$)? Thanks for any help.
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0answers
19 views

Does the following hold for all Hardy-Littlewood Maximal Functions?

Define a function $f:\mathbb{R}^d \rightarrow \mathbb{R}$ that is bounded by some $N>0$, $N\in \mathbb{R}$ such that $|f(x)| \le N$ $\forall x\in \mathbb{R}^d$ i.e. $f(x)$ is bounded by $N$. Define ...
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0answers
22 views

Establish the very useful approximation formula $(1+u)^r\sim1+ru$, where $r$ is any rational exponent and $|u|$ is small compared to $1$. [closed]

Let $\,f(x)=x',\;\;\;x=1,\;\;\;\text{ and }\;\;\; \Delta x = u. \;\;\; f'(x) = r x^{r-1}.\;\;\;$ Note that $\;\; \Delta y=f(1+u)-f(1)=(1+u)^r-1.\,$ By the approximation principle, $\; (1+u)^r-1 \...
-1
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2answers
14 views

write the general term (the (k+1)th term) of the binomial expansion (a+b)^n in terms of n and k, where k < n. [closed]

I'm really unsure of what to do here, but I am trying to find the (k+1)th term or (r+1)th term of the binomial expansion (a+b)^n in terms of n and k, where k is less than n (k < n)
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2answers
53 views

Is it possible to get the area between $y^2 = 3x$ and $y = x^3$? [closed]

I am a Grade 12 student taking Calculus. Our lesson for tomorrow is about area between curves. Our teacher gave this question for us to ponder about and do advanced study. Can someone explain this and ...
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0answers
17 views

Proving a summation formula using the general Leibniz rule

I am trying to prove the following relations: $$ \partial^{N-2}(f^{N-1}g) =\sum_{n+m=N-2}\frac{(N-2)!}{n!\,(m+1)!}\,\big[\partial^{n}(f^{n}g)\big]\,(\partial^{m}f^{m+1}), \qquad N\geq2, $$ and $$ \...
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0answers
18 views

What is the first and second order derivative of the following function with respect to Yi? [closed]

What is the first and second-order derivative of the following function? $$\sum_{l=1}^{k}(\sum_{j\in B_l} Y_j^{\frac{1}{\Theta_l}})^{\Theta_l}$$
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1answer
38 views

Does this hold true

Let $f$ be function such as $f(x)=g(x)$ when $x\in Q$ $f(x)=h(x)$ when $x\in R/Q$ $f$ is continious at $x'$ iff $g(x')=h(x')$ Does this hold true and if so, why?
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0answers
29 views

Should the probability of $f(x)>0$ be the same for any choices of $A,B$?

We have this function for $x>0$ and two constants $B>A>0$ $$f (x)=8 \cos (A x+B x)+19 \cos (A x-B x)+5 \cos (2 A x-B x)\\\quad+8 \cos (A x-2 B x)+2 \cos (2 A x-2 B x)+19 \cos (A x)\\+2 \cos (...
-4
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0answers
36 views

Using the definition of limit to show $\lim_{x\to -3}\frac{x-3}{x^2 -9} = \frac{-1}{6}$ [closed]

I need prove this limit by the formal definition of limit: $$\lim_{x\to -3}\frac{x-3}{x^2 -9} = \frac{-1}{6}$$
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0answers
21 views

xD log method for B(x)=A(x)^k from generatingfunctionology - How to compare coefficients?

The book by Herbert Wilf "generatingfunctionology" has the exercise: Let A(x) be a power series with $A(0) = 1$, and let $B(x) = A(x)^k$. It is desired to compute the coefficients of $B(x)$, ...
1
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3answers
68 views

What is the limit of $\frac{e^n}{(n+4)!}$

How do we compute the limit: $\lim_{n\rightarrow \infty} \frac{e^n}{(n+4)!}$ I can only think of this approach, but I am not sure that it's too valid: We know that the terms are positive, so we would ...
0
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0answers
45 views

Why does the ratio of a chord and arc PQ in this unit circle approach 1 as PQ approaches 0?

I would like some help with this part in my textbook please: It mentions that the chord and arc length of PQ → $1$ as the arc PQ → $0$; however I just can't seem to see it. Thank you in advance.
1
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3answers
34 views

Sequential definition of continuity: What does "all sequences mean

I've been exposed to both the classic and sequential definition of continuity. The sequential definition is the following: A function $f: A \to \mathbb R$ is continuous at a point $a \in A$ if for ...
0
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2answers
40 views

how many ways can a group of 15 people be divided into three groups of 3 and three groups of 2

enter image description hereThat's all. I need help with this question and would prefer the answer in combination notation e.g. 15C3 * 12C3 etc. Thankyou.

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