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.
115,205
questions
0
votes
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 ...
0
votes
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 ...
-4
votes
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}/...
1
vote
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:
$$\displaystyle \lim_{n \to \infty} I_n ...
0
votes
1answer
17 views
Inverse functions when one is given functions $f(x)$ but then integrating with respect to $y$
I know that when finding area between curves, and they're both given as functions $f\left(x\right)$ then the integral becomes something like $\int ( f_1 - f_2 ) dx$ where $f_1$ is the top function ...
-7
votes
0answers
29 views
Prove that the sequence (1 + (−1)^n) is divergent [closed]
Use the definition to show that the sequence (1 + (−1)^n) is divergent.
0
votes
1answer
28 views
Finding Taylor Series of Indefinite Integral $\cos(x^2)$ around $x = 0$
How would I find the Taylor series around $x = 0$ for this integral?
$$\int \cos(x^2)$$
My first point of confusion is if it is around $x = 0$, doesn't that make it a Maclaurin series?
Would I go ...
0
votes
1answer
5 views
For each set of conditions below give a formula for a function that satisfies the conditions or explain why one cannot exist.
(a) A function with exactly two horizontal asymptotes and exactly two vertical asymptotes, but is defined everywhere else on $\mathbb{R}$.
(b) A continuous function on $\mathbb{R}$ with $f(2)=−3$, $f(−...
-1
votes
2answers
48 views
Calculate the following limit: $\lim\limits_{x\to\infty}\frac{\left(1+\cos\left(2^{-x}\right)\right)^x}{2^x}$ [closed]
Hi I've been stuck trying to calculate this limit, I'd appreciate the help.
$$\lim\limits_{x\to\infty}\frac{\left(1+\cos\left(2^{-x}\right)\right)^x}{2^x}$$
2
votes
2answers
56 views
$\lim_{n \to \infty} \sqrt[k]{\prod_{k=1}^n \left(1+ \frac{k}{n}\right)}$
I was studying for my Calculus test and I've got stuck in the following activity:
Compute $$\lim_{n \to \infty} \sqrt[k]{\prod_{k=1}^n \left(1+ \frac{k}{n}\right)}$$
I've tried to solve it using ...
0
votes
3answers
22 views
Confirmation of polar coordinate calculation
$$\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 \...
0
votes
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
votes
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 ...
0
votes
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
votes
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
$$\...
0
votes
0answers
23 views
Statistical efficiency in context of neural networks?
According to https://en.wikipedia.org/wiki/Efficiency_(statistics), an estimator with statistical efficiency is one that "... needs fewer observations than a less efficient one to achieve a given ...
0
votes
2answers
44 views
Derivative of $\int_0^x(f(u) \cdot u)\,\mathrm{d}u$
I have two integrals that I want to calculate its derivative:
$$\int_0^x(f(u) \cdot u)du ~~~~~ , ~~~~~ \int_0^x(f(u))du$$
So from what I understand: $[\int_0^x(f(u))du]'=f(x) \cdot 1 \cdot x = x \cdot ...
2
votes
1answer
55 views
Is there a specific reason why the majority of textbooks say “positive integer” instead of “natural number”?
This question is derived from just curiosity, but is there a specific reason for the situation I stated in the title? I am pretty sure that "positive integer" = {1, 2, 3, 4, ...}, which is ...
1
vote
0answers
22 views
Integration of a shifted Gaussian multiplied by a Bessel function
I am trying to solve an integral like
$$
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
$$
For a similar result, it is well known that $\int_0^{+\infty}...
0
votes
1answer
29 views
Trying to solve an equation involving the floor function for an algorithm
I have a c++ function which performs the following operation:
...
4
votes
2answers
92 views
how to solve $\int\frac{x^2}{x^2+1}dx$ by using series
I want to solve $\int\frac{x^2}{x^2+1}dx$ by using series, but I did something wrong.
Correct solution:
\begin{align*}
\int\frac{x^2}{x^2+1}dx &= \int\frac{(x^2+1)-1}{x^2+1}dx \\
&= \int\left(...
0
votes
0answers
28 views
Monotonicity of this function
Let $a,b,c$ be positive real numbers and $f(x)= \frac{a^x}{b^x+c^x}+\frac{b^x}{a^x+c^x}+\frac{c^x}{a^x+b^x}$. Prove that $f$ is increasing on $[0,\infty)$ and decreasing on $(-\infty,0]$.
Before ...
0
votes
1answer
20 views
Stokes Formula with wedge product
$$\alpha=\sum^m_{i=1} x^idx^1\land...\widehat {dx^i} ...\land dx^m$$
Compute ${\int_S}_{m-1} j^*\alpha$ where $j$ is the inclusion map of $S_{m-1}$ in $\mathbb R^m$.
I'm mostly an intuitionistic proof ...
0
votes
1answer
23 views
Intuitive error in finding Volume of sphere using single integration.
To calculate the volume of a sphere of radius $\mathbf R$, I considered a thin disc of radius $ r= \mathbf Rsin\theta$ ,where $\theta$ is the angle radius vector on the circumference makes with the ...
-2
votes
1answer
20 views
Calculate impulse response of a dynamic system
I am given an exercise where I need to find the impulse response of the system
$$
y_n = 1.02(y_{n-1} + x_n).
$$
I have looked online but haven't been able to find an example with a system with memory.
...
0
votes
2answers
38 views
Estimate sum over squares $f(r)=\sum_{k_1,k_2 \in \mathbb Z; \vert k_1 \vert,\vert k_2\vert \ge r} \frac{1}{(k_1^2+k_2^2-k_1k_2)^2}.$
I want to upper-bound the following sum
$$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}.$$
One simple way I could think of was to use polar ...
0
votes
0answers
11 views
a fully manageable but not fully observable second order system
I am currently trying to understand the concepts of manageability and observability. Can you give me an example of a fully manageable but not fully observable second order system?
0
votes
0answers
28 views
Show that for all integers 𝑛: b) If 𝑛 | 58, then 𝑛+7 and 𝑛^2+9 are coprime.
Part a) stated : If 𝑑 is an integer such that 𝑑|𝑛+7 and 𝑑|𝑛2+9, then 𝑑|58. The next part (part b) is what i need help with. I thought I'd state this here for context.
Working out:
In order to ...
-1
votes
0answers
15 views
Find a Hardy-Littlewood Maximal Function for which the following holds
I want to find a Hardy-Littlewood maximal function for $f:\mathbb{R}\rightarrow \mathbb{R}$ for which we have $|f(x)| \le 1$ for every $x\in \mathbb{R}$ and also $f(0)=0$. But then we would also have
$...
1
vote
0answers
42 views
Is this manipulation of the Taylor series valid?
The Taylor series of a function $f(x)$, (assuming differentiability) is:
$$\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}...
3
votes
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
votes
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 ...
0
votes
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 ...
1
vote
0answers
6 views
Riemann invariants of a special hyperbolic system
How can one compute the Riemann invariants of the following one dimensional hyperbolic system?
$$\begin{pmatrix} u \\ v \end{pmatrix}_t + \begin{pmatrix} -v & -u \\
|v|-k & \mathrm{sgn}(v)u
\...
0
votes
1answer
30 views
Finding a planes equation containing a specific curve
What is the algorithm if I wanna fond a planes equation that contains a specific curve in space (a continuous curve $\gamma(t)$)?
Thanks for any help.
0
votes
0answers
19 views
Does the following hold for all Hardy-Littlewood Maximal Functions?
Define a function $f:\mathbb{R}^d \rightarrow \mathbb{R}$ that is bounded by some $N>0$, $N\in \mathbb{R}$ such that $|f(x)| \le N$ $\forall x\in \mathbb{R}^d$ i.e. $f(x)$ is bounded by $N$. Define ...
-4
votes
0answers
22 views
Establish the very useful approximation formula $(1+u)^r\sim1+ru$, where $r$ is any rational exponent and $|u|$ is small compared to $1$. [closed]
Let $\,f(x)=x',\;\;\;x=1,\;\;\;\text{ and }\;\;\;
\Delta x = u. \;\;\; f'(x) = r x^{r-1}.\;\;\;$
Note that $\;\; \Delta y=f(1+u)-f(1)=(1+u)^r-1.\,$
By the approximation principle,
$\; (1+u)^r-1 \...
-1
votes
2answers
14 views
write the general term (the (k+1)th term) of the binomial expansion (a+b)^n in terms of n and k, where k < n. [closed]
I'm really unsure of what to do here, but I am trying to find the (k+1)th term or (r+1)th term of the binomial expansion (a+b)^n in terms of n and k, where k is less than n (k < n)
0
votes
2answers
53 views
Is it possible to get the area between $y^2 = 3x$ and $y = x^3$? [closed]
I am a Grade 12 student taking Calculus. Our lesson for tomorrow is about area between curves. Our teacher gave this question for us to ponder about and do advanced study. Can someone explain this and ...
0
votes
0answers
17 views
Proving a summation formula using the general Leibniz rule
I am trying to prove the following relations:
$$
\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,
$$
and
$$
\...
-3
votes
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?
$$\sum_{l=1}^{k}(\sum_{j\in B_l} Y_j^{\frac{1}{\Theta_l}})^{\Theta_l}$$
0
votes
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?
1
vote
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 $B>A>0$
$$f (x)=8 \cos (A x+B x)+19 \cos (A x-B x)+5 \cos (2 A x-B x)\\\quad+8 \cos (A x-2 B x)+2 \cos (2 A x-2 B x)+19 \cos (A x)\\+2 \cos (...
-4
votes
0answers
36 views
Using the definition of limit to show $\lim_{x\to -3}\frac{x-3}{x^2 -9} = \frac{-1}{6}$ [closed]
I need prove this limit by the formal definition of limit:
$$\lim_{x\to -3}\frac{x-3}{x^2 -9} = \frac{-1}{6}$$
0
votes
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
vote
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
votes
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
vote
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
votes
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.
-2
votes
0answers
27 views
What is the volume of the solid obtained by rotating the the region between f(x)=x and g(x)=x^2-2x on the interval [0, 3] about the y-axis? [closed]
Picture of the graphs of f(x)=x and g(x)=x^2-2x
