Questions tagged [calculus]
For basic questions about limits, derivatives, integrals and applications, mainly of one-variable functions.
97,637
questions
0
votes
0answers
8 views
Gamma function and limit of integral
For $n \to \infty$, prove
$\int_{2n}^{\infty} t^n e^{-t}\ dt = o(\Gamma(n+1))$.
I have no clue to resolve this.
I started like this :
$\Gamma(n+1) =\int_{0}^{\infty} t^n e^{-t} dt$ and $\Gamma(n+1)=...
-1
votes
0answers
8 views
Vector and Scalar potential conditions
Given that A is a 3-by-3 matrix, with constants a11, a12, a13, ..., a31, a32, a33. And vector F is A*x, where x is a vector. What are the general conditions on vector A for vector F to have a 1) ...
0
votes
0answers
9 views
How to justify this limit (Gamma function)?
For all $x, t \in \mathbb R_*^+$ and $n \ge 2$ such that $t < n$. And from this equation:
$\left(1-\frac t n \right)^n \times \exp\left( n \int_0^{\frac{t}{n}} \left( \frac t n - \sigma \right) \...
0
votes
1answer
22 views
Show that a function is negative over its domain
I would like to demonstrate that the following function is negative
\begin{equation}
-\frac{t}{4\sqrt{x}^3}\bigg[1-\bigg(1+\sqrt{x}\bigg)^{\frac{1}{t-1}}\bigg]+\bigg(\frac{1}{1-t}\bigg)\bigg(\frac{t}{...
1
vote
1answer
26 views
Singularity of holomorphic function
Let $f: \mathbb C \rightarrow \mathbb C$ be a holomorphic function in an open set around some $c \in \mathbb C$, but excluding $c$. Moreover, assume that the Laurent series for $f$ around $c$ ...
3
votes
3answers
42 views
Find $\lim_{n \to \infty} \prod_{k=1}^{n} \frac{(k+1)^2}{k(k+2)}$
I have to find the following limit:
$$\lim\limits_{n \to \infty} {\displaystyle \prod_{k=1}^{n} \dfrac{(k+1)^2}{k(k+2)}}$$
This is what I tried:
$$\lim\limits_{n \to \infty} {\displaystyle \prod_{k=...
0
votes
1answer
15 views
Volume bounded by rotating a curve
Consider the region bounded by the curves $y=\sqrt{4−x^2}$, $y=2$, and
$x=2$. This region is rotated about the $y-$axis. What is the volume of the generated solid?
1
vote
1answer
15 views
Gradient vector parallel to postition vector
Let $f:\mathbb{R}^3\to \mathbb{R}$ be a differentiable fuction such that $\nabla f(x,y,z)$ is parallel to $(x,y,z)$ for all $(x,y,z)\in \mathbb{R}^3$. Show that, $f(a,0,0)=f(-a,0,0)$ for all $a\in \...
0
votes
0answers
34 views
Prove that the second derivative has a zero
my question is: Can I use Rolle's theorem to prove that the second derivative has a zero?
Consider a Real function of Real variable defined by $f(x)=(x+1)\cdot e^{x^2}$.
Prove that the second ...
0
votes
0answers
49 views
Show that $f(x) = \arctan(x)$ is uniformly continuous in R, directly from the definition
I need a proof like this, just for $\arctan x$ (without using derivate)
$ϵ>0$ and $x,y∈R$
$|f(x)−f(y)|<\epsilon$ ⟹ $|\arctan x−\arctan y|<\epsilon$
?? $|\arctan x−\arctan y|$=$|\arctan \...
0
votes
0answers
19 views
proving that a function is continuous using delta and epsilon
Prove that $x^3$ is continuous at $x=-2$
How can we prove that the limit is continuous using epslion delta proof$?$
0
votes
1answer
12 views
x-axis graph question. very short question
I have the following equation
$(x+a)y=0$
where x and y are non-negative variables. and a is strictly positive.
So, the graph go this equation is x-axis graph? Is this right?
Regardless of value ...
0
votes
0answers
20 views
$Ψ(x, y) = (e^x cos y, e^x sin y) =: (u(x, y), v(x, y))$ range and solution check
Define $Ψ:ℝ^2→ℝ^2$ by
$Ψ(x, y) = (e^x cos y, e^x sin y) =: (u(x, y), v(x, y))$
(a) What are the images, under $Ψ$, of lines parallel to the coordinate axes?
(b) What is the range of $Ψ$? Describe ...
0
votes
1answer
5 views
derivative of position vector is perpendicular to position vector
$\Large \vec{r}(s) = x(s)~\hat{\text{i}} + y(s)~\hat{\text{j}} + z(s)~\hat{\text{k}}$
$\Large \frac{\vec{r}(s)}{ds} = \frac{dx(s)}{ds}~\hat{\text{i}} + \frac{dy(s)}{ds}~\hat{\text{j}} + \frac{dz(s)}{...
0
votes
3answers
30 views
sum of continuous functions
Is there any possible way to prove that $\sum_{i=1}^{N}f(x_i)$ is smaller or equal to $0$ if we know that $f$ is smooth monotonously growing function with $f(0)=0$ and $\sum_{i=1}^{N}x_i=0$?
0
votes
2answers
20 views
Multivariable limit with min and max
1) I'm trying to show that
$$
\lim_{(x,y) \rightarrow (0, 0)} \frac{\max(x,y)}{\min(x,y)} = 1.
$$
I'm not totally sure the limit exists, but I couldn't find a counterexample so far.
I've tried the ...
0
votes
1answer
12 views
List the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use
List the critical numbers of the following function in increasing order.
$$f(\theta)=14\cos(\theta)+7\sin^2(\theta),-\pi \leq \theta \leq \pi$$
-1
votes
2answers
41 views
Solve integral $\int_\sqrt{x}^0 \sqrt{1+\sec^2 p}\ \mathrm dp $
How would I go about solving the following integral:
$$T(x) = \int_\sqrt{x}^0 \sqrt{1+\sec^2 p}\ \mathrm dp $$
I went ahead and started by flipping the bounds, but am confused on next steps, this ...
0
votes
2answers
26 views
Proving $ f(x)=x \sin{(\frac{1}{x})}-\cos{(\frac{1}{x})} $ has no limit at x=0
How to prove that $ f(x)=x \sin{(\frac{1}{x})}-\cos{(\frac{1}{x})} $ has no limit in x=0?
One-sided limits in x=0 do not exist, according to WolframAlpha.
0
votes
0answers
20 views
Find $a_0,a_1$, and $a_2$ from the fact that Simpson’s rule is exact for $f(x) = x^n$ when $n = 1,2$, and $3$.
I found the answer on Slader .
I just don't understand the concept behind it and need someone to explain it in a little for detail.
-3
votes
0answers
17 views
0
votes
0answers
8 views
Reverse Lipschitz condition and continuity implies bijection
I am trying to prove the following:
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function with $\mid f(x)-f(y)\mid\geqq c\mid x-y\mid$ for all $x,y\in\mathbb{R}$, where $c>0$ does not depend on ...
0
votes
1answer
20 views
Find $a$ and $b$ such that the improper integral converges
Find $a$ and $b$ so that the integral $$\int_1^\infty \frac{x}{(x-1)^a(1+x^2)^b}\, dx$$ converges.
I know that $\int_1^\infty \frac{1}{x^p}$ converges if $p>1$ and I've tried integration by parts ...
0
votes
0answers
25 views
Prove that the equations have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.
Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove
that the equations $$2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}\\
x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}\\
x+y+2z=\sqrt{a^2+x^2}+\sqrt{...
0
votes
1answer
43 views
Does this double limit of $\cos^{2n}(m! \pi x)$ exist?
I've figured out that the repeated limit exists:
$$
\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases}
1,&x\text{ is rational}\\
0,&x\text{ is irrational}\end{cases}
$...
0
votes
2answers
29 views
finding coordinates with derivatives
Find the coordinates of the point on the curve $x^2+xy+y^2=7$ where the slope is the same as at the point $(2,1)$.
I’ve already found $y’$ to be $\frac{(-2x+y)}{(x+2y)}$ and the slope at the point ...
0
votes
1answer
18 views
Lagrange multiplier volume maximatisation
Using method of lagrange multiplier show that among all rectangular parallelepiped inscribed in a given sphere cube has the maximum volume
2
votes
2answers
60 views
Setting up the volume $\iiint_{?}^{?}dV$
Let $$S = \{ (x,y,z) | x=a+b, y = b+c, z = -b, ac-b^2\ge0, c\ge0, a\ge0\}$$ and
$$x^2+y^2+z^2\le 1$$
Compute the volume of S.
My work:
$$V=\iiint_R 1 dV = \int\limits_{-1}^{1}\int\limits_{-\sqrt{(1-...
1
vote
0answers
37 views
Mechanics question : Tension
I think there are 10 unknowns.
$$A_y,A_x,B_y,B_x,C_x,C_y,D_x,D_y,E_x,T$$
And unable to obtain 10 equations so uanble to solve this question.
The answer is 138 lb according to the book for the ...
3
votes
0answers
75 views
Challenging Sum: compute $\sum_{n=1}^\infty\frac{H_n}{2n+1}\left(\zeta(3)-H_n^{(3)}\right)$
How to prove
$$\sum_{n=1}^\infty\frac{H_n}{2n+1}\left(\zeta(3)-H_n^{(3)}\right)=\frac74\zeta(2)\zeta(3)-\frac{279}{16}\zeta(5)+\frac43\ln^3(2)\zeta(2)-7\ln^2(2)\zeta(3)\\+\frac{53}4\ln(2)\zeta(4)-\...
1
vote
0answers
14 views
What's the connection between Legendre polynomials, Associated Legendre polynomials, and Sphereical Harmonic?
Associated Legendre Polynomials and Legendre Polynomials are strongly connected. From what I read, Legendre polynomials is a special case of Associated Legendre polynomials where $m=0$. Thus the ...
-1
votes
0answers
17 views
How to find direction in which Directional Derivative is zero? [on hold]
Please help me with this question
***Find the direction in which the directional derivative of
f(x, y) = (x^2 - y^2)/xy at (1, 1) is zero.
**
3
votes
2answers
39 views
Stability without Lyapunov methods
I've been having some problems trying to solve a problem which appears in the book im following, so any help would be really appreciated.
Defn. A fixed point $x_{0}$ is asymptotically stable if it is ...
0
votes
3answers
22 views
Find $f(x)$ given $f'$' and two initial values
I was given that $f''(x) = 6x + \frac{1}{x^2}$ , $f(1)=0$ and $f(2) = 6$. I have gotten down to $x^3 + ln(|{x}|)+Cx + D$ and I am confused on how to find the values for $Cx$ and $D$ .
0
votes
1answer
37 views
Improper integrals(Limit comparison test)
Here is the question that someone gave to me.
For the $f(x) = {x-3 \over x^2 \sqrt{\vert x-2\vert \vert x-3 \vert}}$ , Determine $\int_{2}^\infty$$ f(x)dx$ is converge or not
So my colleague ...
1
vote
1answer
25 views
How to turn a limit to a definite integral
I am stuck as to where to go with this problem. What I know now is that this may be a Reimann Sum problem but I am not 100% sure. I also know that the integral is lower bound by 0 and upper bound by 4....
0
votes
2answers
23 views
Applying the Intermediate Value Theorem on periodic functions
I have just recently covered the Intermediate Value Theorem, and I wanted to practice solving problems involving this theorem. However, I encountered a problem that I am not exactly sure how to tackle ...
1
vote
0answers
32 views
intersections of decreasing sequences of convex sets [on hold]
Let $\{C_k\}_k$ denote a decreasing sequences of convex sets, where $C_1 \supseteq
C_{2} \supseteq\cdots $, its intersection $C = \bigcap_k C_k \neq \emptyset$. For any vector $x\in\mathbb{R}^n$, ...
3
votes
4answers
48 views
Find $\lim\limits_{n \to \infty} \sqrt[3]{n^3+2n^2+1}-\sqrt[3]{n^3-1}$.
I have to find the limit:
$$\lim\limits_{n \to \infty} \sqrt[3]{n^3+2n^2+1}-\sqrt[3]{n^3-1}$$
I tried multiplying with the conjugate of the formula:
$$(a-b)(a^2+ab+b^2)=a^3-b^3$$
So I got:
$$\lim\...
0
votes
2answers
26 views
How do I find a function $f(x,y)$ such that $\nabla f = \langle y,-x\rangle$?
I've been stuck on this problem for a while and I am starting to think that it's not possible. Could someone please point me in the right direction? Thanks!
0
votes
2answers
28 views
Use the intersection of the following sets to prove the integral
if $0 \in \{ f(0),g(0)\} \cap \{f(e),g(e)\}$ prove that$\int_0^e f(x)g'(x)= -\int_0^e g(x)f'(x)$.
I am lost on this one this looks very similar the theory of integration by parts
0
votes
2answers
26 views
Attempt of a question dealing with the Intermediate Value Theorem
I'm new to proofs, and I recently got introduced to the Intermediate Value Theorem and decided to practice some questions. So, I was wondering if I got the following question right and if I'm even on ...
1
vote
1answer
19 views
Derivative of a generic polynomial function
Let $P:M_n(\mathbb{C})\to\mathbb{C}$ be a polynomial function, $A=(A_{ij}),E_{ji}\in M_n(\mathbb{C})$,where $E_{ji}$ is defined as matrix filled with zeros except $1$ in j-th row and i-th column and $...
0
votes
2answers
32 views
Question involving the application of Intermediate Value Theorem
I'm new to mathematical proofs, and I have just covered the Intermediate Value Theorem. I have tried to practice my understanding of the theorem, but I have encountered a question that I'm not sure ...
1
vote
2answers
48 views
Use the $\epsilon$ - $\delta$ definition to show that $\lim_{x\to \sqrt2} \frac{1}{2}(\frac{2}{x}+x) = \sqrt2$
Use the epsilon-delta definition to show that $\lim_{x\to \sqrt2} \frac{1}{2}\left(\frac{2}{x}+x\right) = \sqrt2$.
I have been shown the following approach to solve this:
Let first $\epsilon > ...
1
vote
0answers
47 views
Converting an Integral to a Sum
I have an integral over $t$ where I choose $t$ takes discrete values $t=0,1,2,...$ and would like to write the integral as a sum using the same function. I realize that this is not necessarily ...
0
votes
2answers
31 views
Calculation of limit (two sides) [on hold]
I don't have a clue, how to solve this limit from each side (+-).
It is +inf*0 from left side and -inf*0 from right side.
Can ...
4
votes
2answers
60 views
Evaluate $\sum_{n=0}^{\infty} \frac{z^{kn+p}}{(kn+p)!}$
$$\sum_{n=0}^{\infty} \frac{z^{kn+p}}{(kn+p)!}$$
I have found out that
$$\sum_{n=0}^{\infty} \frac{z^{kn}}{(kn)!} = \frac{1}{k} {}\sum_{i=0}^{k-1} e^{z\varepsilon_i}$$
where $\varepsilon_i$ is the ...
3
votes
2answers
57 views
What is the supremum of the set?
What is the supremum of the set consisting the following real numbers:
• 0.200 . . .
• 0.2500 . . .
• 0.25200 . . .
• 0.252500 . . .
Would it just be 0.252500?
0
votes
4answers
72 views
Solve the ODE $y''-x^{-1}y'+x^{-2}y=0$.
Consider ODE $y''-x^{-1}y'+x^{-2}y=0$.
I have found the values of the constant $n$ for that $y(x) = x ( \ln x)^n $ satisfies the ODE as $0$ or $1$. How do you calculate the general solution I'm ...
