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

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3answers
46 views

Solve $\int_{0}^{1}y(xt)dt=ny(x)$ [on hold]

Solve $\int_{0}^{1}y(xt)dt=ny(x)$ Could someone help me to solve this?
1
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1answer
28 views

What trig identites were used in rewritting this equation

The undamped response for a system is: $$x(t)=x(0)e^{-\zeta \omega t}(\cos \omega_d t+ \frac{\zeta}{\sqrt{1-\zeta^2}} \sin \omega_d t)$$ In the book they claimed using trig identities they were able ...
1
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1answer
31 views

The function $\lim_{n\to\infty}({4^n+x^{2n}+\frac{1}{x^{2n}})}^{1/n}$ is non-derivable at

The function $$f(x)=\lim_{n\to\infty}\left(4^n+x^{2n}+\frac{1}{x^{2n}}\right)^{\frac{1}{n}}$$ is non-derivable at how many points? The limit is of $\infty^0$ form. Is it an indeterminate form or ...
1
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1answer
31 views

Evaluate $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{k^p}{n^{p+1}}$

Let $p$ be a real number. Evaluate $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \dfrac{k^p}{n^{p+1}}$. I think this depends on the value of $p$ because then large $n$ would mean small ...
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1answer
28 views

Surface integral of the hemisphere $z^2 = 16- x^2 - y^2$

solve the surface integral $\iint_S z^2 dS$ $z = \sqrt {16-x^2-y^2}$ $dS = 4/\sqrt {16-r^2} $ $4\int_0^{2\pi}\int_0^4 ((16-r^2)/\sqrt {16-r^2})rdrd\theta$ What am I doing wrong with my integral ...
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2answers
71 views

What is the $\lim _{x→\infty}(0.5)^x$? [on hold]

I need to verify the answers to two questions: 1)What is the $\lim_{x→∞}$ $(0.5)^x$? A: $.25$ 2)What is the value of $\lim_{h→0}$ $(8^h−1)/h$ A: $-1$ Are my answers correct? If not, what are ...
2
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3answers
48 views

Derivatives: If $f(x)= 1/e^x $ then

If $f(x)= 1/e^x $ then $ƒ′(x) = ?$ A: $1/e^x⋅ln(e^x)$ If $ƒ′(x) = e^x$ then $ƒ(x) = ?$ A: $x$ Are my solutions correct?
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1answer
16 views

Is $f(x,y) = (x^2-y^2,xy)$ lipschitz on $\mathbb{R}^2$?

How can I show that the $f$ is lipschitz? I try to calculation such that $|f(x_0, y_0) - f(x_1,y_1)|^2 = ((x_0^2 -y_0^2)-(x_1^2 -y_1^2))^2 +(x_0 y_0 -x_1y_1)^2$ and $|(x_0-x_1,y_0-y_1)|^2 = ...
2
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1answer
29 views

Continuity of the integral of a continuous function composed with a measurable one

For a continuous function $g:\mathbb{R} \rightarrow \mathbb{R} $ and functions $x\in L_1[a,b]$, is it true that the functional $$ F(x)=\int_a^b g(x(t)) \,d\mu(t) $$ is a continuous function from ...
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4answers
49 views

right circular cylinder inscribed in a sphere

Find the dimensions of the right-circular cylinder of greatest vloume that can be inscribed in a sphere with a radius of 6 $in$ I think I need help visualizing, and maybe the solution. I've ...
1
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0answers
14 views

Cardinality of strict extrema of a real function

I recently encountered a problem seeking to prove that a real function can only have a maximum number of #$ \mathbb{N}$ strict maximums. It may be that I have copied the problem improperly since there ...
1
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2answers
19 views

Find all Points on the Surface at which the Tangent is Parallel to the Plane

The problem: Find all points on the surface $z=x^3+xy^2$ at which the tangent plane is parallel to the plane $2x+2y+z=0$ So I established $f(x,y,z)=x^3+xy^2-z$ and the normal vector determined from ...
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3answers
39 views

Implicit Differentiation - What am I doing wrong?

I need to find $y'$for the following equation: $$ e^{\frac{x}{y}} = x-y $$ Before differentiating I decided to perform a quick rewrite: $$ \begin{align*} e^{\frac{x}{y}} &= x-y \newline ...
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1answer
31 views

If f(x) is continuous on $[a,b],$ differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b),$ then f'(x) is stable. [on hold]

If $f(x)$ is a continuous function on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b)$ then $f'(x)$ is stable $\left(\text{i.e.},\ f'(x)<0\ \ \text{or}\ \ ...
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1answer
34 views

Integrate $\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$ [on hold]

I'm having a trouble with this integral expression: $$\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$$ I want to solve to using residue but it seems hard.
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3answers
83 views

Maximum and minimum of of $f(x)=|x-1|+|x-2|+|x-3|$

I am trying to find the maximums or minimums of $$f(x)=|x-1|+|x-2|+|x-3|$$ (if there exist). My attempt: First I compute the derivative and tried to find critical point, i.e, $f'(x) = ...
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0answers
47 views

Some basic manipulations with infinite series

In a linear algebra book that I am studying they prove that $$ ...
1
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0answers
33 views

Pendulum equation

If we denote by $\phi^{-1}$ the inverse of the function that satisfied the equation $\phi'=\sqrt{(1-k^2\sin^2\phi)}$, we obtain $$ \left( \phi^{-1}\right) ^{\prime}\left( t\right) =\frac{1}{\sqrt ...
2
votes
2answers
29 views

Showing a recursively defined sequence is convergent

For $a_1=1$ and $a_{n+1} = 1 + \frac{a_n}{3+n}$, I want to show that the sequence $a_n$ converges. I will use the Monotone Convergence Theorem. Of course, the sequence is bounded below by $1$. Now I ...
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1answer
19 views

Laplace pairs - proof of summation transform

I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end. Here's the question: If $F_i(t)$ ...
0
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1answer
34 views

Find the value of a such that F(a) achieves its minimum value

Find the value of a such that $F(a)$ achieves its minimum value. $$F(a)=\int_{0}^{\pi/2} \left|\sin x - a\cos x \right| dx $$ I'm trying to use following fact to solve the problem but then I need ...
0
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1answer
24 views

Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

I came across this inequality today: $$\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$$ I realise if we let $h \to 0$ we obtain the derivative on the left hand side so I can see it has ...
0
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2answers
19 views

Find critical point by graph observation

I can easily see that points 1 and 5 and 6 are critical points by observation. I can see that since the function is not defined at point 3, there can be no critical point. However, I don't see why ...
1
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1answer
30 views

Integral with Dirac delta function

We are given that: $$u(x,t)=\frac{2}{\pi} \int_0^\infty e^{-k^2t}G_s(k)\sin {kx}\space\text{d}k,$$ where $G_s(k)$ is the Fourier sine transform of $g(x)$. Find the solution $u(x,t)$ when ...
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0answers
27 views

Is there a way to calculate RMS value continuously?

Using that the RMS by definition is: $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ which can be calculated by using Riemann sums in the following way: $\sqrt {\frac 1N\sum_0^Nf[i]^2} $ I've tried that in ...
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1answer
46 views

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? [on hold]

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? Thank you very much.
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0answers
16 views

Proving a functional series does not uniformly converge within a range

Prove that the functional series $\sum_{n=1}^{\infty}x^{n}$ does not uniformly converge where $\left|{x}\right| < 1$, but does converge uniformly where $\left|{x}\right| \le q < 1$. I don't ...
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1answer
54 views

Why doesn't L'Hospital's rule work for this limit?

Let $a,b,A,B$ be positive real numbers, with $a>b$ and $A>B$. Consider the limit: $$\lim_{x\to\infty}\frac{ax+b\sin(x)}{Ax+B\sin(x)}$$ By the squeeze theorem, the limit exists and is equal to ...
0
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4answers
54 views

Why does the sequence $a_n = (n^2)(1 - \cos(\frac{4.4}{n}))$ converge to 9.68?

Find the limit of the sequence whose terms are given by $a_n = (n^2)(1 - \cos(\frac{4.4}{n}))$. The given answer for this problem is $9.68$. What rules about sequences, and steps, should be taken ...
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0answers
20 views

Plotting an implicit function

I have a function $u(\theta)=f(\theta)$, where $f$ is known, and I also have a relation between $\theta$ and $x$, of type $G(\theta)=x$. The problem is that I can not solve for $\theta$ as ...
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3answers
55 views

Why does this Calculus II sequence diverge?

What steps should I take to understand why this sequence diverges (specifically, not to +∞ or -∞)?
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4answers
67 views

How to evaluate $\lim _{x\to \infty }x\left(\arctan x-\frac{\pi }{2}e^{1/x}\right)$?

I have a problem with this limit, I have no idea how to compute it. Can you explain the method and the steps used (without L'Hôpital if is possible)? Thanks $$\lim _{x\to \infty }x\left(\arctan ...
1
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1answer
56 views

$a_n = b_n -b_{n-1}$ Prove that $\sum_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists

Let $\{b_n\}$ be a sequence Let $a_n = b_n - b_{n-1}$. Prove that $\sum\limits_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists. I am extremely stuck on this homework problem and ...
0
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1answer
35 views

Derivative of log-likelihood cost function with respect to a matrix

Recently, I am learning derivative method to a function and thanks to @hans help, I can solve those which can be expressed by Frobenius product. But for the log-likelihood function, I do not how to ...
1
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1answer
25 views

Mean value theorem, Wierstrass theorems

I have a question that is related to these theorems I tried to tackle but got stuck Please let me know if it is the proper way to go ? The question is: Let $f:[0,1]\rightarrow\mathbb{R}$ be a ...
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0answers
13 views

newton raphson method- root outside the interval

I am using Newton Raphson method to show that there is a root in the interval [-2,0] $f(x)=x^4 + x^2+8x-1 $ then we need: $ f(x)=0$ so: $x^4 + x^2+8x-1=0$ the derivative is $f'(x)= 4x^3+2x+8$ I ...
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1answer
15 views

How to prove that the right derivative of a convex function is right continuous?

let $f$ be a convex function and $D^+f$ be the right-derivative of $f$, I want to show that $D^+f$ is right continuous. first I want to use the $\epsilon-\delta$: by the definition of $D^+f$, ...
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1answer
35 views

Area between two curves (Demidovich)

I'm trying to solve some problems on definite integrals from Demidovich's book and I'm stuck on calculating the area between two curves defined by: $$y_1 = \frac{a^3}{a^2 + x^2}, y_2 = 0$$ Any hints ...
2
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1answer
113 views

$\frac{0}{0}$ Indeterminate

I confused myself a little bit the other day explaining to someone why $\frac{0}{0}$ is defined, but not well-determined. I know the standard explanation is $y\cdot 0=0$ for all real $y$. First, why ...
1
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4answers
58 views

How to evaluate $\lim _{x\to 1}\left(\frac{x+2}{x-1}-\frac{3}{\ln x}\right)$?

I have a problem with this limit, I have no idea how to compute it. Can you explain the method and the steps used (without L'Hôpital if is possible)? Thanks $$\lim _{x\to ...
0
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0answers
64 views

Limit involving integrals [on hold]

Given $$I_n=\int_{0}^1\frac{x^n}{x+2014} dx$$ Solve $$\lim_{n\to\infty}nI_n$$
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0answers
29 views

Having trouble understanding what exactly I am integrating for a hydrostatic force/pressure calculus question

So the question is: a trough has vertical ends that are trapezoids with parallel sides of length 4m (top) and 2m (bottom) and a height of 3m. If the trough is filled with water to a depth of 2m, find ...
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1answer
14 views

Significance of derivative in finding square free decomposition

If $gcd(f(x),`f(x))=1$ then f(x) is square free. But what is the reason behind taking derivative of f(x)? How one came to this conclusion?
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0answers
42 views

Hint for solving an indefinite integral

I would like to solve the following integral $$\int \frac{dx}{x^2 \sqrt[4]{(a-x^2)(b+x^2)}},$$ where $a$ and $b$ are the real constants. My attempt: $$\sqrt[4]{(a-x^2)(b+x^2)} = \sqrt[4]{-\bigg[x^4 ...
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1answer
67 views

For what x the series converges?

I have the following problem: For what values of x the following series: $$ \sum_{n=1}^\infty n! \cdot (x-4)^n$$ a) Converges absolutely. b) Converges conditionally. I started by using the ratio ...
5
votes
1answer
88 views

Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
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votes
1answer
24 views

Forms of functions in dynamical systems

I wanted to read some introductory material about dynamical systems since I might need a basic understanding of them in a related task. So, as far as I see, in a continuous time dynamical system, we ...
2
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0answers
56 views

Graphing the surface $z = xy$

Let the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the ...
0
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1answer
36 views

Show that there is a continuous function $h$ over $[a,b]$ for which $h(x) \leq f(x)$ and $\int_{a}^b (f(x)-h(x))dx < \epsilon$

Assume $f$ is integrable over $[a,b]$ and $\epsilon > 0$. Show that there is a continuous function $h$ over $[a,b]$ for which $h(x) \leq f(x)$ for all $x \in [a,b]$ and $\displaystyle \int_{a}^b ...
1
vote
1answer
23 views

euler's theorem of homogeneuos function

suppose that $f$ is a mapping from $\mathbb R^n \to \mathbb R$. a positively homogeneous of degree $n$ and suppose $f_1, f_2, .... f_N$ are continuous for $a \neq 0$. then $$\sum_{i=1}^n a_if_i(a)= ...