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

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3
votes
3answers
75 views

Integral of $\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx$

I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it? \begin{equation} ...
1
vote
2answers
51 views

Integrating triangle in a 2D plane

I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$. I know this loop forms a triangle with all three sides ...
10
votes
5answers
2k views

Why can a derivative be non-linear?

A definition of the derivative is that it is the slope of the tangent line. For example, $x^3$ has a quadratic derivative. How could the slope of the tangent line be non-linear?
1
vote
2answers
50 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
4
votes
2answers
74 views

Find the integral $\int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx$

The integral can be represented as $$ \int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx= \int \left(\frac{1+x}{1-x}\right)^{1/2}\mathrm dx $$ Substitution $$t=\frac{1+x}{1-x}\Rightarrow ...
0
votes
1answer
43 views

Maximum of polynomial [on hold]

I was studying statics and came across this problem: Find the value $\beta$ such that $P$ has a maximum value in $R^2 - 1000^2 = P^2 + 2000P\cos(75^{\circ}+\beta)$. When $R$ is constant, the ...
2
votes
1answer
18 views

Calculate the surface integral of bounded cylinder

Evaluate $$\int\int z^2\,dS,$$ where $S$ is the part of outer surface of cylinder $x^2+y^2=4$ between the planes $z=0$ and $z=3$. The answer given in book is $\pi$ but I am not getting this ...
2
votes
1answer
31 views

Finding the maximum value of a divergent series [on hold]

I came across this divergent sum- $$\sum_{n=1}^\infty\frac{1}{n+1}$$ Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that ...
1
vote
1answer
36 views

the maxima of given function

What's the maxima of $$2^{\sin(x)}+2^{\cos(x)}$$ I found max by taking logs and then differentiating and equating to $0$ at $x=45°$ so the answer is $2^{\frac{\sqrt{2}+1}{\sqrt{2}}}$ am I right or I ...
0
votes
2answers
30 views

Find the area bounded by function $y^2=16-2x$, the tangent to the curve at the point $(6,2)$ and the $y$-axis

First we find a tangent line of function $y^2=16-2x$ at $T(6,2)$: $y_t-y_0=f'(x_0)(x-x_0)$ where $x_0=6,y_0=2$ $y^2=16-2x\Rightarrow y=\sqrt{16-2x}$ or $y=-\sqrt{16-2x}$ Derivative of ...
3
votes
0answers
31 views

Why would an equation switch signs when something becomes independent of time? (Traffic Flow)

EDIT: I'm too tired for math and the answer to my question is explained in a comment below. Should this post be removed? Not sure if it adds much to the community given that it was all just me ...
2
votes
3answers
40 views

How to derive the equation of tangent to an arbitrarily point on a ellipse?

Show that the equation of a tangent in a point $P\left(x_0, y_0\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, could be written as: $$\frac{xx_0}{a^2} + \frac{yy_0}{b^2} = 1$$ ...
7
votes
3answers
87 views

$f'(x) = g(f(x)) $ where $g: \mathbb{R} \rightarrow \mathbb{R}$ is smooth. Show $f$ is smooth. [duplicate]

Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is differentiable and $g: \mathbb{R} \rightarrow \mathbb{R} $ is infinitely differentiable, i.e. $ g \in C^{\infty}(\mathbb{R})$, where we know ...
-1
votes
0answers
12 views

Gradient descent for loss function and parameter update [on hold]

How exactly do i derive the gradient function? I know what the gradient does, however, i don't know exactly how to go about finding how to derive the gradient function. problem
0
votes
0answers
39 views

I have a question about calculating integral

I need some help to solve this integral: $$\int_{4x}^{\infty} \frac{w^{\frac{m}{2}}e^{-a\sqrt{w+2\sqrt{xw}}}}{\sqrt{w^2-4xw}}\mathrm{d}w.$$ Thank you
1
vote
1answer
39 views

Find the partial derivative of a sphere with equation $x^2+y^2+z^2=4$

We have a sphere with the following equation: $x^2+y^2+z^2=4$ We seek to find the partial derivative, with respect to $x$, of this equation. We think of this equation as a function of three ...
0
votes
2answers
52 views

Evaluate the limit or prove that it does not exist [on hold]

I want to evaluate $\displaystyle \lim_{(x,y)\to (0,0)}\frac{\ln(1-x^2-y^2)}{x^2+y^2}$. Any idea how to prove the answer is -1? I don't see an easy way to simplify this.
2
votes
2answers
37 views

Evaluate this limit of inverse trigonometric and radical functions without l'Hospital

How can I solve this using only 'simple' algebraic tricks and asymptotic equivalences? No l'Hospital. $$\lim_{x \rightarrow0} \frac {\sqrt[3]{1+\arctan{3x}} - \sqrt[3]{1-\arcsin{3x}}} ...
2
votes
1answer
44 views

Find the radius of convergence and interval of convergence of the series

Find the radius of convergence and interval of convergence of the series: $\sum_{n=1}^{\infty}n^n x^{n^4}$ I'm really lost as to how to approach this problem. The other power-series problems were ...
0
votes
2answers
25 views

Show that for any $x_0\in \mathbb{R}$, the one sided limits exist and that $f^+(x_0)\geq f^-(x_0)$.

Suppose $f(x)$ is a monotone increasing function defined for all $x\in \mathbb{R}$. Show that for any $x_0\in \mathbb{R}$, the one sided limits $$f^+(x_0)=\lim_{x\to x_0^+}f(x) \text{ and } ...
0
votes
1answer
26 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
0
votes
0answers
18 views

What is the largest t-interval on which guarantees a unique solution? [on hold]

What is the largest t-interval on which guarantees a unique solution for this equation? $$y'' + y'+ 3ty = \tan t,\quad y(\pi) = 1,\quad y'(\pi) = -1$$
1
vote
2answers
47 views

Infinite closed subset of $[0, 1]$ that does not have any subset of the form $[a, b]$ for $a< b$?

What is an infinite closed subset of $[0, 1]$ that does not have any subset of the form $[a, b]$ for $a< b$?
32
votes
5answers
2k views

Which derivatives are eventually periodic?

What derivatives are eventually periodic? I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$. If $f(x)$ was a polynomial, and ...
2
votes
3answers
75 views

How do I finish this trig integral $\int_0^{\pi/4}\frac{\sin^2 \theta}{\cos \theta}d\theta$?

I got up to the part where it's $$\frac{9}{125}\int_0^{\large \frac{\pi}{4}}\frac{\sin^2\theta}{\cos\theta}\,\,d\theta$$ but I can't figure out how to finish it off. By the way the original problem ...
0
votes
1answer
24 views

Help to evaluate integral in cartesian and cylindrical

I want to solve $$\iint_{R} (x+z)dR$$ where R is the first octant of the cylinder $x^2+y^2=9$ and between $z=0$ and $z=4$ I think it could be done in either cylindrical or Cartesian. I am having ...
4
votes
2answers
65 views

Finding limit via Sandwich Theorem: $\lim_{n\to\infty} n\sum_{n+1}^{2n} \frac{1}{i^2}$

Question: Use the Sandwich Theorem to find $$\lim_{n\to ∞} n\sum_{n+1}^{2n} \frac{1}{i^2}$$ Appreciate any guidance.
-1
votes
0answers
23 views

Volume of Revolution (Semi-Circle, Line)

This is purely for academic curiosity and is not part of any homework assignment, quiz, or exam. Suppose $R$ is the region bounded by the curves $x^2 + y^2 = 36$ on $[-6,0]$, $y = 2x-6$ on $[0,6]$ ...
1
vote
3answers
37 views

Limit of derivative does not exist, while limit of difference quotient is infinite

Can anyone show an example of a function $f$ of a real variabile such that $f$ is differentiable on a neighborhood of a point $x_0 \in \mathbb{R}$, except at $x_0$ itself; $f$ is continuous at ...
0
votes
0answers
36 views

How to square a number that got more digits than search results “digits” on Google.

I am implementing the quadratic sieve algorithm. And I got run in unexpected problem. Take a look at those two final steps of the algorithm as described in wiki. Use linear algebra to find a ...
0
votes
0answers
16 views

Extending the unit tangent to an analytic function

Suppose $\Gamma\in\mathbb{R}^2$ is a smooth, simple closed curve and denote its unit normal vector(say outward) at each point $z\in\Gamma$ by $T(z)$. Under what assumption on the boundary curve, $T$ ...
0
votes
1answer
35 views

Prove, that for any $0<a<1$, $-\frac{a}{1+a}<\ln(1-a)<-a.$

Prove, that for any $a>0$, $$a>\ln(1-a)>\frac{a}{1+a}.$$ Prove, that for any $0<a<1$, $$-\frac{a}{1+a}<\ln(1-a)<-a.$$ Proof of 1: We will prove (1) by doing a proof by ...
1
vote
0answers
110 views

Another way of doing integration

What's your option for calculating this integral? No full solution is necessary, it's optional as usual. Calculate $$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) ...
2
votes
2answers
43 views

Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$.

I have tried to solve this limit : $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. Where n $\in\mathbb{N}$. I have understood that the limit exists and goes to 0 if the argument becomes ...
0
votes
2answers
18 views

Finding common tangent line to two functions

Sometimes you want to find the common tangent line of two functions. The first thing that comes to mind to a person that is learning basic calculus is that you should equal the derivatives of those ...
3
votes
3answers
127 views

Is there any way to solve integral of $\sqrt{8-x^{2}}$ without using $\sin$ or $\cos$ formulas?

I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with ...
-1
votes
1answer
40 views

Parametrization of two curves. [on hold]

I have an assigment to parametrize the edge of the volume which is given by the intersection of the two curves $x^2+y^2+z^2=2$ and $z=x^2+y^2$. I really have no idea how i can parametrize this? I know ...
2
votes
0answers
36 views

Reciprocal of a limit that goes to infinity

Lets say we have a limit $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = +\infty$, then is it safe to assume that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n} = 0$?
-1
votes
1answer
23 views

Is this sum differentiable?

Let $R$ be an infinite set of complex numbers $\rho$ with $0<\Re(\rho)<1$, $x$ be a nonzero real number and consider the summation $\sum_{\rho} x^{\rho}$. Is this sum differentiable with respect ...
0
votes
1answer
25 views

Non-monotonically decreasing flow whose limit is $\vec{0}$

I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but ...
0
votes
1answer
45 views

Multivariable Calculus, Parametrization and extreme values

I want to find the extreme values of the function $f(x,y,z) = 2x + 2y + z$ under the constraints $x^2+y^2+z^2 \le 2$ and $x^2 + y^2 \le z$ The task is to use a parametrization of the two ...
3
votes
3answers
51 views

Are these $3$ functions linearly independent or dependent?

Are the functions $f, g, h$ given below linearly independent?, If they are not linearly independent, find a nontrivial sol'ns to the equations below $$f(x)=e^{2x}- \cos(9x), \quad g(x)=e^{2x}+ ...
2
votes
3answers
58 views

integrate $\int \frac{dx}{(9+x^2)^2}$

$$\int \frac{dx}{(9+x^2)^2}$$ $x=3\tan\theta$ $dx=\frac{3}{\cos^2\theta}d\theta$ $$\int\frac{\frac{3}{\cos^2\theta}}{(9[1+\tan^2\theta])^2} \,d\theta = ...
0
votes
1answer
83 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving with respect to $k$, transforming into an integral over the whole real ...
0
votes
1answer
20 views

Find the particular solution for the following differential equation when z=z1, x=0

$\displaystyle \frac{\text{d}z}{\text{d}x} = \frac{2a(x-z)}{1-ax}$ where $a$ is a constant. $(x>0, a\ne 0)$ So after rewriting the above as $\displaystyle \frac{\text{d}z}{\text{d}x} + ...
3
votes
2answers
19 views

Finding a root approach with a polynomial

So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to ...
2
votes
2answers
61 views

What is the partial derivative of $f(x,y(x))$?

What is the total derivative of $f(x,y(x,z))$ with respect to $x$? Is it $$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}?$$ If this is correct, what is ...
2
votes
2answers
70 views

integrate $\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$

$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$$ $$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx=\int \frac{3\left(\frac{16}{9}-x^2\right)^{\frac{3}{2}}}{x^6}dx$$ $x=\frac{4}{3}\sin\theta$ ...
1
vote
2answers
49 views

Problem with Indefinite Integral $\int\frac {\cos^4x}{\sin^3x} dx$

I'm stuck with this integral $\int\frac {\cos^4x}{\sin^3x} dx$ which I rewrote as $\int \csc^3x \cos^4xdx$ then after using the half angle formula twice for $\cos^4x$ I got this $\frac 14\int ...
1
vote
1answer
50 views

Evaluate $\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x) \sin x} $

Evaluate $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx \:\:\: n \in \mathbb{N}$$ $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx + ...