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

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4
votes
2answers
53 views

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

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 ...
-2
votes
0answers
10 views

Gradient descent for loss function and parameter update

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
27 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
0answers
24 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 ...
-1
votes
2answers
38 views

Evaluate the limit or prove that it does not exist

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.
1
vote
2answers
27 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
25 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
21 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
22 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
14 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
43 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 $a< b$?
9
votes
4answers
419 views

Are derivatives 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
57 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
57 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.
0
votes
0answers
14 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
30 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
28 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
14 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
34 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 ...
0
votes
0answers
83 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
37 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
17 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
4answers
116 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
33 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
34 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 $\lim_{n \rightarrow \infty} \frac{b_n}{a_n} = 0$?
0
votes
0answers
14 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
22 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
31 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
44 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
55 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
0answers
44 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 wrt $k$, transforming into an integral over the whole real line... with ...
0
votes
1answer
19 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
18 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
56 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
68 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
48 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
46 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 + ...
1
vote
2answers
60 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
-5
votes
0answers
19 views

Calculus: Proving Continuous Function by Intermediate Value Theorem [duplicate]

Prove step by step: Let $f(x)$ be a continuous function from the closed interval $[a, b]$. Use the Intermediate Value Theorem to show that $f(x)$ has a fixed point, that is, there is a point $x \in ...
1
vote
0answers
49 views

Sum involving binomial coefficient and gamma function

I was wondering if anyone has ever seen the following sum: \begin{equation} \sum_{j=0}^{n} \left(-1\right)^{j} \binom{n}{j}\frac{\Gamma\left(\mu+j\right)}{\Gamma\left(\mu+j+n+1\right)} ...
1
vote
2answers
95 views

How to differentiate this integral?

Given $$g(x)=\int_{0}^{x} (x-t)e^{t}dt$$ find out $g''(x)$ I thought of using Lebnitz theorem to differentiate it but using Lebnitz I get this $g'(x)=1\cdot (x-x)e^{x}=0$ I don't know how to find ...
0
votes
0answers
33 views

On summation of series [on hold]

Consider the equality of summations $\sum_{a} f(a) = \sum_{a}f(1-a)$ where both sums are convergent. What conditions need to be satisfied such that $f(a) = f(1-a)$ for all $a$, where $a$ is a ...
0
votes
1answer
28 views

Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even?

"Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even, and at most 3 roots if $n$ is odd?" To do this, I assume I must show that there are certain values of ...
0
votes
3answers
56 views

Proving equation has only one solution

So i want to prove that $$x^2e^x=1$$ has at least one solution for $$x\in\mathbb{R}$$ I am kinda lost and would appreciate any help. This is suppose to be solved using basic calculus but i am not ...
0
votes
2answers
25 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
2
votes
2answers
17 views

Property of real functions when derivative approaches zero

This is a question from my exam in Calculus 1. Problem 6 Let $f: [0,\infty[ \to \mathbb{R}$ be continuously differentiable and $\lim_{x \to \infty} f'(x) = 0$. a) Show that for $n \in ...
2
votes
4answers
59 views

compute the value of an indefinite integral

Help me please with this indefinite trigonometric integral. How can I solve this kind of integrals? $$\int\limits \frac{1}{\left(\cos^4(x) \cdot \sin^2(x)\right)}dx$$
1
vote
1answer
16 views

Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$

Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer. Show that $$T_2 (x) = 2x^2 – 1.$$ My attempt: $T_2 (x) = \cos (2 \arccos x)$ ...
8
votes
2answers
61 views

Solving an exponential equation with different bases

Solve the equation $2^x + 5^x = 3^x + 4^x$. I can figure out two special solutions $x=0$ and $x=1$, and I try to prove that they are the only two solutions. However, I find it hard to do so because I ...