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Questions tagged [calculus]

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

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6 views

How can you tell if an equation represents an exponential growth or decay?

How can you tell if an equation represents an exponential growth or decay? Hello, I am a Pre Calc/Trig student and I would appreciate the help.
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0answers
18 views

Fibonacci spiral on sphere surface

I Need to figure out how the Fibonacci spiral on sphere surface can be drawn with given number of points which are evenly distributed over sprial, calculate to Radius with respect to given ...
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1answer
18 views

Finding the function when given definite integral output

Find a function $f$ and a positive number $a$ such that: $$\int_{\sqrt y}^a f(t)\ln(t)\, dt= {\exp(y)\over2}-\ln\left({\sqrt y\over a}\right)-\pi$$ for all $y>0.$
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5answers
66 views

How to prove that $\left(\ln(\ln(x)) \right)^2 \lt \ln(x)$

How to prove that $\left(\ln(\ln(x)) \right)^2 \lt \ln(x)$ for sufficiently large $x$ This is what I did. Using L'Hopital's rule we have $$\lim_{x\to\infty}\frac{\left(\ln(\ln(x)) \right)^2 }{ \ln(...
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0answers
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How to find the Current in terms of C for the equation in description

For $${V=C\cdot sin(120\cdot \pi\cdot t)}$$ where $$t$$ is time (in seconds) and is a constant. The square root of the average value of $$V^2$$ over one period (or cycle) of $$V$$ is called the root-...
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0answers
13 views

Proving bound on the solution of a first order linear ODE.

Given, $d’ + fd = -E$, where $|E(t)|\leq e^{-\delta t}$, $\delta >0$. $0<c_0<f(t)<C_0$ for all time $t>0$. Prove that $|d|<e^{-\delta t}, \forall\ t>t_0,$ for some $t_0$ ...
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1answer
39 views

Show $\int \frac{\sin(x^p)}{x} dx = \frac{\operatorname{Si}(x)}{p} $

I was messing around with the Fresnel integral and the Sine integral and found that $\int_{0}^{\infty} \frac{\sin(x^2)}{x}dx=\frac{\pi}{4}$ but I dont see how to extend to irrational powers.
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3answers
54 views

Can you spot the mistake?

Let $$I=\displaystyle\int\sin^2x \ dx$$then $$I=\displaystyle\int(1-\cos^2x) \ dx=x+C-\displaystyle\int \cos^2x \ dx$$ Using the substitution $x=x+\frac{\pi}{2}$ we get $$I=x+C-\displaystyle\int\...
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0answers
8 views

Integrate a Generalized incomplete gamma function

My question is about an integral but not an ordinary one $\int_{le^{-bt'}}^{l}s^{l-1}e^{-s}ds$ I have the idea of use the leibniz integral rule because I don't want to use the Generalized ...
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6answers
40 views

How do you differentiate $x^{\cos(x)}$ [on hold]

How do you differentiate $x^{\cos(x)}$ I encounter this problem in my homework but I don't know how to differentiate it. Do you need use logarithmic differentiation to do this? Thank you!
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0answers
43 views

Stuck on an integral while solving an ODE

I was solving $$\left(1+3e^{y/x}\right)\,\mathrm dx+3e^{y/x}\left(1-\dfrac xy\right)\,\mathrm dy=0\\ \implies \dfrac{\mathrm dy}{\mathrm dx}= \dfrac{1+3e^{y/x}}{3e^{y/x}\left(\dfrac xy-1\right)}$$ ...
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1answer
35 views

Evaluating limit with 2 unknown parameters

I have another question involving limits with unknown parameters: In this one, the limit approaches infinity: $\lim \limits_{x \to 1 } {{\frac{x^2+ax+b}{(x-1)^2}}}= \infty $ This one is really ...
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1answer
25 views

How do you use logarithmic differentiation to differentiate $y=\frac{e^{-x}\cos^2x}{x^2+x+1}$

How do you use logarithmic differentiation to differentiate $y=\frac{e^{-x}\cos^2x}{x^2+x+1}$. I can take the logarithm on both sides $\ln(y)=\ln(\frac{e^{-x}\cos^2x}{x^2+x+1})$ That would be $\ln(y)...
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3answers
28 views

Find $a$ such that $\lim \limits_{x \to 1 } {{\frac{x^2+(3-a)x+3a}{(x-1)}}}= 7$

I have this limit equation below: $$ \lim \limits_{x \to 1 } {{\frac{x^2+(3-a)x+3a}{(x-1)}}}= 7 $$ One needs to determine the value of $a$ (where $a$ is a real number) such that as $x$ approaches $1$...
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0answers
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Using the Fundamental Calculus Theorem for two variables to prove smoothness.

There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters: $Pr(v)=\int_{G(v)} dP(v)$, where $G (v)$ is the "goal" region, the probability of getting ...
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2answers
48 views

An Integral Error

I was studying the derivations for the volume and surface area of a sphere . One derivation , for the volume of the sphere is the disk-method . A circle of radius $r$ is considered , centred at ...
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1answer
24 views

Questions on the proof that $f$ below is $f\in C^\infty(\mathbb{R})$

Given the function, $$f(x)=\begin{cases}\exp\left(-\frac 1 x\right)&x>0\\ 0&x\leq 0,\end{cases}$$ it is a common exercise to show that $f\in C^\infty(\mathbb{R})$. An attempt to this ...
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1answer
70 views

Finding the integral $\int_0^\infty\sin(x^n)dx$ [duplicate]

Is there a non-complex number involving method to find the following integral $$\int_0^\infty\sin(x^n)dx$$ Maybe something using the idea similar to that for evaluating $\frac{\sin x}{x}$ under ...
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0answers
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If $f(x)$ is a differentiable function amd $g(x)$ is a double differentiable function such that $|f(x)| \leq 1$ and $f'(x) = g(x)$. [duplicate]

If $f(x)$ is a differentiable function amd $g(x)$ is a double differentiable function such that $|f(x)| \leq 1$ and $f'(x) = g(x)$. If $f^2(0) + g^2(0) = 9$. Prove that there exists some $c \in (-3,...
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5answers
87 views

Evaluating $\lim_{(x,y)\to(0,0)}\frac{x^2+y^2}{\sin^2y+\ln(1+x^2)}$

$$\lim_{(x,y)\to(0,0)}\frac{x^2+y^2}{\sin^2y+\ln(1+x^2)}$$ If I use a specific path I know I can use Cauchy Theorem to get a number, but how do I prove this for all paths? Thank you!
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2answers
17 views

Questions on the proof of $f*g\in C^\infty(\mathbb R)$ when $f\in L^2(\mathbb R)$ and $g\in C_c^\infty(\mathbb R)$

I am working through the proof that the convolution of a square integrable function with a compactly supported continuously differentiable function is itself continuously differentiable: "Let $f\in L^...
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1answer
24 views

Please what has gone wrong with these transformations?

Let the equality $$x^2-y^4=6$$ define $y$ as a function of $x,$ where both variables are real. The problem I set out to solve was to find the second derivative of $y$ with regard to $x,$ and I found ...
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0answers
30 views

Asymptotic stability of a nonautonemous and non linear system

I have been asked to decide if solutions to the system $ \dddot{x} - \ddot{x} + cos(tx + \dot{x}) = 0 $ are asymptotically stable. Tried linearization but think it's wrong. Is it true that i should ...
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0answers
18 views

compute the derivative of the function F(t)

$$F(t)=\iint {e}^{\frac{tx}{p}}dA$$ where the $$0.1\le{x}\le{t}$$and$$0.1\le{y}\le{t}$$ please help me to find$${F}'(t)$$
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1answer
39 views

Integrating wrt y first and x first gives different answers.

I am solving trying to solve for the CDF of $$f(x,y)=2(x+y) , 0 \leq x \leq y \leq 1$$ I am doing it this way$$\int_{x}^{y} \bigg(\int_{0}^{x}2(x+y)dx \bigg) dy$$ And this way $$ \int_{0}^{x}\bigg(\...
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1answer
27 views

Can improper integral of polynomial be convergent?

Is there any polynomial function whose improper integral is convergent, i.e. having finite value?
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1answer
23 views

Algorithm or formula for shortest direction to travel between two points relative to a point

Given point $(x_0,y_0)$ representing a laser gun, I must find the direction (clockwise or anti-clockwise) to rotate the laser to get it from pointing to point $(x_1,y_1)$ to pointing to point $(x_2,...
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2answers
89 views

Integrate $\frac{1}{x\,\log{x}}$ by parts

A naive indefinite integration of the function $\dfrac{1}{x\,\log{x}}$ can be performed as follows: Let $ \begin{eqnarray} I &=& \int\dfrac{dx}{x\,\log{x}}\\ \therefore I &=& \...
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3answers
35 views

Natural logarithm approximation of numbers greater than 2 for computers

I am interested in approximating the natural logarithm for implementation in an embedded system. I am aware of the Maclaurin series, but it has the issue of only covering numbers in the range (0; 2). ...
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0answers
26 views

Velocity vector angle against a meridian

For a parametric curve $\alpha(t)$ given by: $$\begin{pmatrix} x=\Big(R_1+R_2\cos(t)\Big)\cos\biggr(c\arctan\Big(\dfrac{\tan(t/2)}{\sqrt3}\Big)\biggr) \\y=\Big(R_1+R_2\cos(t)\Big)\sin\biggr(c\arctan\...
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2answers
26 views

If $f(t)$ is continuous for $t$ $\in [0,1]$, $f' > 0$ and $f''(t) > 0$ for $t \in (0,1)$, do we have that $f'(t)$ is strictly increasing on $[0,1]$.

If $f(t)$ is continuous for $t$ $\in [0,1]$, $f' > 0$ and $f''(t) > 0$ for $t \in (0,1)$, do we have that $f'(t)$ is strictly increasing on $[0,1]$? Here is what i think: Since $f(t)$ is ...
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1answer
24 views

How do I find the answer to this differential graph?

This isn't very advanced calculus, but I've been having trouble and can't figure out where to start on this problem. How do I proceed?
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3answers
45 views

Cancelling out constants with u-substitution integration

When using u-substitution to integrate, I tend to think about adding constants to make my $dx$ match my $du$. I don't have a basic enough understanding to grasp why it won't work if my $du$ contains ...
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1answer
40 views

How do I find the local minima of a function from $\mathbb{R}^3\to\mathbb{R}$?

For a function $f:\mathbb{R}\to\mathbb{R}$, the way to find the local minima of a function is to find all $x$'s such that $f'(x) = 0$ and $f''(x) > 0$. Notice that finding the second derivative is ...
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1answer
24 views

Solving ODE G(ax+by)

$\frac{dy}{dx} = (x+y+2)^2$ $let z= x+y+2$ $\frac{dz}{dx} = 1+ \frac{dy}{dx}$ $\frac{dy}{dx}= \frac{dz}{dx} -1$ substitute back in: $z^2+1 = \frac{dz}{dx}$ $\int \frac{dz}{z^2+1} = \int dx$ $...
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1answer
34 views

I don't understand understand this differential equality

I don't understand the following equality: \begin{equation} x'\frac{\partial x'}{\partial q'}+y'\frac{\partial y'}{\partial q'}+z'\frac{\partial z'}{\partial q'}=\frac{\partial}{\partial q'}(\frac{x'^...
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2answers
50 views

Binomial expansion of $(2\cos(x))^m$

This is a follow-up from the question:How do you obtain these results from $(2\cos(x))^{m}$ Let $u=\cos(x)+i\sin(x)$ $v=\cos(x)-i\sin(x)$ Then $u+v$ = $2\cos(x)$ $2^{m}\cos^{m}=(u+v)^{m}$ Using ...
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0answers
33 views

Optimization- Cost minimizing in calculus

There is a baker who needs to make $3000$ cupcakes in total and decides to hire extra help. Workers cost $\$10$ to train and are paid $\$15$ per hour Each helper can make 100 cupcakes per hour. A ...
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1answer
52 views

What does these symbols mean? [on hold]

I am learning mathematics didactically, reading some scientific articles I find those symbols and I do not understand what they do, neither do I find informaion n googl what they mean This is not ...
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4answers
40 views

If $f(x) \rightarrow c > 0$ when $x \rightarrow \infty$, is it true that its anti-derivative $F(x) \rightarrow \infty$ when $x \rightarrow \infty$?

If $f$ is a continous function and $f(x) \rightarrow c > 0$ when $x \rightarrow \infty$, is it true that the function's anti-derivative $F(x) \rightarrow \infty$ when $x \rightarrow \infty$? ...
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4answers
65 views

How to handle the “$2$” in an expression such as “$\sin\left(2\arcsin\frac{x}{4}\right)$”

When an inverse trig function has a constant in front of it being multiplied how do you look at it? Do you distribute the 2 to the whole Pythagorean theorem? I can't really wrap my head around how to ...
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1answer
37 views

Equations of Motion in Cylindrical Co-ordinates

I've run into an interesting set of differential equations, that I'm not 100% sure where to begin- I'm not looking for a 100% complete solution, more just a push in the right direction of where I can ...
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3answers
32 views

Power series representation of $f(x) = $ln$(5-x)$

This was part of an answer posted as a solution to the problem. Of course after differentiating you would need to integrate to get back to $f(x)$ but this part has been left out as it's the ...
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2answers
40 views

Computing this limit: $ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y} = g(x)$

If $f(x,y) \in \mathbb{R^2}$ and $g(x) \in \mathbb{R}$. Assuming $\frac{f(x,y) - f(x-y,y)}{y} = g(x); \forall y \in \mathbb{R}$ $$$$ Can we do the following: $$ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y}...
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0answers
11 views

Supermodularity and increasing differences: Am I correct?

For background, I have been given the following definition of increasing differences: $$ \text{A function }F : X \times T \rightarrow \mathbb{R}\text{ has increasing differences in }(x,t)\text{ if and ...
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1answer
41 views

When are we not allowed to replaced $\sin{x} $ , $\tan{x}$… by $x$ in a limit where $x\to 0$?

In what situation, can we not replace for example $e^x$ with $x$ when when $x\to 0$, in a limit. Sorry if the question is extremely vague , English is not my native language. Thank you in advance
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3answers
87 views

Can someone help me solve this limit? [on hold]

Can someone help me solve this limit? Thank you in advance. $$\lim_{x\to 0} \bigg( \frac{1+\tan{x}}{1+\sin{x}} \bigg)^{\frac{1}{\sin{x}}}$$ If possible without L'Hospital, the exercise gives more ...
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2answers
26 views

Problem understanding a step of a short proof

I'm having some trouble understanding a step of the proof of the following theorem: if $f$ is continuous at $g(c)$ and $g$ is continuous at $c$, then $fog$ is continuous at $c$. Proof: Step 1: ...
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0answers
15 views

Finding the direction vector of a tangent line given one point

I have read example in a math text book with the following problem: The surface $z=f(x,y)=\sqrt{9-2x^2-y^2}$ and the plane $y=1$ intersect. Find parametric equation for a tangent line at $(\sqrt{2},1,...
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2answers
25 views

Question about a Bernoulli's integration.

This is taken from bernoulli's book HYDRODYNAMICS chapter X, p.258. My question is about the integration. Can it be correct or there is an error?