Questions tagged [calculus]

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

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2answers
43 views

What would be the step by step solution of this double integral by changing it to Polar coordinates?

$$\int_0 ^ {1} \int_{-\sqrt{x-x^2}} ^ {\sqrt {x-x^2}} (x^2+y^2) ~dy~dx$$ My findings are: $$\int_?^? \int_?^? r^3 ~dr~d\theta$$ Region is the circle of radius $~\frac{1}{2}~$ centered at $~(\frac{...
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2answers
56 views

how to apply “$ \varepsilon$- $\delta$” definition of limit

I am self studying calculus, and i came across a University quiz that required me to: Use the $ \varepsilon$- $\delta$ definition of limits to verify that the $\lim\limits_{x\to\pi} 3 = 3$? how to ...
2
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1answer
57 views

Find the solution of $~\dfrac{dy}{dt} + \sqrt{1+ t^2}~e^{-t}~y=0$ , $~y(0)=1~.$

Find the solution of the initial value problem $$\dfrac{dy}{dt} + \sqrt{1+ t^2}~e^{-t}~y=0~ ,~~~ y(0)=1~.$$ It seems like 'separable equation' so I tried $$-\frac{1}{y} \frac{dy}{dt} = \sqrt{1+ t^2}~...
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1answer
56 views

Minimize $\sum_{i=1}^{n}a_i^2\sigma_i^2$ subject to $\sum_{i=1}^{n}a_i=1$

I am having a little issue with this problem. I know that if we were to only have $$\sum_{i=1}^{n}a_i^2$$ subject to $$\sum_{i=1}^{n}a_i=1$$ then we can use Lagrange multipliers, and we get $a_i=1/n$ ...
2
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4answers
125 views

How to solve for $~ 2x - \tan(x)=0~$

I need to find the roots for this function $$~ 2x - \tan(x)=0~$$ in order to graph it. I have found the one root $~(x=0)~$ but there are two more $~(x= -1.164 ,~ x= 1.164)~$. How can I find these ...
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2answers
30 views

Linear Approximation — horizontal change in position of tip of a clock hand.

The problem is: A clock's minute hand has a length of $r$ meters, linearly approximate the change in the horizontal distance of its tip as the arm moves from 6.01 am to 6.02 am. I've worked it out ...
2
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3answers
105 views

Inverse of rational function $y= \frac{3-x}{1+x^2}$

I have the function $$y= \frac{3-x}{1+x^2}$$ and I want to find the inverse of this function. I know that $$x= \frac{1 \pm \sqrt{1-4y(3-y)}}{2y}$$ My question is how do I find the domain where the ...
4
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4answers
125 views

Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$

Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$ I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence ...
2
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4answers
77 views

Limit of a function in which square roots are involved

$$\lim_{x \to 0}\frac{x+2-\sqrt{2x+4}}{3x-1+\sqrt{x+1}}$$ Could someone please help me solve this problem. I tried multiplying by a unity factor but I end up stuck.
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0answers
46 views

Snell's Law using Calculus for smooth curves

I hope this does not conflict with the purpose of this site, so I will be as clear as I can. So I know this seems like a simple equation, I know that it revolves mostly with refractive index and ...
3
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1answer
49 views

How to find the value of Grandi's series using Ramanujan's summation

I can't figure out how to solve the infinite sum of $\sum ^{\infty }_{n=0}\left[( -1)^{n}\right]$ I know that Srinivasa Ramanujan solved it and I couldn't figure it out with Ramanujan's summation. ...
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0answers
18 views

Relative position of a function's surface and a tangent plane

Let $f$ be a function such that \begin{align*} f(x,y)=&f(x_0,y_0)+(x-x_0)\frac{\partial f}{\partial x}(x_0,y_0)+(y-y_0)\frac{\partial f}{\partial y}(x_0,y_0)+ \\ &\frac{1}{2}\left[ (x-a)^2\...
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0answers
56 views

Find how much substance is formed in 4 minutes if the rate of formation is given by the function $f(t) = 9 - \frac{1}{2}t^2$

In a chemical reaction two substances are combining to form a third one. The rate at which the third substance is being formed in grams/min is given by the function $f(t) = 9 - \frac{1}{2}t^2$ ...
2
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1answer
60 views

Generalizing the “The Volume of a Cone is a Third that of its Bounding Cylinder” fact

The ancient result is that a right-circular cone of height $h$ and base-radius $r$ will have volume $\frac{1}{3} \pi r^2h$, which is $1/3$ the volume of the cylinder with same base and height. And the ...
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5answers
60 views

Infinite sequence

One of the major distinctions between a set and a sequence is that the order of terms matters in a sequence. Looking at a set of integers, am I right to say that we cannot have a sequence whose terms ...
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3answers
32 views

Sinusoidal curve average decrease between sections - simple but need help

New here and would like to ask you clever people something. I have what I’m sure is a simple question and I will look silly asking this but I haven’t done math in over 15 years so I’m really rusty. ...
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0answers
48 views

Uniform convergence of the power series on [-R,0]

I need to determine whether the power series $$ \sum _{n=2}^{\infty} \frac{x^{n}}{\ln \left(n\right)}\quad x\in R, $$ converges uniformly or not on the interval $[-1,0]$. I know already that the ...
4
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2answers
79 views

Limits and Continuity in Multi variable calculus.

Checking continuity of $f(x,y)$ at $(0,0)$: $$ f(x,y)=\begin{cases}\dfrac{x^3+y^3}{x-y}\ \ ,x\neq y\\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,x=y \end{cases}$$ Using polar coordinates $x=r\cos\theta$ and $...
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3answers
68 views

Is there a difference operation $\frac{d}{dx}$ and $'$?

I have a question I've been wondering about for a long time. Is operation $«\color{red}{\frac{d}{dx}}»$ mathematically equal to operation $«\color{red}{'}»$ ? Is there any difference between them?
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2answers
38 views

non-decreasing sequences

I have the following sequence $\{n^4-6n^2\}$ I have to determine if the sequence is non-decreasing, increasing or decreasing. In my opinion, the sequence is neither, decreasing, non-decreasing nor ...
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1answer
102 views

Generalizing Archimedes' “The Quadrature of the Parabola”

In the third century BC Archimedes discovered that The area enclosed by a parabola and a line (left figure) is 4/3 that of a related inscribed triangle (right figure). Consequentially, the area ...
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2answers
54 views

Can you tell me how to solve with process?

What formula can I use for this question?
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2answers
82 views

Find a general solution for $y'=\frac{3x+4y+2}{2x+y+3}$

$$\frac{dy}{dx}=\frac{3x+4y+2}{2x+y+3}$$ I tried to solve this problem as following, like 'exact form problem', $$(2x+y+3)dy - (3x+4y+2)dx = 0$$ This equation is not exact yet, so i tried to find ...
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3answers
71 views

A hyperplane is a subspace whose dimension is one less than that of its ambient space. Why? [closed]

If a space is $3$-dimensional then its hyperplanes are the $2$-dimensional planes, while if the space is $2$-dimensional, its hyperplanes are the $1$-dimensional lines. why? furthermore, how does a ...
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2answers
55 views

Meaning of absolute value in squareroot

Recently I've encountered this problem: $y = \sqrt{x}$ $y = \sqrt{|x|}$ We all know $\sqrt{-x}$ is invalid so this is quite confusing to use $\sqrt{|x|}$. Is |absolute| being used here to "...
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3answers
84 views

Show that $f(x) = xe^{x^2}$ is invertible and determine the 6th degree maclaurin polynomial of $f^{-1}(x)$

Show that $f(x) = xe^{x^2}$ is invertible and determine the 6th degree maclaurin polynomial of $f^{-1}(x)$. I can see that $f$ is invertible since $x$ is strictly increasing and when $x < 0$ $e^{...
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0answers
59 views

Integration by substitution which involves trigonometry [closed]

Need help on how to do this integration by substitution. $$\int (\sqrt{\sin x} + \sqrt{\cos x}) dx$$
3
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7answers
107 views

Problem with $x^{6} - 2 = 0$ compute roots in $\mathbb{C}$

I have problem with simple equation $x^{6} - 2 = $ compute roots in $\mathbb{C}$ I will try compute roots of $x^{6} - 2 = (x^{3}-\sqrt{2})(x^{3}+\sqrt{2})=(x-2^{1/6})(x^{2}+2^{1/6}x+2^{1/3})(x^{3}+\...
-3
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1answer
24 views

TRUE/FALSE: “Every divergent improper integral has an unbounded integrand over the interval of integration.” [closed]

TRUE/FALSE: "Every divergent improper integral has an unbounded integrand over the interval of integration." I think this one is supposed to be True for every divergent integral. But correct me if i'...
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6answers
88 views

Prove that $x^n(1−x)^n$ has its maximum at x=1/2.

I was trying to prove that $f_n(x)=x^n(1-x)^n$ converges uniformly on [0,1], and found this: Convergence of $f_n(x)=x^n(1-x)^n$ But I've got stuck on how to prove that $f_n(x)$ has a maximum on 1/2. ...
0
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1answer
43 views

Compute the Taylor series for $f(x)= \frac{1}{2 - x}$

Where does this series converge? Knowing $\frac{1}{2 - x} = \frac{1}{2}\frac{1}{1 - x/2}.$ My taylor series is $\frac{1}{2}+\frac{1}{4}x+\frac{1}{8}x^2+\frac{1}{16}x^3+\frac{1}{32}x^4$ with center $...
0
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1answer
31 views

TRUE/FALSE: Function f is integrable on [a, b] provided that f is uniformly continuous on [a, b]

Statement: "TRUE/FALSE: Function f is integrable on [a, b] provided that f is uniformly continuous on [a, b]" I'm not sure if I understand what "provided that" means. Does this mean that the proof ...
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2answers
40 views

True or false : Let $b \in \mathbb R, b>0$ and let $B:=\{x^2 :\, x \in (−b,0)\}$. We then have that $\inf(B) = 0$ and $\sup(B) = b^2$ [closed]

Statement: Let $b \in \mathbb R, b>0$ and let $B:=\{x^2 :\, x \in (−b,0)\}$. We then have that $\inf(B) = 0$ and $\sup(B) = b^2$. I'm not sure where to start with this question. Can anyone ...
0
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1answer
39 views

For which values of $a$, $f(x)=1+a(a-1)(a+2)x+ax^2+x^3+o(x^3)$ has a minimum in $x=0$?

I would like to find for which values of $a$, $f(x)=1+a(a-1)(a+2)x+ax^2+x^3+o(x^3)$ has a minimum in $x=0$. At first glance, I would take the derivative of $f(x)$ and do $f'(x)=0$. Then, check where ...
0
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1answer
41 views

Find the length of a curve given by equations

Find the length of a curve given by the equations $x^2+y^2+z^2=1$ and $(x^2+y^2)^2=x^2-y^2$ I tried with polar parametrization and with spherical parametrization but I can't solve the definite ...
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2answers
48 views

Applying the chain rule: differentiating $f(x+s)$

This might sound trivial, but I'm confused about using the chain rule on univariate real-valued functions. On Wikipedia, I see that $${\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}$$ My ...
0
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1answer
58 views

True/False: For all real $x$ and $y$, $|x - y| \leq |x| + |y|$ [duplicate]

For all real $x$ and $y$ we have that $|x-y| \leq |x| + |y|.$ I think the statement is False because it looks like a Triangle Inequality, except the triangle inequality is in the form of $|x+-y| \leq ...
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2answers
44 views

Formal Definition of Exponentiation (Rational Exponent)

This question has bothered me for a long time... I know some people define $(-8)^\frac{1}{3}\approx 1 +1.73i$ referring to root with the minimum argument by De Moivre's Theorem. For this question, I ...
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1answer
51 views

Is my demonstration for $ \lim_{(x,y)\to(0,0)}(x^2-y^2)\sin(\frac{1}{x^2+y^2})=0$ correct?

I need to prove that $$ \lim_{(x,y)\to(0,0)}(x^2-y^2)\sin(\frac{1}{x^2+y^2})=0$$ But I'm not sure about my demonstration. I see that $\left|\sin(\frac{1}{x^2+y^2})\right| \le 1$ So I write that $$\...
4
votes
3answers
66 views

Solving $e^{ix}=i$

I was assigned this problem: $$e^{ix}=i$$ I understand that with Euler's formula, $e^{ix}=\cos x+i\sin x$. I then set up the problem as $$i=\cos x +i\sin x$$ This means that $\cos x = 0$ and $\sin x =...
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2answers
39 views

Lim of a function - excercise

Hi i try to do a question on limits and i want to ask if the way i calculated the answer was right. Here is the question and then my answer. thank you! Question: for $n \in \Bbb N$, let $f_n(x) = \...
0
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2answers
43 views

When can we use the integral test?

I was going over my notes and I found that I wrote that we cannot use the integral test on the following series, why is that? $$ \sum \frac{5}{k^2 \ln(k)} $$ Isn't it both decreasing and positive? ...
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2answers
43 views

Find $\lim_{x \to \infty} x^2\big(\ln(x\cot^{-1}(x))$

$$\lim_{x \to \infty} x^2\big(\ln(x\cot^{-1}(x))$$ I tried using the Series Expansion of the $\ln(x)$ but then got stuck in between. I also tried using the L'Hopital but the expression got quickly ...
0
votes
1answer
86 views

Integral $\int_{0}^{\infty}\frac{dy}{1+y^{2}}\log\left(\sqrt{1+y^{2}}+\sqrt{x+y^{2}}\right)$

I need help finding an analytical expression for the integral $$I(x)=\int_{0}^{\infty}\frac{dy}{1+y^{2}}\log\left(\sqrt{1+y^{2}}+\sqrt{x+y^{2}}\right), $$ where $0<x<1$. The expression can be ...
0
votes
2answers
37 views

Finding the area of a curve where one of the curves is a line

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves. $$ y^2 = 4x, y = 4x - 2 $$ Answer: \...
0
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0answers
36 views

Continuous $f$ with $\int_0^1 x^n f(x) dx=0$ implies $f \equiv 0$ [duplicate]

Suppose $f$ is a continuous real-valued function on $[0,1]$ and $$ \int_{0}^{1}x^n f(x) dx=0 $$ holds for all non-negative integers $n$ How to prove that $f(x)=0$ for all $x \in [0,1]$?
3
votes
1answer
106 views

$\frac{d}{dt}$ represents an operator or infinitesimal change?

In high school physics, they teach us a little bit Calculus as it would be done months later in maths. And many times my physics teacher wrote things like$$\frac{dv}{dt} =\frac{dv}{dx}\frac{dx}{dt}$$ ...
0
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2answers
60 views

How to prove functions to be Lipschitz only in bounded closed domain

Question. If $X\left(x,t\right)$ has continuous partial derivatives on a bounded closed area/set $\mathscr{D}\subset\mathbb{R}^n\times\mathbb{R}$, then prove or disprove that $X$ is a Lipschitz ...
1
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2answers
26 views

$M-N$ Unbounded Sequence Proof

Did I do this correctly? I want to prove the sequence does not converge. $$a_n = \{\frac{(n+1)!}{n!}\}$$ $$a_n = \{n+1\}$$ WTS: $\forall M>0, \exists N>0 | \forall n \in$ N, if $n>N$ then $...
-1
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1answer
33 views

If $f'(c)>f'(x)$ for all $x\in(c-\epsilon,c)\cup(c,c+\epsilon)$, then $f'(c)>\frac{f(x)-f(c)}{x-c}$ for all $x\in(c-\epsilon,c)\cup(c,c+\epsilon)$? [closed]

Is the above claim true? If yes, how might we prove it? If no, what is a counterexample?