Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
128,624
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Calculate the volume bounded by surfaces $\frac{x^2}{25}+\frac{y^2}{9}-z^2=1,z=0,z=2$
Question
Calculate the areas bounded by the curves: $\frac{x^2}{25}+\frac{y^2}{9}-z^2=1,z=0,z=2$
I've watched several videos on youtube on this topic, but I couldn't figure out what I can do with it....
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N th degree polynomial integral
Consider an n-th degree polynomial
p(x) = $a_{0}$ + $a_{1}$x+ $a_{2}$$x^2$+...+$a_{n}$$x^n$
and compute $$\int_a^b p(x) dx=1$$
I'm stuck on this part and please help
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Differentiable and Lebesgue’s Criterion for Riemann Integrability to show that $xf''(x)$ is Riemann integrable on $[a, b]$
Function $f(x)$ is twice differentiable on [a, b], and suppose that its
second derivative $f''(x)$ is bounded and Riemann integrable on [a, b]. Use Lebesgue’s
Criterion for Riemann Integrability to ...
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1
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Math 110C, Multivariable Calculus, Abs. max. and min. for $2$ variables [duplicate]
Find the abs. max. and min. values of $𝑓$
on the set $D$
for
$𝑓(𝑥,𝑦)=2𝑥^3+𝑦^4$
, $𝐷=\{(𝑥,𝑦) \mid 𝑥^2+𝑦^2 \leqslant 1\}$
First, I found the C.P.(s) on the inside:
$𝑓_𝑥=6𝑥^2$
and $𝑓_𝑦=4y^...
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2
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Multivariable Calculus, abs. max. and min. for $2$ variables
Find the abs. max. and min. values of $f$ on the set $D$ for
$f(x,y)=2x^3+y^4$ , $\;D=\left\{(x,y)\,|\;x^2+y^2\leqslant1\right\}$
First, I found the C.P.(s) on the inside:
$f_x=6x^2$ and $f_y=4y^3$
...
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1
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Maximise area of rectangle with fixed perimeter
I've got a problem where a rectangle's area must be maximised given a fixed perimeter of $60$m.
Assuming a length of $x$ and height of $y$ I wrote an equation $y = 30x - x^2$ which i differentiated, ...
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On the commutativity of integral and limit
Suppose that $f\in C^1$
I try to prove that $\frac{d}{dt}\int_{0}^{t}h(t,x)dx=h(t,t)+\int_{0}^{t}\frac{\partial h}{\partial t}(t,x)dx$.
However this depends on the fact that $\lim_{\Delta t\to0} \...
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Proof that $0^0=1$?
So we know that $$\sum_{k=0}^\infty\frac{x^k}{k!}=e^x$$Letting $x=0$: $$\sum_{k=0}^\infty\frac{0^k}{k!}=e^0=1$$The first term of the sum is $0^0$ and the other terms are $0$, so $0^0=1$. What am I ...
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Partial derivatives and saddle points
Is the following statement true or false?
Suppose $f:\mathbb{R}^2\to\mathbb{R}$ is a differentiable function and $(c,d)\in\mathbb{R}^2$. If $f_{xx}(c,d)f_{yy}(c,d)<[f_{xy}(c,d)]^2$ then $f$ has a ...
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Can I split this limit into two like this? [closed]
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$, and $f \in C^{1}$ (continuous and derivative continuous). Let $p,e,v \in \mathbb{R}^{n}$.
This is supposed to be true (is part of the proof of ...
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On an exotic type of "mathematical mean"
Background, motivation and the problems
Most of us are familiar with the arithmetic mean (AM) - the simplest and most basic type of mathematical mean. The arithmetic mean can be said to be based on ...
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Expanding two iterated limits
I have always had a little trouble fully understanding iterated limits. So I thought it might be a good exercise to expand it in quantifiers.
Suppose we want to expand $\displaystyle \lim_{x \to a} f'(...
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Good materials on calculus and linear algebra [closed]
I have a master's degree in distributed networks and applications, and I graduated with a bachelor's degree in applied mathematics. I wouldn't say that I studied well, because my knowledge of calculus,...
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Length Of a Tangent from contact to intercept
If I was Given a Function y and Some x value how to find the length of the tangent from the contact to x axis intercept,you can use any example except for quadratic equations .
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Double Series for the Bernoulli and Euler Numbers
This refers to the following paper:
Higgins, J. -- Double Series for the Bernoulli and Euler Numbers,
J.London Math. Soc. (2), 2 (1970), 722-726.
I ...
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Show that $f\mapsto \exp(f)$ is (locally) Lipschitz on $H^1(\Omega)$
Let $f,f' \in H^2(\Omega)$ with $\|f\|_{H^2(\Omega)},\|f'\|_{H^2(\Omega)}\leq M$, $\Omega$ smooth and bounded subset of $\mathbb{R}^2$. I want to show that
$$\|\exp(f)-\exp(f')\|_{H^1(\Omega)}\leq C(M)...
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2
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Confusion with implicit function theorem
According to my understanding of the implicit function theorem, if $F(y_0,x_0)=c$ and $\frac{dF}{dy}$ at $(y_0,x_0)$ is nonzero then there is a unique function $y=F(x)$ for $x \in B$ where B is an ...
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Does usage of partial derivatives requires apriori knowledge about arguments of the function
I want to check if I understand the concept and notation of partial and total derivatives correctly.
Is it true that if I have some multivariable function f(x,y,..) I can only use the partial derivate ...
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norm of gradient of multivariate function
I know that the gradient of a scalar function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ is the vector
$$\nabla F(x_1,\dots,x_n)=(\partial_{x_1} F,\dots,\partial_{x_n}F).$$
I'm reading a paper where ...
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Possible typo in a text using Fourier transform properties
I am working my way carefully through an article in Analytical Chemistry on round-off error. I'm a scientist but not a professional mathematician so it is rather slow-reading for me and none of it is ...
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Condition for $f^\prime$ to be absolute integrable
Suppose $f(x)$ is the probability density function of a random variable $X$, which means:
$$\int_{a}^{b} f(x) dx = 1$$
Also suppose $f$ is continuous and differentiable.
Provide a non-trivial ...
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Equative integration term to prove
I've find a statement of equation like this:
I justified the cases that N=1,2,3, which seem to be correct. However, when I was trying to prove it with induction, I failed to reduce the repetitious ...
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Calculus For Engineering [closed]
I tried this question, but couldn't find the correct answer,
a.Find d/dx(1n1n2x) and obtain an expression for dy/dx such that y=1n1n1n2x.
b.If y=ln(2x+(x^2+2)) ,find dy/dx in terms of x and show that(...
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What is the concept of a limit in simple terms
I'm 14 just starting high school and I got interested in all the new math so I took a little dive into calculus but I can't wrap my head around the concept of a limit and for which uses it is applied ...
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What am I doing wrong? I tried everything
I left an image of the question below.
enter image description here
enter image description here
It says my image too big to include, but the second picture I plugged back into the constraint to solve ...
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1
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Proof for Calculus: Maxima and Minima
I would like help with the following problem.
Let $a, b \in \mathbb{R}$ with $a < b$. Suppose that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose further that $f$ has local ...
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Given that $f(x)=g(x^2)/x , g'(5)=3$ and $g(5)=5$ find the value of $f'(9)$ [closed]
I tried this question but the answer is differ from answer given in the book.Given that $f(x)=g(x^2)/x , g'(5)=3$ and $g(5)=5$ find the value of $f'(9)=$
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Math 110C, Multivariable Calculus, Absolute Max and Min for 2 Variables
Find the absolute maximum and minimum values of $f$ on the set $D$, where $f(x,y)=x^2+y^2-2x$, and $D$ is the closed triangular region with vertices $(2,0),(0,2),$ and $(0,-2)$.
First, I found the ...
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$\sum \geq \int \mathrm{ or } \sum \leq \int?$
If I have a function that is continuous, differentiable and positive, then would its sum be greater or its integral? Or is there no fixed answer? If so, under what factors is what greater?
$$\sum_{x=1}...
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Math110C, Multivariable Calculus, Second Derivatives Test
Find the local maximum and minimum values and saddle point(s) of the function $f(x,y)=y^2-2y \cos x$, -1<=x<=7:
First, I found the first partial derivatives:
$f_x=-2y(-\sin x)=2y \sin x$ and $...
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Proving my function is optimized in a particular regime
Consider the optimization problem:
$$
\max_{x \geq 0} (1-x)f\left(g\left(x\right)\right)
$$
where $f'(x)< 0, f(1) = 0, f(0) = 1$, and $f$ is differentiable. In addition, $g$ has the form:
$$
g(x)=
...
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Math 110C, Multivariable Calculus, Second Derivatives Test
How can I find local maximum and minimum values and saddle points of $f(x,y)=xy+e^{-(xy)}$?
First I found the first partial derivatives:
$f_x=y-ye^{-xy}$ and $f_y=x-xe^{-xy}$
and then got $0=y-ye^{-xy}...
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Bounds on the maximum real root of a polynomial with coefficients $-1,0,1$
Suppose I have a polynomial that is given a form
$$
f(x)=x^n - a_{n-1}x^{n-1} - \ldots - a_1x - 1
$$
where each $a_k$ can be either $0,1$.
I've tried a bunch of examples and found that the maximum ...
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I derived a formula for $[x!]^\prime$. Is it correct?
The starting point was that $ \Gamma'(x+1)=\Gamma(x+1)\psi(x+1)$ where $\psi(x+1)=-\gamma+H_{x}$ . Hence $$ [x!]' = x!\biggl[-\gamma+\sum_{k=1}^{x}\frac{1}{k}\biggl]$$ For example $ [4!]' = 24[-\gamma+...
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projectile motion : maximum angle for launch [closed]
Hello everyone I am working on the following problem (I have attached my working). I have worked out the range but not sure where to go from here. I think I may need to work out $\frac{dR}{d\theta}$ ...
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Using Root Test to see whether $\sum_{n=1}^{\infty}\frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^{n}}$ converges
This exercise specifically requires that we use the root test to determine whether the series converges or not.
All I've done so far is get the sequence in this form:
$$\sqrt[n] \frac{n^{n+\frac{1}{n}}...
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What exactly does the Remainder Estimate for Integral Test actually mean? $R_n \le \int_{n}^{\infty}f(x)dx$
$$\int_{n+1}^{\infty}f(x)dx \le R_n \le \int_{n}^{\infty}f(x)dx$$
What does this actually mean?
Let's use n=5.
The $R_5$ is the error of the partial sum $S_5$
That error is less than the sum of the ...
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curious GoldenRatio identity [closed]
I would like to verify the following identity but I don't know how mathematics says that it is equal to numerically.
$$\prod _{k=0}^{\infty } \sqrt{\frac{\phi ^{2^{-k-1}} \left(\phi ^{2^{-k}}+1\right)}...
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Prove that a series is divergent, with $\epsilon,N$
I have this question:
Prove that $a_n=\frac{(-1)^nn+1}{n+2}$ is divergent, without proving it by contradiction, or by using any theorems from the book. Prove it by using $\epsilon$ and $N$ notation.
...
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If the length of a curve on $[0,s_0]$ is $s_0$ then $\|\gamma'(s)\|=1$ for all $s\in [0,b]$
Let $\gamma :[0,b]\to \Bbb{R}^d$ be $C_1[0,b]$ (I am not sure about the range, since the question didn't specify, but this is Calculus 4(which is in EN sources III-IV) so I assume the range is $\Bbb{R}...
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Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums
Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums
For example, is it true that $\sum\limits_{n=1}^\infty (k\cdot a_n) = k\left(\sum\limits_{n=...
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2
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Finding the largest square that can be inscribed inside the astroid curve $x^{2/3}+y^{2/3}=4$
Finding the largest square that can be inscribed inside the astroid curve $x^{2/3}+y^{2/3}=4$
A square is described by four vertices. There will be one vertex of the square in each quadrant. I ...
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Help with Calculus Problem: Finding Maximum Value of Integral [closed]
I'm currently struggling with a calculus problem and would appreciate any help or guidance. The problem is as follows:
The figure shows the graphs of the functions $F$ and $G$. For the functions $f$ ...
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Dealing with the $dx$ that is in the denominator of a definite integral.
I’m not sure if I should specify between a definite or an indefinite integral here but that’s the instance I’m specifically interested in.
I was thinking about an integration rule that states that if ...
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Is the derivative of the exponential family's density function absolute integrable?
Consider the exponential family of distributions, whose density can be written as:
$$f_\theta(x)=h(x)\exp\left[\eta(\theta)^T T(x)-A(\theta)\right]$$
where $h: \mathbb{R}\to \mathbb{R}^+$, $\theta$ is ...
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Calculate the areas bounded by the curves: $y=\frac{1}{x^2}e^\frac{1}{x}, y=0, y=1, x =2$
Question
Calculate the areas bounded by the curves: $y=\frac{1}{x^2}e^\frac{1}{x}, y=0, y=1, x =2$
I have little to no experience in finding areas with such curves , I've watched a lot of videos on ...
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2
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How can I solve $\int_{0}^{\infty}{\frac{\log(x)}{x^2+4}}dx$ using $2\tan\theta$ as substitution for x? [duplicate]
$$\int_{0}^{\infty}{\frac{\log(x)}{x^2+4}}dx$$
-from the book Advanced Problems in Mathematics by Vikas Gupta
So I tried to solve this integral by substituting $x=2\tan\theta$ so that we get the upper ...
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3
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Given a variable $x$, is its "growth rate" (or "rate of change") $\frac{\mathrm{d}x}{\mathrm{d}t}$ or $\frac{\frac{\mathrm{d}x}{\mathrm{d}t}}{x}$?
I think it is (or should be) the latter (reasoning below). But many writers (e.g. Stewart, Calculus) use the former instead.
So I'm confused and hence this question. (Maybe this is just one of those ...
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how do you prove that sin is a function of arclength(radian) to y coordinate?
I was watching https://youtu.be/TpWQlKHPyJ4
what it does is the following
calculate the arclength of a unit circle in terms of x
integrate $(1-x^2)^{-1/2}$
you get arclength = $sin^-1$(x)
and the ...
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Limit of $\sum_{t=0}^n a_n(t)$ for n tends to infinity
I am wondering about how to deal with a limit of the following form
$$ \lim_{n \rightarrow \infty} \sum_{t=0}^n a_n(t),$$
with the function $a_n(t)$ and the sum being dependent on $n$.
Consider the ...
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