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

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1answer
46 views

using Maclaurin series to estimate $\frac{1}{e} $

The problem I'm trying to solve is: Determine how many terms of the Maclaurin series of $f(x) =e^{-x}$ should be used to estimate $\frac{1}{e} $ with an error of magnitude less than $5 \times ...
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31 views

Taking limits on integration limits.

For some function $f$ and $g$ lets say that I have an integral which looks like, $\int ^{f(\epsilon)}_0 g(t,\epsilon) dt$. So if I want to compute this to zeroth order in $\epsilon$ can I just ...
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1answer
38 views

Sequential Criterion Trouble

Define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = 5x, \; x\in \mathbb{Q} \; \text{ and } x^2 + 6\; x\in n\mathbb{Q}$. Using the sequential Criterion, show that $f$ is discontinuous at $1$, but ...
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2answers
40 views

How to find if the sum of periodic function is periodic?

Basically, I am suppose to check if $f(x)=f(x+T)$. however my function is a bit complex: $x(t)=10\cos(20000\pi t)+0.5\cos(24000\pi t)+0.5\cos(16000\pi t)$ How shall I check if this function is ...
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1answer
27 views

calculus volume problem

I need to set up an integral for the volume of a solid obtained by rotating about the y-axis the region bounded by the curves $$y=x^2 \quad \text{and} \quad x^2+y^2=2$$ Now, I know that $x^2+y^2=2$ ...
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11 views

Question about multivariable calculus and conservative vector fields

Suppose $f: R^n \to R^n$. $f(x) = (f_1(x),...,f_n(x)) $. Let $\psi :[0,\infty) \to R^n $ be differentiable on $(0,\infty)$. Put $f(x) = \psi(||x||) = ( g_1(||x||),...,g_n(||x||))$. Suppose $f$ is ...
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18 views

Differential of $ \int_{0}^{t} e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau} c(s)ds $

I think -- using the chain rule -- it's $$ e^{\int_{t}^{t}\cdots d\tau} c(t)dt \cdot e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau}\cdot ...
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23 views

Jacobian in Change of Variables

Let us consider an integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f (k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. We want to ...
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1answer
33 views

Locally minimizing a concave function

What will happen if we minimize a concave function via gradient descent? Where does it get stuck? Intuitively a concave function has more structure than an arbitrary function, and seem to be easier ...
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2answers
45 views

Cross Product of Vectors

Hey everyone, I'm having trouble with this problem. I'm aware that the length is equal to the norm of v and w. And that the cross product is orthogonal to the vectors. I'm just not sure how to use ...
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1answer
47 views

Line Integral Help (Vector Calculus)

I'm currently revising for a maths module that I am taking as part of my physics degree. I'm taking the exam tomorrow and I'm feeling pretty confident although upon attempting this line integral I ...
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1answer
76 views

double integral.

I just received this problems from a friend, and I think its a HW problem. its: $$ \int_1^e \int_{1+y^2}^5 \cos (x- \ln x) \ dx \ dy $$ I looked at it, and If I did graph the the region right then ...
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0answers
52 views

Closed form of $\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1)$

Is there a known closed form of the series below? $$\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1]$$
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25 views

computing length of a curve given as set of points

Given a set of points $(x_i,y_i)$ from a simple curve. How can the length of the curve be computed approximately?? I understand these points can be connected by line segments and the sum of the ...
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2answers
50 views

calculus radical integration question

Does anyone know how to calculate integral of $\sqrt{ 1-\cos (x)}$ ? I tried several methods resulting in $-2\sqrt2 \cos (x/2) + c$, but this is wrong in accordance with the text book, so i dont know ...
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28 views

Calculating Flux with Gauss's Theorem

I'm having a difficult time answering this problem with a path integral. Let $F(x,y) = (x, 3y)$ be a fluid flow. Compute the net flow out of the region below $y = 1 - x^2$ and above $y = 0$ using ...
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1answer
47 views

Volume of revolution

I'm reviewing for a final exam and found a great question that I would like some assistance with! Write two integrals representing the volume of the solid obtained by rotating the region ecnlsoed by ...
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1answer
55 views

how to determine delta epsilon figures for an equation involving the constant e

For the problem $\lim_{x\to 0} {e^x-1 \over x}=1$, I need to solve for x in order to solve using the delta epsilon definition. How do I go about solving for x in the equation $f(x)={e^x-1 \over x}$? ...
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3answers
69 views

reverse the order of integration

How do you reverse the order of this integral into $dy\,dx$? I feel like you need two separate ones but I don't know how to do it: $$\int_0^3\int_\sqrt{y}^3 f(x,y) \, dx \, dy$$ thanks
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1answer
26 views

Limits of triple integration (prism)

Integrate $f(x,y,z) = x^2 + y^2$ over the prism shown My problem isn't the integration process but just to determine what the limits are. $$\int_0^4 \int_{0}^{1-x} \int x^2 + y^2 dzdydx $$ I think ...
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1answer
41 views

Find deltas algebraically for given epsilons

The exercise below is from the book Calculus by Thomas / Finney (9th edition): Prove that $\displaystyle \lim_{x\rightarrow 2}f(x) = 4$ if: $f(x) = x^2$ when $x \neq 2$ $f(x) = 1$ when ...
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68 views

Triple integral of a regular otctahedron

I have to express the volume of a regular octahedron as a single triple integral in rectangular coordinates, with vertices $(\pm 1, 0, 0),(0,\pm 1, 0 ), (0,0, \pm 1)$ centred at the origin. From ...
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1answer
93 views

Find the area of the region bounded by: $r^2=50\cos(2\theta)$

How can I do this? I can do Find the area inside one leaf of the rose: $r=\sin(6\theta)$ but cannot figure this one out. Please I need help!
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57 views

Periodic solutions to Riccati equations

Suppose $\alpha, L>0.$ Under what conditions (between $\alpha, L$) the Riccati equation $d\Phi/dz=2i[\Phi(z)^2+\alpha Cos(2\pi z/L)\Phi(z)+1]$ can have a periodic solution with period $L$ (under ...
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1answer
34 views

First derivative test and uniqueness of local extrema

This is the context in which my question lies. See below for the actual question. Let $f(x)$ be differentiable everywhere and have a minimum at $x^*$. Then for every $x$ in a proper neighbourhood ...
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1answer
33 views

Proving that A'(x) = f(x)

So imagine a curve f that is continuous in the interval [a,b]. I now define the arealfunction A(x) as the area under the curve from a to x. I also define A(x + dx) as the area under the curve from a ...
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1answer
24 views

Limit Problem-To determine the value of a and b.?

$$\lim_{n \to \infty} \sqrt{n^2-n+1}-an-b=-\infty$$ $$\lim_{x \to -\infty} \sqrt{x^2+x+1}-ax+b=0$$ Is not the answer of the second one $a=-1$ and $b=\frac12$? But if,I put $a=-1$ and $b=\frac12$, the ...
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1answer
67 views

What is the proof of integral theorem i.e area under curve is given by anti derivative?

I have learnt integration as well as differentiation. In the early days I learnt a very simple proof for why the derivative of sin(x) is cos(x) and that of tan(x^2) is 2x*sec(x^2). This basically ...
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40 views

question about green theorems

I want to compute the line integral $$ \int_{\gamma} (e^x \sin y) dx + (e^x \cos y) dy $$ where $\gamma$ is one loop counterclockwise on the ellipse $x^2 + xy + y^2=1 $ My effort: Since we are on ...
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44 views

Justifying that the derivative doesn't exist

Is the right way of justifying that the derivative of $\sqrt[3]{x}$ at $x=0$ calculting, by th definition, $$\lim_{x\to0} \frac{1}{\sqrt[3]{x^2}}$$ via left and right havd limits? I mean, it is ...
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47 views

Wedge volume of sphere problem

The Clare College bridge at Cambridge is decorated with 14 stone spheres, but one of it missed a wedge. I took a photo to estimate the volume of the missing part of the sphere. I am not confident ...
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1answer
102 views

Finding the value of one-sided limits and greatest integer function.

$$ \lim_{x \to 0} \frac{a}{x} \left\lfloor\frac{x}{b} \right\rfloor $$ The $\lfloor \rfloor$ stands for the greatest integer function. I have calculated and the left-hand limit is coming as ...
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2answers
51 views

Is a bounded continuous function defined on $\Bbb R$ differentiable?

Is a bounded continuous function defined on $\Bbb R$ differentiable? Why so? The query is fueled by the following question: Let $f : \Bbb R \rightarrow \Bbb R$ be a bounded continuous function. ...
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18 views

Is this the correct setting for the following volume?

LEt $\Omega = \{ 1 \leq x^2 + y^2 + z^2 \leq 4 \} $, and let $f(x,y,z) = z^2$, then does $$ \int_\Omega f = \int_0^\pi \int_0^{2 \pi} \int_1^2 r^2 \cos^2(\phi)r^2 \sin( \phi) \, dr \, d \theta \, ...
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36 views

Prove that limit $ a^x \text{is } a^c$

Prove that $\lim\limits_{x\to c} a^x = a^c$ strictly by the epsilon-delta definition. I know it is quite easy to prove this using logarithm but that is assuming that we already know the ...
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1answer
30 views

Calculate the volume of the region

Calculate the volume of the region inside the sphere $x^2+y^2+z^2=a^2$ and outside the cylinder $x^2+y^2=b^2$, where $a > b,$ by using an appropriate double integral. I was using polar ...
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33 views

Epsilon-Delta proof of the limit of an arbitrary line

Consider the line $f(x)=mx+b$, where $m\neq 0$. I have to use the epsilon-delta definition to show that $$\lim_{x\to c}f(x)=mc+b$$ So given $\varepsilon >0$ ...
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1answer
58 views

What does this mean: the magnitude of the rate of change of $\theta$ increases without bound

(Advance note: I'm not looking for the answers to this question. I want to understand what "the magnitude of the rate of change of $\theta$ increases without bound" means.) Here's the problem I'm ...
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1answer
42 views

The chain rule, partial derivatives and general functions

I am revising for my first year Calculus examination. The following question is on a past paper and I am given the solution, however I am struggling to make sense of it: Let $V(x,y)$ be a ...
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2answers
97 views

Reduction formula tricky problem (Further Maths: F3)

$\int \:e^{ax}\cos ^n\left(x\right)dx$ I just cannot get it to reduce, I keep ending up with too many species in the next integral to use parts again. I have important Further Pure F3 exam in a month ...
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51 views

Prove the following: $\int_{a}^{b}|f\left(t\right)|dt\leq\left(b-a\right)\int_{a}^{b}|f'\left(t\right)|dt$

I have this homework question and I'm in need of some assistance: "Let there be a function $f:\left[a,b\right] \rightarrow\mathbb{R}$ continuously derivatable (every derivative is continuous), and ...
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1answer
43 views

Evaluating stationary points with null Hessian determinant

Given the function $f(x,y)=x^2y^3$ I'm asked to evaluate all the stationary point. My work: I started calculating the derivatives: $f_x=2xy^3$ and $f_y=3x^2y^2$ then I looked for the point such that ...
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21 views

How to find new region of integration after changing variable?

I'm not quite sure how to go about finding the new region of integration. The evaluation of an integral over a region D can be done by changing variables and integrating over a new region and ...
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49 views

bilinear form, anti symmetric part

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
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1answer
16 views

Finding the slope using the implicit function theorem

If $F(x,y)=x^2+xy^3$ how can one find the slope of $F(x,y)=2$ at the point where y=1 and x>0 using the implicit function theorem?
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1answer
66 views

convergence of $\frac{x^n}{1+x^n}$

How do I check convergence/ uniform convergence of $\sum\frac{x^n}{1+x^n}$. Also for series $\sum \sin \left(\frac{x}{n^2}\right)$, can I use that $\sin x \leq x$?
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30 views

area in polar coordinates

Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this particular question. To be completely honest I am not sure how to even approach this problem, ...
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1answer
50 views

Finding volume of the solid of revolution?

Can anyone help me with finding the volume of a solid of revolution of f(x) about the x axis for the interval [1,6]. It's supposed to be able to be done without needing calculus but I am having ...
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27 views

Total derivation

$$\pi_1(p_2)=\max_{x_1}p_1(x_1;p_2)x_1-MC_{x_1}x_1.\tag{2}$$ The first order condition is $$\phi(x_1;p_2)=\dfrac{\partial p_1}{\partial x_1}(x_1;p_2)x_1+p_1(x_1;p_2)-MC_{x_1}=0$$ and the second ...
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1answer
35 views

How to find Area and Perimeter

hey guys can you solve these 2 questions for me shown in the image below. I've done this but I think it's not correct could you guys just show me the solution with working. Thanks in Advan =) God ...