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

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7 views

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to ...
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2answers
25 views

If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
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1answer
23 views

Expected Value of Two Random Variables

X is a random variable with a probability density function $f(x)$, g(x,y) is a function of two variables one of them is the random variable. I have \begin{equation} \int_{-\infty}^{\infty} ...
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3answers
46 views

Derivative of $f(t)=\frac {1}{\rho}\log (1+\rho t)$

Could you have me to find the ferivative of $$f(t)=\frac {1}{\rho}\log (1+\rho t)$$ with repsect to $t$? And Is it $$\lim_{t\to \infty} \frac {f'(t)}{t}=1?$$ Update: Based on Hint of Surb: ...
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0answers
16 views

Variant of CMVT

If $\Phi(x)$ and $\Psi (x)$ are continuous for $a \leq x \leq b$ , differentiable for $a \lt x \lt b, $ and $\Psi'(x)$ never vanishes , then for some $\xi$ in $(a,b)$ ...
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1answer
46 views

Area of a shape bounded by two curves

was given a shape that could be modeled by two sin curves. I was then required to find the two equations that would model the shape. I found GRAPH A could be modeled by $y = -2\sin(1.14(x-0.5))$ and ...
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1answer
15 views

find the equation of a tangent surface which is parallel to a plane surface

How to get the equation of the tangent surface of $F=x^2 + 2y^2 +3 z^2=21$, which is also parallel to the plane surface $ x+ 4y + 6z=0 $? Here is what I've tried: $n = \{1, 4, 6\}$ from $ x+ 4y + ...
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0answers
43 views

Is $i^i$a real number or not? [duplicate]

How might we go about proving $i^i$ is not a real number? I don't know in general how to exponentiate to complex powers, I found this question in an introductory calculus course, so maybe there is an ...
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0answers
66 views

An alternative proof of Cauchy's Mean Value Theorem

Let's focus on the following version of Cauchy's Mean Value Theorem: Cauchy's Mean Value Theorem: Let $f, g$ be functions defined on closed interval $[a, b]$ such that 1) Both $f, g$ are ...
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1answer
36 views

Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$

This is Problem 4-30 from Spivak's Calculus on Manifolds: If $\omega$ is a $1$-form on ${\bf R}^2 - 0$ such that $d\omega = 0$, prove that $$\omega = \lambda \,d\theta + dg$$ for some $\lambda ...
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1answer
53 views

Why can I multiply both sides by $dx$? [duplicate]

When we start learning about differential equations sometimes we "multiply" both sides of the equation by a differential and then integrate. Example: $\frac{dy}{dx}=x$ then $dy=x*dx$ and so on. I ...
2
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0answers
18 views

Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
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0answers
22 views

Is this upper bound ok to use when bounding the error between the Riemann sum and its integral?

I found this on some class notes, which gives several different estimates of the error term, when going from the Riemann sum to its corresponding Riemann integral: $$\frac{b-a}{n}[f(b)-f(a)]$$ Does ...
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3answers
88 views

Show that this difference goes to zero,

$$\frac{1+\sqrt{2} + ... + \sqrt{N}}{N} - \frac{2}{3}\sqrt{N} \to 0.$$ The hint given in the question is this: choose appropriate Riemann sums and estimate the approximation error. My current work: ...
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2answers
78 views

Is Spivak wrong here, or am I just missing something?

Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so ...
3
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1answer
28 views

Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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1answer
71 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\mathrm{d}t$

Suppose $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, \mathrm{d}t$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I ...
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1answer
43 views

Evaluations of a Definite Integral with cosine function

How do you evaluate this integral? Does it involve an elliptical integral? What technique do I use to evaluate this integral? $$\int _{ 0 }^{ 2\pi }{ \sqrt { 5-4\cos { \theta } } d\theta } $$
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1answer
34 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
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5answers
65 views

Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
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1answer
78 views

Why prove that area is unique?

in the book Apostol's Calculus Volume 1, in the proof of the area of under of the parabola $x^2$ from $x=0$ to $x=b$ it is shown that the area $A$ must satisfy ...
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0answers
57 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
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39 views

I need help on problem 54 and 55 (the way to solve this kind of quiz.) [on hold]

I need help in problem $54$ and $55$. How do I solve these kinds of questions? I know how to find gradients and how to find the tangent plane equation and the normal line too.
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2answers
46 views

Maximum value of expression

Let the maximum value of the expression $y=\frac{x^4-x^2}{x^6+2x^3-1}$ for $x>1$ is $\frac{p}{q}$,where p and q are relatively prime positive integers.Find (p+q). My ...
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3answers
43 views

Add or subtract something to a number to reduce it to the range 0 to 24

I'm developing a C++ program and I need to find a formula that given a number to reduce and a limit number, get a value between 0 and this limit number. I don't know if it is allow to put C++ code ...
5
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0answers
32 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
5
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3answers
65 views

Showing that the sequence $x_n=\frac{1}{3}x_{n-1}(4+x_{n-1}^3)$ where $x_0=-0.5$ quadratically converges

I am stuck at a point in solving this problem: Show that the sequence defined by: For all $n\in\mathbb{N}, x_n = \begin{cases} -\frac{1}{2}, & \text{if $n=0$} \\ ...
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0answers
26 views

To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) f_y$ exists.

Let $f : \Bbb R^2 → \Bbb R$ be defined by $f(x, y) := x^2 + y^2$ if $x$ and $y$ are both rational, and $f(x, y) := 0$ otherwise. To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) ...
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2answers
497 views

A curious proof of L'Hospital's rule

I quote P. Nahin When Least is Best (2004), pp. 173-174 "Since $g(x)=R(x)h(x)$, then differentiation of both sides gives $$g'(x)=R(x)h'(x)+R'(x)h(x).$$ Since $\lim_{x \rightarrow 0} h(x)=0$, and we ...
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0answers
52 views

List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
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2answers
24 views

Convergence when the derivative is uniformly continuous

Let $f: \Bbb R \to \Bbb R$ be a derivable function. $f'$ is uniformly continuous in $\Bbb R$ Prove that $[n(f(x+1/n)-f(x))]$ converges uniformly to $f'(x)$ I'm having a hard time seeing why does ...
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1answer
33 views

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$?

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$? Its clear to see that the point-wise convergence is to $0$. By finding the derivative I obtained that the maximum of ...
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4answers
51 views

What does it actually mean if a cost function is differentiable?

I am just learning about optimization, and having trouble understanding the idea behind differentiating cost functions. I have read that for standard optimization problems, the cost function needs to ...
6
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1answer
99 views

Closed-form of $\int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

I've conjectured the following closed-form: $$ I = \int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx = -2\,G\,\ln2, $$ where $G$ is ...
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1answer
22 views

How do I take the derivative of this vector valued function?

Problem: Find the velocity at time $t$ of the particle whose position is $\hat{r}(t)$: \begin{align*} \hat{r} = e^{-t} \cos(e^t) \hat{i} + e^{-t} \sin(e^t) \hat{j} - e^t \hat{k} \end{align*} This is ...
2
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0answers
19 views

Determine Critical points in optimisation problem

So I have this problem where I am supposed to calculate the max and min value of a function $f(x,y)=x+2y$ restricted by the disk $x^2+y^2\le 1 $. I have calculated the $df/dx $ and $df/dy$ and they ...
0
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1answer
26 views

How to evaluate dh/dt giving dV/dt?

Water evaporates from an open bowl of unspecified shape at a rate proportional to the area ofthe water surface; that is, $$\frac{dV}{dt} = -cA(h)$$ where V is the volume of water, A(h) is the area of ...
4
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1answer
64 views

Calculation of integral using two different methods? [on hold]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...
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2answers
27 views

Why is this counting function finite? (It is used Probability)

Why is this counting function finite? I don't understand this interpretation of the author. Can you explain more about this? Please.
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0answers
12 views

Find Intersection of Two Circle given Lat/Lon and radius

I am attempting to calculate the intersection of two circle on the Earth with a given latitude, longitude and radius. I started with this post. While I am using this in the context of programming, it ...
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2answers
35 views

Linear combination of basis function in logarithm space. Is it possible?

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
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1answer
30 views

What condition do I have set to have $x_j$ for $j=1,…,n$ to be non-negative?

I have $\sum_{j=1}^n q_{jj}(x_j-y_j)^2\le1$ What condition do I have set to have $x_j$ for $j=1,...,n$ to be non-negative? The book I am reading says $\sqrt{q_{jj}}\ge1/y_j$ but why? edit: ...
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2answers
24 views

Finding the bounds for a triple integration

I'm currently working on a problem stating: $\iiint_Q y*dV$, where Q is the solid that lies between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$, above the xy-plane, and below the plane z=x+2. My ...
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2answers
50 views

Solving $y' + \frac{1}{2}xy + y^{2} = 0$

I am trying to solve the ODE $$y' + \frac{1}{2}xy + y^{2} = 0.$$ Mathematica gives that the answer is $$y(x) = \frac{e^{-x^2/4}}{C + 2\int_{0}^{x/2}e^{-t^{2}}\, dt}.$$ Of course, if I take this answer ...
3
votes
3answers
188 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
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1answer
19 views

How to distinguish between global maxima/minima and local maxima/minima of a function?

How to distinguish between global maxima/minima and local maxima/minima of a function (when the graph is not provided)? For instance:I find all the points having f'(x)=0 and do second ...
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vote
2answers
21 views

Radius of a wheel based on parametric equations

I am working on a question and I don't have the slightest idea where to begin. Any nudge in the right direction would be very helpful. Here is the question: A bicycle wheel has radius R. Let P be ...
0
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0answers
14 views

Help in calculating the Hessian Matrix from the log-likelihood

I am trying to find the Fisher Information Matrix for a univariate linear linear Moving Average model: \begin{align} z(n) &= h_1 u(n-1) + h_2 u(n-2) + u(n) \tag{1} \\ y(n) &= \mathbf{h^Tz(n)} ...
2
votes
4answers
60 views

Why $\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$?

I'm reading Nahin's: Inside Interesting Integrals. I've been able to follow it until: $$\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$$ I ...
2
votes
1answer
43 views

Proving the continuity of these maps

Backstory: I am having an exam soon, and these are the assignments that keep coming up, I cannot finish any of them to the end, but have ideas about solving them, and would like to hear your thoughts ...