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

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0
votes
2answers
49 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: ...
1
vote
2answers
63 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 ...
1
vote
0answers
15 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 ...
0
votes
1answer
32 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? $$\int _{ 0 }^{ 2\pi }{ \sqrt { 5-4\cos { \theta } } d\theta } $$
3
votes
1answer
25 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 ...
1
vote
5answers
61 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).
1
vote
0answers
60 views

Why prove that area is unique?

in the book Apostol's Calculus Volume 1, in the proof of the area of a parabola $x^2$ from $x=0$ to $x=b$ it is shown that the area $A$ must satisfy ...
3
votes
0answers
51 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 ...
-8
votes
0answers
36 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.
1
vote
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 ...
0
votes
3answers
42 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
votes
0answers
31 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
votes
3answers
62 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 $n=0$} \\ ...
0
votes
0answers
23 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) ...
3
votes
1answer
401 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 ...
3
votes
0answers
50 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 ...
0
votes
2answers
23 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 ...
1
vote
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 ...
0
votes
4answers
48 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 ...
5
votes
1answer
75 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 ...
0
votes
1answer
21 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
votes
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
votes
1answer
24 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
votes
1answer
62 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 ...
0
votes
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.
0
votes
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 ...
1
vote
2answers
34 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 ...
0
votes
1answer
27 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: ...
0
votes
2answers
23 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 ...
2
votes
2answers
49 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
183 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 ...
1
vote
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 ...
1
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
votes
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
56 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
41 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 ...
1
vote
6answers
71 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
2
votes
1answer
33 views

Proof of integration of parts.

I can almost see how to derive the formula of integration by parts by this extremely helpful picture: But Im having trouble figuring out how the limits of integration simplify so nicely to how they ...
0
votes
2answers
41 views

Integration by Partial Fractions $\int\frac{1}{(x+1)^3(x+2)}dx$

I'm trying to do a problem regarding partial fractions and I'm not sure if I have gone about this right as my answer here doesn't compare to the answer provided by wolfram alpha. Is it that I can't ...
0
votes
3answers
51 views

Find all solutions to the equation. $7 \sin^2x - 14 \sin x + 2 = -5$

I got this question wrong on a test and I want to see what I did wrong so I don't get this type of question wrong again.
2
votes
1answer
45 views

washer method calculus help [on hold]

The question is: "Find the volume of the solid obtained by rotating the region bounded by the line $y = 5$ using washer method outer - inner formula the functions to graph are: $y = x^2$, $y = 2x$. ...
4
votes
4answers
35 views

Eliminate $t$ to give an equation that relates $x$ and $y$

I am having problems understanding how to solve the following parametric equation. I have achieved an answer, but am unsure if my answer is correct or not. Eliminate t to give an equation that ...
3
votes
1answer
32 views

Are the stationary points of a strongly convex function unique in each dimension?

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$. I am wondering: if I have another point $y \in ...
3
votes
0answers
26 views

Helicity is Conserved

In fluid mechanics, the helicity is defined as $$\int_{R^3} u(x,t)\cdot \omega(x,t),$$ where $u(x,t)$ is a smooth solution of the Euler equations $$\partial_tu + (u \cdot \nabla) u = -\nabla p$$ ...
2
votes
2answers
46 views

Finding the volume of a solid bounded by a sphere and a paraboloid

I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 2$ and the paraboloid $x^2 + y^2 = z$. I know that to do this, I must use triple ...
-1
votes
1answer
29 views

Supremum vs Integral [on hold]

Let $h$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}h(t)\leq\int_{r}^{\infty}h(t)\frac{dt}{t} $$
6
votes
3answers
134 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
0
votes
4answers
109 views

calculate-binomio-newton

i am help Calculate: $$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$ PD : use $(1+x)^{16}$ and binomio newton
1
vote
0answers
22 views

Calculus book for computer science students

I'm going to teach calculus I and II to undergraduate computer science students and I would like to know if someone here knows some book or site with easy calculus applications in computer science. ...
0
votes
1answer
22 views

Finding a tangent line with implicit differentiatio

Find the tangent line on point $P$ for this curve $(x + 2)^2 + (y - 3)^2 = 37$ on $P(4,4)$ I tried implicit differentiating $2(x + 2) + 2(y - 3)y' = 0$ I'm not sure if solving for $y'$ is the ...