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

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2
votes
3answers
42 views

Interpretation of the curl of a vector field

Let us assume the curl of a vector field is $$ P=(xy)(a_x)+ (y z) (a_y) +(z x) (a_z) $$ Where $ a_x, a_y, a_z $ are unit vectors along x y and z . Then is the curl at a point in the field the ...
2
votes
0answers
28 views

find the total differential of this equation $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $

How to calculate the total differential of $ z= z(x,y)$, which is $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $ at point (1, 0, -1)? The evaluation of mine seems wrong, $ dz= \frac{\partial ...
6
votes
1answer
43 views

For nonnegative continuous $f$, if $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0$, find the value of $f(1)$. [duplicate]

Let $f(x)$ be a non-negative continuous function such that $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0,$find the value of $f(1)$. $f'(x)-f(x)\leq 0$$\Rightarrow f'(x)\leq f(x)$$\Rightarrow ...
1
vote
1answer
36 views

Tough problem on sum of infinite series [on hold]

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
2
votes
1answer
32 views

Evaluating a limit I

Consider the limit \begin{align} \lim_{x \to \infty} \left[ \frac{(x+a)^{x+1}}{(x+b)^{x}} - \frac{(x+a-n)^{x+1-n}}{(x+b-n)^{x-n}} \right]. \end{align} It is speculated that the resulting value is ...
2
votes
4answers
65 views

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? [duplicate]

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? I've tried the ratio test, but i'm unsure if I can continue this way. ...
1
vote
1answer
11 views

Anti-deriving composition of a non-linear activation function on Fourier series?

My pea-brain is not commensurate with the big words in the title. But I'm working on a project where I need to compute definite integrals of the composition $f(g(x))$, where $f(x)$ is any non-linear ...
1
vote
2answers
28 views

Minimize the area of a wire divided into a circle and square.

A wire is divided into two parts. One part is shaped into a square, and the other part is shaped into a circle. Let r be the ratio of the circumference of the circle to the perimeter of the square ...
7
votes
1answer
90 views

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

Please help me to evaluate this integral in a closed form: $$I=\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$$ Using integration by parts I found that it could be expressed through ...
0
votes
0answers
36 views

Evaluating triple integrals that are bounded

I'm slowly learning how to bound triple integration problems, but this one has me a little confused. $\iiint_D(x+2y)dV$, where D is bounded by the parabolic cylinder, $y = x^2$, and the planes x=z, ...
2
votes
1answer
29 views

Minimize distance between polynomials, of a certain form, with Laguerre polynomials

A typical problem that I may encounter on an upcoming test looks like this: Find the polynomial $P(x)$ of a degree less than or equal to three that minimizes $$\int_0^\infty (x^4 - ...
0
votes
0answers
22 views

Closed form solution (formula) for possible events

Let's have 100 time units and 4 possible events A1, A2, B1, B2 that might occur within the 100 units. A1 always occurs before A2, B1 always occurs before B2, t1 < t2 < t3 < t4. There are 2 ...
1
vote
1answer
34 views

Finding the volume of a solid region

I'm trying to find the volume of the solid region inside the sphere $x^2+y^2+z^2=4$, and the upper nappe of the cone $z^2=3x^2+3y^2$ (I only have to set up the triple integral itself, not evaluate ...
2
votes
1answer
54 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
-1
votes
1answer
42 views

Calculating the limit of a quotient with exponential functions using exponent rules

I need to calculate the following limit $$\lim_{n \to \infty}\frac{3\cdot2^n - 2\cdot3^n}{ 5\cdot2^n - 6\cdot3^n}.$$ Any way, back to our topic, according to my book and wolframalpha, the answer is ...
1
vote
2answers
50 views

What is a real world example of “zero work” done by a conservative vector field?

I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the ...
6
votes
0answers
106 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
2
votes
2answers
46 views

Evaluate $\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$

I need to evaluate the following integral using Green's theorem $$\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$$ $C$: from point $E \to F\to G\to H$ ...
0
votes
1answer
48 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
2
votes
2answers
80 views

Find the limit of an infinite series

My intuition was to try and see if the series is a Riemann Sum of a function and then see what happens but I can't really see which function fits here. Thanks!
1
vote
1answer
29 views

confusion about change of variable

If you are integrating $f(x,y)$ over a region and you do a change of variable to $f(u,v)$. The jacobian gives $dx\,dy = du\,dv (\partial x/ \partial u\ \partial y/\partial v - \partial x/\partial v\ ...
0
votes
1answer
32 views

Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
0
votes
0answers
32 views

When is using the gradient to calculate distance not accurate?

I've read that if you have a function like $y=f(x)$ or $z=f(x,y)$, that you can get the distance from a point $P$ to the closest point on the function by using the gradient. Specifically, you plug ...
1
vote
4answers
51 views

Trigonometric substitution and triangles

I'm learning trigonometric substitutions and am having a bit of trouble understanding the intuition behind the conversions (why do most use secant?). If you could explain the conversions geometrically ...
11
votes
2answers
312 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
1
vote
2answers
72 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 ...
1
vote
2answers
29 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 $f([a,b])\subseteq [a,b]$ ...
-1
votes
4answers
73 views

The sum of the series $\sum_{n=1}^{\infty}\sin^n(k)$

What is $$\sum\limits_{n=1}^{\infty}\sin^n(k)?$$ Can you find what is the sum of that series. It is convergent not divergent. What if $k=\frac{\pi}{6}$?
1
vote
1answer
27 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} ...
4
votes
3answers
52 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: ...
0
votes
0answers
21 views

Variant of CMVT [duplicate]

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 $(a,b)$ ...
0
votes
1answer
16 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 + ...
0
votes
0answers
48 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 ...
5
votes
0answers
85 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 ...
2
votes
1answer
50 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 ...
4
votes
1answer
56 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
votes
0answers
21 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 ...
0
votes
0answers
25 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 ...
4
votes
3answers
105 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
79 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 ...
2
votes
1answer
39 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
75 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 ...
0
votes
1answer
47 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
36 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
69 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
1answer
87 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 ...
3
votes
0answers
61 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
40 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
52 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
47 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 ...