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

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1
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3answers
67 views

$\lim_{y \rightarrow b} \lim_{x \rightarrow a} f \neq \lim_{(x,y)\rightarrow (a,b)} f \neq \lim_{x \rightarrow a} \lim_{y \rightarrow b} f$

Can someone give me an example to show that in general $\lim_{y \rightarrow b} \lim_{x \rightarrow a} f(x,y) \neq \lim_{(x,y)\rightarrow (a,b)} f(x,y) \neq \lim_{x \rightarrow a} \lim_{y \rightarrow ...
0
votes
0answers
21 views

Sketching varieties.

So I'm trying to learn by myself about Varieties using a book called Ideals, Varieties, and Algorithms - Springer, that someone recommended me and I just started yesterday, but I noticed that I'm the ...
0
votes
2answers
25 views

Help me find my mistake for the population solution function

Question A pond is initially stocked with 100 fish. It can carry a maximum population of $C=3,500$ fish. The population of fish, $P(t)$, where t is measured in months, grows according to the standard ...
2
votes
3answers
26 views

Help me find my mistake in this question on finding the population after 2 years from a model

Question Suppose a population is modelled by the logistic equation such that the population after t years is given by $P(t)=\frac{1,000}{1+5e^{−kt}}$ for some constant k. If the initial population ...
0
votes
1answer
28 views

Choosing a technique for solids of revolution

Is there a heuristic to choose between the disk method and the washer method? To take the simplest example $y=x$ can be revolved around the $y$ axis using $R=x$ or $r=x$ and $R=$ a constant $C$.
2
votes
1answer
57 views

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$) I was ...
-8
votes
1answer
74 views

The greatest book on calculus ever written, if Spivak didn't exist [closed]

If Spivak, supposedly, did not exist....who would replace him as the author of the greatest book on calculus ever? And please, don't just list a name of books like an absolute imbecile. Give us your ...
7
votes
2answers
174 views

Does anyone have a proof that the intersection and union of two compact sets is compact.

I have my take on it. It is quite informal and don;t know where it would be evaluated correctly on an exam. Since the sets are compact that means for every open cover there is a finite cover. When ...
4
votes
1answer
31 views

What are the maps of these closed sets in $\mathbb R^3 \mapsto \mathbb R$

What is the map of an elipsoid(closed) if $H(x,y,z)=x+y+z$ This is kind of an tricky question, because I am not sure precisely what the answer is. I think when it says closed elipsoid it can be ...
4
votes
3answers
74 views

Implicit Differentiation. Please help me understand why!

I am trying to understand implicit differentiation; I understand what to do (that is no problem), but why I do it is another story. For example: $$3y^2=5x^3 $$ I understand that, if I take the ...
0
votes
1answer
28 views

Is there proof anywhere of the continuity of spherical coordinates and cylindrical coordinates?

Im thinking they are continuous as a composition of continuous functions, but then again. I don't know exactly which specific(precisely speeking) functions are in question.. Any thoughts on this?
4
votes
1answer
23 views

How would I make continuous functions to form these sets? Parametarizing of sets

How would I make continuous functions to form these sets?(So the domain is connected) I need continuous functions that map connected sets to these in question. $1. \text{Cone}$ $$(x,y,z)| \ ...
1
vote
1answer
66 views

-relationship between a function and a tangent line

$f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function at $x=a$. Show that $f$ has derivate at $x=a$ iff there's only a $L(x) = m(x-a)+b $ such that $$ \lim_{x \to a}\frac{f(x)-L(x)}{x-a} = 0 ...
0
votes
1answer
31 views

Bounding the derivative of a $C^2$ functions

Assume $f:\mathbb{R}\rightarrow [0,\infty)$ is $C^2$ and $|f''(x)|\leq A$ for all $x\in \mathbb{R}$. Show that the inequality $$f'(x)^2\leq 2Af(x)$$ The hint is to use Taylor's theorem I havent got ...
-3
votes
0answers
57 views

How do i prove that $f(x) = \sin\frac{1}{x}$ is continous for all $x \in \mathbb{R}$ except $0$. [closed]

How do I prove that $$f(x) = \sin\frac{1}{x}$$ is continuous for all $x \in \mathbb{R}$ except $0$. At $0$ I can show it is discontinous, but how to show its continuity at other points
0
votes
0answers
41 views

integration by parts of double integral

I have a question for integration by parts. If we have $$\int\int_Uf(x,y)\frac{\partial}{\partial x}g(x,y)dxdy,$$ how can we integrate by parts? I guess it might be $$\int_{\partial ...
1
vote
3answers
73 views

Prove an improper double integral is convergent

I need to prove the following integral is convergent and find an upper bound $$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$ I've tried integrating $\frac{1}{1+x^2+y^2} \lt ...
4
votes
2answers
89 views

Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$

One of the integrals I came across these days (during my studies) is $$\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x) \ dx \ dy$$ that can be turned into a series, or can be approached by ...
-2
votes
2answers
51 views

$n$-th derivative of $(ax+b)^{-m}$ [closed]

How to find the $n$-th derivative of $(ax+b)^{-m}$ ?
-1
votes
1answer
21 views

finding linear intersection points [closed]

How do you find the intersection points of a straight line and a polynomial function of degree 28? I know there will be a maximum of 28 intersection points, but how do I find all 28? In other words, ...
1
vote
2answers
29 views

Confused in limit ratio

Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences with all positive terms. If $\sum_{n=1}^{\infty}b_n$ converges and $\lim_{n\to∞} \frac{a_{n}}{b_{n}} = \infty$ , what does the limit comparison test ...
1
vote
1answer
58 views

Correctness of the definite integral

Consider the integral \begin{eqnarray*} I & = & \int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}\\ & = & ...
6
votes
0answers
105 views

prove that $\displaystyle \sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right]$

Using the relation $2(1-\cos x)<x^2,x\neq 0$ or otherwise, prove that $$ \sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right] $$ My Attempt: Let $f(x) = \sin (\tan x)-x$. Then ...
2
votes
1answer
43 views

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$.

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$. Can I use Taylor's polynomial?
2
votes
1answer
63 views

Integral Evaluation.

How can we justify the fact that some integrals can't be evaluated? It's like we can't sum up a function within two bounds or we are unable to find the area under the curve of a function. How's that ...
0
votes
1answer
32 views

Help with finding a general method to solve this function [closed]

Hi guys i am trying to find a function with, s1 < s2, f(s1) , f(s2) [an increasing function] Domain:[20,70] Range: [50,b], b <= 90 these are the the following requirements : a fair ...
2
votes
1answer
40 views

Estimate integral $\,\displaystyle\int_{0}^{\infty}\operatorname{sech}\left(\varepsilon x\right)\cos\left(kx\right)\,dx,\,$ with $\,k,\varepsilon>0$

$ \newcommand{\sech}{\operatorname{sech}} $ Is there any analytic/asymptotic way to estimate the value of the integral: $$ \int_{0}^{\infty} \sech\left(\varepsilon x\right)\cdot ...
5
votes
3answers
120 views

Prove the monotony of $\frac{\sin x}{x}$

I was wondering whether it is possible to prove that $f(x)=\frac{\sin x}{x}$ is strictly decreasing in $(0, \pi]$ and strictly increasing in $[-\pi , 0)$ without using any derivatives at all. If it is ...
1
vote
0answers
13 views

Homogenous Linear ODE from standard to normal form. Justification.

I have this ODE, I am going from standard form $$p(x) y'(x)+Q(x) y(x)+y''(x)=0$$ to normal form: $$q(x) u(x)+u''(x)$$ Question 1 Now it is this way of defining y(x) that I do not get. Can I ...
1
vote
1answer
23 views

1-1 Function Composition

I am asked to consider f(x) and g(x) which are both 1-1 functions on their respective domains. I have been asked to show f ∘ g is a 1-1 function and then a followup statement of if f ∘ g is 1-1 does ...
2
votes
1answer
28 views

Differential geometry: restriction of differentiable map to regular surface is differentiable

From Do Carmo: Let $S_1$, $S_2$ be regular surfaces. Suppose $S_1\subset V\subset \mathbb{R}^3$ and $\varphi:V\rightarrow \mathbb{R}^3$ is a differentiable map such that $\varphi(S_1)\subset S_2$. ...
0
votes
0answers
31 views

Compensating Variation

I have a compensating variation problem to ask. Two goods, $X$ and $Y$(composite good) $MUx = 1$ , $MUy = \frac{10}{\sqrt{y}}$ $P_{x_1} = \$1.50$ , $P_{x_2}= \$2.00$ , $Py= \$1$ $U(x,y) = x + 20 ...
0
votes
2answers
41 views

Why I'm getting wrong answer? What is wrong in my solution?

I'm learning single variable calculus right now. Right now trying to understand integration with partial fraction. I'm confused in a problem from sometime. I think I'm doing right but answer in my ...
-2
votes
2answers
77 views

parametric equations $(y-1)^2=x-49$. [closed]

Find three sets of parametric equations for the curve whose equation is $(y-1)^2=x-49$. Can anyone explain how to find it? Thanks
1
vote
0answers
42 views

Finding rate of change with integration (my solution correct?)

The rate of change in a person's body temperature, with respect to the dosage of $x$ milligrams of a drug, is given by $D'(x)=\frac{7}{x+8}$. One milligram raises the temperature 3.7 C. Find the ...
2
votes
1answer
44 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
2
votes
0answers
32 views

What calculus material can prepare me for MCMC?

I am looking to revise calculus from scratch to move on to Monte Carlo Markov Chain Methods and Quasi Monte Carlo Methods. I studied calculus properly a couple of years ago however it was back in high ...
18
votes
4answers
194 views

How to Prove : $\frac{2}{(n+2)!}\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^{n+2}=\frac{n(3n+1)}{12}$

While I calculate an integral $$ \int\limits_{[0,1]^n}\cdots\int(x_1+\cdots+x_n)^2\mathrm dx_1\cdots\mathrm dx_n $$ I used two different methods and got two answers. I am sure it's equivalent, but ...
2
votes
1answer
87 views

$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log(2)\log(2n^2)}{2n}$ [on hold]

Does anyone prove the following definite integral ? $$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log2(\log2+2\log(n))}{2n}$$
-4
votes
1answer
87 views

Prove rigorously: $x^2=2^x$ has exactly $3$ real solutions [duplicate]

I am not sure how to prove rigorously (using calculus) that $x^2=2^x$ has exactly $3$ real solutions.
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votes
0answers
35 views

Using the taylor theorem to approximate the image value at $x=1$ [closed]

Use Taylor’s theorem to calculate $\exp(1)$ with a remainder no greater than $.01$. Explain your approach carefully, especially the remainder term.
3
votes
2answers
69 views

What is the significance behind taylor series?

Why does taylor series have ample amount of importance in calculus? I like to know some insights behind taylor series.
3
votes
2answers
103 views

Maximize $xy^2$ on the ellipse $x^2+4y^2=4$

I was using Lagrange multiplier, any steps gone wrong? $$f(x,y)=xy^2$$ $$c(x,y)=x^2+4y^2$$ Partial Derivatives $$\frac {\partial f}{\partial x} = y^2 $$ $$\frac {\partial f}{\partial y} = 2xy $$ ...
0
votes
0answers
7 views

How to determine the Quasi Convexity and Quasi Concavity in a Univariate Case?

What exactly defines upper contour for Univariate case?
0
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2answers
47 views

A certain partial derivative

Hi I am reviewing partial derivatives. For the question below, I am not sure why $(x-1)$ appears. Could anyone give me a explanation on this? $y = x\sin(z)e^{-x}$ $\partial y/\partial x = ...
1
vote
2answers
57 views

Show that a function defined by an integral is differentiable

Define $$g(a)=\int_{0}^{\infty}\frac{\sin(ax)}{x}e^{-x}dx,\ \ \ \ \ \ a\in\mathbb{R}$$ a) Show that $g(a)$ is differentiable and compute $g'(a)$. b) Use this to compute $g(a)$. I have tried various ...
1
vote
0answers
30 views

Integral-Summation Problem (Mathematical Physics problem)

given, $ψ_{k}$ =$\sqrt{\frac{2}{q}}$ $\sin \frac{kπx}{q}$ we have, $g_{kj}$=$q\int^q_0 ψ_j\frac{\partial ψ_k}{\partial q} dx$ verify, $\sum_{k}g_{jk}g_{lk}=q^2\int^q_0 ...
5
votes
4answers
102 views

Limit of definite integral of $f(x)\cos(mx)$

Source: Old comp./preliminary exam. Let $f(x)$ be a Riemann integrable function on $[0,1]$. Prove that $$\lim_{m\to\infty}\int_{0}^{1}f(x)\cos(mx) \, \,dx=0$$ Thought $(1)$ Because we don't know if ...
24
votes
0answers
263 views
+50

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
2
votes
1answer
32 views

How to integrate: $\int \frac{\sec x}{\sqrt{\sin(2x + A) + \sin A}} dx$?

How do I integrate: $$\int \frac{\sec x}{\sqrt{\sin(2x + A) + \sin A}}\, dx?$$ First, I tried to substitute $t^2$ for the denominator, but it was really a great flop. I then removed $\sin A$ since ...