Tagged Questions

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

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2
votes
1answer
19 views

Formal explanation for this change of integration [duplicate]

Formally, why is true that $$\int_0^\infty \int_{x}^\infty f(x,y)dy dx =\int_0^\infty \int_{0}^y f(x,y)dx dy $$ ? I know and understand perfectly the geometric interpretation, and with that, I´m well ...
0
votes
1answer
9 views

Indefinite Integral of a Function multiplied by Heaviside

I want to do the following integral: $$\int^\ x*H(x-a) dx $$ In mathematica I get that $$\int^\ x*H(x-a) dx = \frac{1}2*(x-a)*(x+a)*H(x-a)$$ where H(x-a) Is the heaviside function. But by hand I ...
1
vote
1answer
38 views

I am not understanding this step

I am starting the chapter on differential equations and have this example to work through but I do not understand a few things Solve $dy=\frac{dy}{dx}=\frac{2x(y-1)}{x^2+1}$ solution: note that ...
20
votes
2answers
183 views

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ...
5
votes
3answers
89 views

How to calculate $\lim_{x \to0} \dfrac{f(x)-f(\ln(1+x))}{x^{3}}$

$f$ is a differntiable function on $[-1,1]$ and doubly differentiable on $x=0$ and $f^{'}(0)=0,f^{"}(0)=4$,how to calculate $\lim_{x \to0} \dfrac{f(x)-f(\ln(1+x))}{x^{3}}$,I have tried the ...
1
vote
2answers
17 views

Find the centroid of a lamina

Region is bounded by : $y=x+4$, $y=5$, $y=-x-4$, $x=3$ Graph is here: https://www.desmos.com/calculator/cwmlwknywe Can I just split the shape at the x-axis, find the individual centroids, and then ...
0
votes
1answer
22 views

Inverse Matrix Methods to find Nash Equilibriums

The board of directors of two companies determines the salary of its CEO according to the following reaction functions: S1 = 100,000 + (1/2) S2 S2 = 70,000 + (2/3) S1 Where Si is salary of company ...
0
votes
1answer
18 views

Derivatives using the chain rule

$F(x) = 16.5x^2$ $x(Y) = 0.065Y + 0.68$ find $dF/dY$ derivative of $F(x) = 31x$, derivative of $x(Y) = 0.065$ I don't understand how to find $dF/dY$.
0
votes
2answers
26 views

How to evaluate this path integral?

So I know that the integral is $$\int_1^e f(x(t), y(t), z(t))(||c'(t)||) \:dt$$ I set this to$$\int_1^e\frac{1}{t^3}\sqrt{\frac{1}{(ln10*t)^2}+1}\; dt$$ I found this too hard to integrate by hand, ...
-2
votes
4answers
102 views

How to prove $x^2+x$ is not uniformly continuous on $\mathbb{R}$? [on hold]

I think it is continuous? How to prove $x^2+x$ is not uniformly continuous on $\mathbb{R}$?
0
votes
1answer
9 views

Related Rates Problem about moving shadow

I have another question about related rates. I have been asked the following question about related rates. It's been a while since I looked at related rates. I appreciate if anyone can help me with ...
5
votes
0answers
26 views

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. [duplicate]

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. MY ATTEMPT SO FAR: If $f^3$ is holomorphic, then we can ...
0
votes
3answers
35 views

How to show whether three points in $\mathbb{R^4}$ lie on a straight line?

If you are given the coordinates of three points in $\mathbb{R^4}$. (Call these three points: A, B and C). How do we know whether these three points lie on a straight line or not? One way I can think ...
0
votes
2answers
33 views

Related Rates Problem involving two runners on a circular path

Problem: There was a typo in the original statement. I fixed it now!! Two runners start running (from the same point) in opposite directions along a circular path of radius $100\ m$ at a speed of ...
0
votes
2answers
40 views

Optimizing a box

I'm learning the use of derivatives and I have found a problem: Supposing we want to build a box of $4000\, \textrm{cm}^3$ of volume without top and a square base. Which are the measures so we ...
0
votes
0answers
41 views

How to do this? Where to replace the $A$? I'm trying to use integration by parts but still not getting it.

Problem: if $$ A =\left(\int_0^{4} e^{(x-2)^ 4} \,dx\right)\ $$ find $$ \left(\int_0^{4}x e^{(x-2)^ 4} \,dx\right)\ $$
0
votes
0answers
16 views

What does G represent in the Position Function?

I believe my teacher has missed G in the notes, I have no idea what it means. The position function that I have- $$s(t)=\frac{1}{2}Gt^2+V_0(t)+S_0$$ Where $t$ is initial time of object, $V_0$ is ...
1
vote
1answer
14 views

Find global and local maxima and minima, given the graph of the function

My attempts were local max: 3,8 - local min: 5 - global max= 3, global min= 5 Module is giving me incorrect. No partial credit. So I can't tell where the problem lies. local max: 6, 4.?? local ...
0
votes
1answer
25 views

Find the mass of the disk. - Double Integration Problem - Calculus 3

A disk of radius 5 cm has density 10 g/cm2 at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. my answer: 157.08g ...
1
vote
1answer
20 views

I don't understand this step in this exercise involving algebra

In the solution of an exercise they go from $e^{2y} = x + x\cdot e^{2y}$ straight to $e^{2y} = x / (1-x)$. Could anybody explain to me how/why this is possible?
1
vote
0answers
17 views

Torn between plugging back into the original vs. an intermediate equation…

Let's say I am starting with this equation $$y^2-x^2=2xy-2x+1$$ It can then be determined that $$\frac{dy}{dx}=\frac{x+y-1}{y-x}$$ Let's say I want to find all points on the curve where the tangent ...
0
votes
0answers
18 views

What dosage maximizes sensitivity?

The sensitivity of the body to the drug is defined as $dT/dD$. For some positive constant $C$, a patient’s temperature change, $T$ , due to a dose, $D$, of a drug is given by $$T = \left(\frac C2 − ...
1
vote
1answer
17 views

Monotonically decreasing sequences

Suppose $(a_n),(b_n)$ be positive sequences such that $(a_n)$ decreases to $0$, monotonically. If lim$_{n\rightarrow \infty}\frac{b_n}{a_n}=1$, does it imply that $(b_n)$ decreases to $0$ ...
1
vote
2answers
30 views

Evaluatinga triple integral

we have a half-sphere $$V:x^2+y^2+z^2\leq 1,\ z\geq 0$$ for the function $f(x,y,z)=z$ $$\iiint_V f(x,y,z) \,dx\,dy\,dz$$ in my solution I tried to use sphere coordinates but I'm not able to get ...
0
votes
0answers
17 views

Taylor polynomial for $\sin(2x)$ about $x=(\pi/2)$, order $2n-1$.

In my assignment I have to find the Taylor polynomial for $\sin(2x)$ about $x=(\pi/2)$, order $2n-1$. And I just made $P_4(x)$ to figure out ...
1
vote
0answers
29 views

Isoperimetrics question

Here's an Algebra 2 problem that has been giving me fits. The question is: A 50-inch length of string is cut and one length is formed into a circle and the other into a square. Where should you cut ...
1
vote
1answer
30 views

The dilogarithm function. Question on an identity of it

Upon reading a journal article about manipulating series using the dilogarithm function, I have a few questions. But before I ask them, let me give the information the article provides. Consider the ...
3
votes
1answer
53 views

Multiple integral differential notation

When writing a multiple integral, I have noticed there is sometimes used a shorthand for writing the differential in the integral. For example in $\mathbb{R}^3$ instead of writing $\mathrm{d}x\ ...
0
votes
1answer
17 views

Divegent series test

I have this infinite series. What test would you use to show that it is divergent? I tried limit, ratio and tried to compare it with $2^{1/n}$ $\sum_{n=1}^\infty {2^{1/n}}-1$ Thanks
0
votes
1answer
25 views

Apostol Calculus volume 1, Archimedes method of finding the area under the curve

So I just began reading Apostol's Calculus, and, discouragingly, I've found myself confused by the very first proof. (Skip to Figure $1.5$) What I don't get is why the horizontal component of ...
1
vote
2answers
25 views

Implicit differentiation involving exponential functions [on hold]

How do I find $dy/dx$ for $xe^{-y/2} + ye^{-x/2} - 2 = 0$?
1
vote
3answers
55 views

If $f(1) > p >1$ then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
1
vote
1answer
33 views

Differential equation $x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x)$

In a proof of the series expansion of the Lambert-W-function, I need that it is the only non-zero function satisfying: $$ x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x) $$ Is it true?
4
votes
1answer
32 views

How can I prove/disprove that the function $f(x)$ satifies $f'(c)=1$ for certain conditions

The function is defined on the interval $[0,1]$ with following conditions: 1) $f(0)=1$, 2) $f(1)=2$, 3) $f(x)$ is continuous on $[0,1]$, Prove or disprove: There exists some $c$ from $(0,1)$, such ...
-2
votes
1answer
32 views

Using both inverse method and cramers rule to figure out output [on hold]

Model A requires 30 min of E-assembly, 40 minutes of M-assembly, and 30 min of testing Model B requires 20 min of E-assembly, 50 minutes of M-assembly, and 30 min of testing Model C requires 30 min ...
4
votes
1answer
39 views

Why can we only talk about derivatives on an open interval?

For instance, in my calculus class, all theorems are in the following form: For example, Rolle's theorem: If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$ ... (etc) My question is, when ...
1
vote
2answers
39 views

Showing $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}z^{k-1} = - \displaystyle\frac{\ln(1-z)}{z}$

Given the sum $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}z^{k-1} = 1 + \frac{z}{2} + \frac{z^2}{3} + ...$ How does one show that this sum is equal to $- \displaystyle\frac{\ln(1-z)}{z}$?
1
vote
0answers
30 views

Verify the following rate question?

Can someone help point out where I went wrong? Question: Sand is being poured from a conveyor belt at the rate of 100 cubic feet per minute. The sand form a pile that is the shape of a cone whose ...
0
votes
2answers
53 views

Differentiate $y =\sin(1+x^2)^{1/2}$

I've tried differentiating $y= \sin(1+x^2)^{1/2}$ using the chain rule, but I keep getting the wrong answer. Can anyone give me a step by step so I can see what I'm doing wrong? Thanks.
3
votes
2answers
68 views

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

I have got a trouble with integral $$\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \, \mathrm{d}x,\quad |\alpha|<1.$$ My teacher said there are two ways of solving such ones, if there is ...
0
votes
1answer
35 views

Finding derivative of integral

I know about basic of fundamental theorem of integral calculus but i cannot think of how to go on with this question
5
votes
0answers
70 views

Finding the closed form of $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k n}$

A while ago I computed pretty easily the series $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k+ n}$ and then I thought of tackling the case where we have the product instead of ...
7
votes
1answer
100 views

Divergence/convergence of a series

how can I prove this series diverges/converges? I used all adequate criteria but nothing useful came out..any ideas ? $$\sum\limits_{n=1}^\infty{\frac{{\sin (n)\cos (n^2 )}}{\sqrt n +\sqrt[3]n}}$$
0
votes
1answer
24 views

Statıstıc problem

Will I use binomial distribution for this question? Can you help me please thnk you
0
votes
0answers
1 views

Compute elasticity then find approximate percent change in demand

I am given this homework problem and I do not know how to solve part b. Here is the problem: Economist use demand functions to describe how much of a commodity can be sold at varying prices. For ...
-2
votes
1answer
38 views

Can anyone help me to find the result of $(a+bi)^n$ by three methods,when $n$ is a positive integer number? [on hold]

Can anyone help me to find the result of $(a+bi)^n$ by three methods,when $n$ is a positive integer number?
2
votes
1answer
37 views

Prove a set $I=[0,\infty)$ is not an open interval.

$(\exists y \in I)$ $(\forall x \in \mathbb{R})$ $x<y\Rightarrow x\notin I$. I am trying to prove that the set $I=[0,\infty)$ is not an open interval. So I know there exists a $y$ such that the ...
2
votes
3answers
52 views

Can a first derivative of a function have more roots than the original function?

This is a general question. Function is to be considered differentiable on some domain. More specifically, I am given a function $f(x)$ which is twice differentiable and has three distinct real ...
1
vote
1answer
26 views

Ways to sum a series. Questions

I am reading a journal on ways to sum a series and the author considers the following: Suppose one wants to sum the alternating series $\displaystyle\sum_{k = 1}^{\infty} (-1)^{k-1}\frac{1}{k} = 1 - ...
0
votes
0answers
18 views

Statistic problem

Can you help me to solve this problem pls,I have exam and I am studyıng. What wıll I use, bınomial or Other thing ? Thank you