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

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4
votes
1answer
32 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
0
votes
1answer
18 views

Maximum volume of a cuboid with constraints

Find maximum volume of cuboid for which sum of three dimensions (x,y,z) is not greater then 108. I am looking for the most straightforward approach to the question. Thus the volume will be $xyz$ and ...
0
votes
1answer
33 views

Integral giving wrong values

Given following function: $x^3 + 1$ Find the area that is selected in red lines. So I solved the root $\sqrt[3]{-1} = -1$ so $x = -1$ So now I have to create the 3 integrals: $S_1 = ...
1
vote
4answers
67 views

Proving that if $f$ is continuous then $f(a_n)$ converges

Suppose $f$ is continuous on the closed interval $[0,1]$ such that $f(0)=f(1)$. Prove that if $$(a_n)=\left(\frac{n(1+\cos(\pi n))}{2 n+1}\right)$$ then the sequence $(f(a_n))$ converges. I noticed ...
0
votes
2answers
18 views

Using substitution to make an equation into a separable differentiable equation

I have the question: By making the substitution $y = t^nz$ and making a cunning choice of n, show that the following equations can be reduced to separable equations and solve them. $$\dfrac{dy}{dt} = ...
1
vote
1answer
23 views

Two simple statements about continuous and monotonic functions

Suppose $f: \mathbb{R} \to \mathbb{R}$. I must determine whether the following statements are true: If $f$ is continuous on $\mathbb{R}$ and not bounded then $\lim_{x\to \infty} f(x)$ is either ...
2
votes
1answer
32 views

Mass of a wire: intersection of surfaces

So I got this mass problem to solve: Find the mass of the wire formed by the intersection of two surfaces whose density is $\phi=x²$ $\underset{C}\int \phi ds $ along the curve: $$ C:\left\{ ...
0
votes
1answer
8 views

I am having difficulty finding, making a triple integral of the space z< |x-y|. Can someone recommend a technique most effective fore such things?

I am having difficulty finding, making a triple integral of the space $z< |x-y|$. Can someone recommend a technique most effective fore such things? I try to somehow draw the graph and see the ...
1
vote
0answers
36 views

Integral of surface

$$\iint_\limits{S}\sqrt{\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}}dS$$ where $$ S: \ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ I tried to solve it : $$\iint_{S^+}P(x,y,z)\ dydz = ...
1
vote
3answers
36 views

Derivative of $f(A) = \|A x\|^2$ with respect to the Matrix

Suppose I have $A \in \mathbb{R}^{n^2}$ and $x \in \mathbb{R}^n$ where $A$ is interpreted as a matrix. We can define $f(A) = ||A x||^2$ for some constant $x$. What is the derivative of $f$, written ...
3
votes
1answer
67 views

Solving an equation including $e^{-x}$ with the Lambert W function

Given two functions of $x$, namely $f(x)$ and $g(x)$, where $$f(x)=x^2-4x+8$$$$g(x)=3xe^{-x}$$ the shortest distance between the graphs of the functions is sought. I begin by defining a function ...
0
votes
1answer
18 views

continuity problem

how to answer the question in the image in steps as all I know is that the denominator has no real zeroes but i don't know what to do next !!!!
0
votes
0answers
34 views

Calculus integration latest

Use a double integral to find the volume of the solid bounded by two surfaces $$x^2+y^2=4,$$ and $$x^2+z^2=4.$$ is it using one way to integrate or there is using separation way?
-1
votes
1answer
32 views

Calculus differentiation [on hold]

The height of the right circular cylinder decrease at rate of $0.5cm/s$. and the radius increase at the rate of $1.5cm/s$. What is the rate of change of the volume of the cylinder when the radius is ...
0
votes
1answer
87 views

Inverse of $x^x$ [duplicate]

Since $x^x$ grows very fast, its inverse should accordingly grow very slow, possibly slower than $\ln(\ln(x))$. I am troubled with finding such an inverse: I only get to the point: $\ln(x)x=\ln(y)$ ...
2
votes
1answer
41 views

Trouble solving extremely simple integration by parts

This is a very basic question so it's kind of embarrassing but I can't seem for the life of me to get the right answer for some reason. I want to find $\int\frac{x}{(1+x)^2}dx=\int ...
1
vote
2answers
115 views

Cannot understand an Integral

$$\displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } =\frac { \sqrt { a } -b }{ 2 } +\frac { \sqrt { c } }{ 2 } \log(\sqrt { d } +\sqrt { e } -\sqrt { f } -g)$$ I had to solve ...
0
votes
2answers
26 views

Plot non-linear relation as a straight line

I have this relation between the data $x$ and $y$: $$y = a + \frac{b}{x}$$ and I would now like to form a straight line from the data. When I have for example $y = a \cdot r^b$ then one can plot ...
0
votes
3answers
33 views

Parametric Curves and Tangents

I am struggling with a question regard parametric curves and finding tangents to them but something is going wrong somewhere in the process and I cannot figure out why. The question asks: consider ...
0
votes
0answers
34 views

intuitive fundamental theorem of calculus

I read in "Joy of X" by Steven Strogatz about an analogy of fundamental theorem of calculus which made intuitive sense to me He explains fundamental theorem using an analogy: He says : Imagine a ...
0
votes
1answer
18 views

A particle moving in a straight line has an acceleration given by $a(t)=2t$. The initial velocity of the particle is $ 2 $ cm/sec.

A particle moving in a straight line has an acceleration given by $a(t)=2t$. The initial velocity of the particle is $2$ cm/sec. How far does the particle move between $t=1$ and $t=2$ seconds. $2t$, ...
0
votes
2answers
27 views

Limit with parameter

I have problem with following limit: Let $x\in[0,1]$ and $f_n(x)=\displaystyle n^3x^n(1-x)^4$ find $\displaystyle \lim_{n \to \infty} f_n(x)$
0
votes
0answers
18 views

Curvature of strictly concave function: scaling of derivative

I am looking at a strictly concave function $f: R_+ \to {R}$ with $f' >0$ and $f'' < 0$. I require the following property which I don't fully understand: $$ k f'(kx) \geq f'(x) \quad \forall ...
1
vote
1answer
52 views

What's in the middle of the arithmetic and geometric mean?

Arithmetic mean of two positive reals is $A(x,y) = \frac{(x+y)}{2}$. Geometric mean of two positive reals is $G(x,y) = \sqrt{(x*y)}$. Is there an easy way to compute the limit of the iteration $(x,y) ...
2
votes
2answers
66 views

Find the value of undefinite integral

Find $$\int \frac{dx}{(x+1)^{1/2}+(x+1)^{1/3}}$$ I have tried with let $u=(x+1)^{1/2}+(x+1)^{1/3}$ but I have nothing to solve that undefinite integral. please give me a clue for solve it.
0
votes
1answer
25 views

polar coordinates question

I was tasked with writing $\iint_D f(x,y) \,dx \,dy$ for $ [ D:{4\leq x^2 + y^2 \leq}9]$ through ''reoccurring integrals'' in polar and Cartesian systems? what are ''reoccurring integrals''? and how ...
3
votes
3answers
36 views

$\sqrt{y}+\sqrt{x}=\sqrt{A}$ … prove that x-intercept + y-intercept of any tangent = constant [on hold]

This is equation of a curve $\sqrt{y}+\sqrt{x}=\sqrt{A}$ $A$ is constant $T$ is a tangent of the curve from any point on it $B$ is y-intercept of $T$ $C$ is x-intercept of $T$ ...
1
vote
0answers
27 views

polar system in a plane?

what is a polar system in a plane and how it helps in calculating integrals in certain areas? I'm looking for a good explanation/a fair/ readable source on the matter.
1
vote
1answer
29 views

equivalent expressions for curvature

The curvature $\kappa$ can be written as $\frac{d\theta}{ds}$, where $\theta$ is the angle between the tangent and a fixed axis, and $s$ the arclength. I cannot understand why $\kappa$ is equivalently ...
1
vote
2answers
28 views

Change of variables in multivariable differential equations

This is a very easy question about how to justify the change of variables. Let $f$ be a $C^1$ function of two variables $x,y$. Introduce the variables $s,t$ as: $$\begin{cases} s=x+y \\ t=x-y ...
2
votes
4answers
76 views

Convergence of $\sum_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$

I have encountered the following problem: Determine whether $$\sum \limits_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$$ converges or diverges. What I have tried so far: Assume that $a_n = ...
1
vote
1answer
13 views

ratio between volumes in $\mathbb{R}^n$

Let $[-a_n,a_n]^n$ be the largest cube that fits into the n-sphere $S^{n-1}.$ Can we say what $a_n$ is? I mean, for $n=1$ we have $a_1=1$ and for $n=2$ we have $a_2 = \frac{1}{\sqrt{2}},$ so does ...
0
votes
0answers
24 views

Explain why $I$ is a function from $P$ to $P$ and determine whether it is one-to-one and onto.

The question and the solution are:( uploaded a photo so it is easier to see the formulas) So I am confused about the formula of p(x). P is the set of polynomial of x. OK, but why it makes p(x) = ...
1
vote
3answers
62 views

Integration, of $x^a$ where $a$ is an irrational number.

Can we integrate $x^{\sqrt{2}}$, or one can integrate $x^q$ only when q is some rational number. Integration of $x^q$ is $(x^{q+1})/(q+1)$.
2
votes
4answers
42 views

Gradient of a curve $y=\ln \sqrt{x+y}$

Find the gradient of the curve $y=\ln \sqrt{x+y}$ at the point when its y-coordinate is 1. My attempt, I differentiated and I got $\frac{dy}{dx}=\frac{1}{2x+2y-1}$. But I've problem in finding the ...
1
vote
2answers
48 views

limits, $x \to 1$ and $n \to \infty$ [on hold]

If $x \to 1^+$ and $n \to + \infty$ ,what will be the answer to $x^n$ How is it different from $\large{(1+x)^\frac{1}{x}}$ where $x \to 0$.
1
vote
1answer
53 views

Evaluate $\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$

I am trying to evaluate $$\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$$ where $s>0$, $M$ and $N$ are positive integers. But seem that the above integral ...
1
vote
1answer
47 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
-1
votes
1answer
24 views

Polya maths enrichment [on hold]

Let $p_1$, $p_2$, $q_1$ and $q_2$ be real numbers such that $p_1p_2 = 2(q_1+q_2)$. Prove that at least one of the equations has real roots: $$ x^2 + p_1x + q_1= 0,\quad x^2 + p_2x + q_2= 0. $$
1
vote
2answers
58 views

Example 2, Sec. 11.1 in Apostol's CALCULUS, vol. 1: How to calculate this limit?

For each $n \in \mathbb{N}$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined as $$f_n(x) \ \colon= \ nx (1-x^2)^n \ \mbox{ for all x } \ \in [0,1]. $$ Then $f_n(0) = 0 = f_n(1)$. So let $0 < x ...
1
vote
3answers
51 views

$\sum_{n=0}^{\infty}(2x^{n}-x^{2n})$

Consider $$\sum_{n=0}^{\infty}\left(2x^{n}-x^{2n}\right)$$. Find its set of convergence and and function defined as $f(x) = \sum_{n=0}^{\infty}(2x^{n}-x^{2n})$ on that set. As far as I ...
1
vote
4answers
55 views

How to solve $ \int \tan{\theta}\sec^4{\theta}d\theta $

I got the answer of: $$\large{\frac {\tan^2{\theta}}{2} + \frac {\tan^4{\theta}}{4} + C}$$ Wolfram Alpha got: $\large{\frac {sec^4{\theta}}{4}}$. I don't know what I did wrong: I set $\large{u = ...
0
votes
0answers
83 views

How to improve my these specific math skills? [on hold]

I am student of CS. Problem is, I feel that I don't have enough math knowledge to solve mathematical problems. When some programming problems arises which needs some math skills to solve then despite ...
-2
votes
0answers
22 views

is 1/x a lipschitz continuous function? [on hold]

Actually, I would like to know if the dynamics associated, with initial condition $x(0)=1$, has any solution in $\mathbb{R}^2$. I tried to find the solution and I got that $x=\pm\sqrt{2t+1}$, but then ...
-1
votes
2answers
40 views

Is there has a smart way to compute the 1order derivative of the circle equation? [on hold]

I have encountered a compute problem. This exercise has given the circle equation and a para-curve equation with unknown parameters, the para-curve and circle has the same radius of curvature, and ...
1
vote
3answers
173 views

Integration of an equation that looks like inverse tangent

$$\int\frac{1}{1+x^{-2}}dx$$ I thought that this equation would become $$\tan^{-1}(x^{-1})+c$$ What is the proper way to take the integral?
1
vote
1answer
45 views

Why Does The Taylor Remainder Formula Work?

I've been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the ...
0
votes
2answers
37 views

Derive the solution to $\frac{dQ}{dt} = kQ$

Derive the solution to $\frac{dQ}{dt} = kQ$ in terms of $Q_0$ Here is my work: $\frac{dQ}{dt} = kQ$ $\frac{dQ}{Q} = kdt$ $\int\frac{dQ}{Q} = \int kdt$ $lnQ = kt + C$ $Q = e^{kt}e^{C}$ Did I ...
0
votes
1answer
40 views

Application of Complex Variables

By considering the integral of: $$\left(\dfrac{\sin\alpha z}{\alpha z}\right)^2 \dfrac{\pi}{\sin{\pi} z},\quad \alpha \lt \dfrac{\pi}{2}$$ around a circle of large radius, prove that ...
0
votes
1answer
17 views

Population Growth Word Problem Using the Law of Natural Growth

The problem is included in the image below. There are three parts to the problem, and all three are on the same page. I am looking for solution verification on all three parts, but I have a specific ...