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

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0
votes
4answers
95 views

The sum of two variable positive numbers is $200$. Find the maximum value of their product.

The sum of two variable positive numbers is $200$. Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$. NB: I'm currently on the ...
1
vote
3answers
51 views

How to prove that $ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$

How to prove $$ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
0
votes
1answer
28 views

To prove:$b^{\frac{1}{p}}-a^{\frac{1}{p}} \le (b-a)^{\frac{1}{p}}$

if$$p\ge1,0<a<b.$$How to prove?thx~
8
votes
7answers
240 views

Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
2
votes
1answer
38 views

partial derivative of function with a matrix

Let $A$ be a $n\times n$ matrix. Let $f\in C^1(\mathbb R^n)$ and $g:\mathbb R^n\rightarrow\mathbb R, g(x)=f(Ax)$. What is the partial derivative $\partial_{x_i} g(x)$? So $Ax=(\sum_{l=1}^n ...
1
vote
4answers
106 views

How do I show that $\lim_{x \rightarrow \infty}(1+\frac{a}{x}+\frac{b}{x^{3/2}})^x =e^{a}$?

How do I show that $\lim_{x \rightarrow \infty}(1+\frac{a}{x}+\frac{b}{x^{3/2}})^x = e^a$? Actually, I had to deal with something similar yesterday and after thinking about it for quite a while I did ...
1
vote
0answers
56 views

Questions about the divergence theorem

I am looking at the proof of the divergence theorem and I have some questions. The proof of the divergence theorem $$\iiint_D \nabla \cdot \overrightarrow{F} dV= \iint_S \overrightarrow{F} \cdot ...
2
votes
1answer
117 views

Tricky improper integral

How to check convergence of $$\int_{-\infty}^{2}\frac{e^x}{\sqrt[3]{x^2-1}}\ dx \ ?$$ No convergent integral that could bound this one came to my mind, nor any which would fit asymptotic criterium. ...
2
votes
3answers
140 views

Differentiate the function into the simplest form

My question: $y=\sin^{2}(x)$ My attempt: Is $\sin^{2}x$ the same as $(\sin(x))^2$? By rearranging the function I came up with the following. $$ \begin{align} u = \sin(x), \ & y=u^2 \\ ...
57
votes
2answers
6k views

The Monster Integral

Compute the following integral \begin{equation} \int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, ...
19
votes
3answers
617 views

Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$

How to compute the following integral \begin{equation} \int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx \end{equation} I have been given two integral questions by my teacher. I cannot answer ...
0
votes
1answer
87 views

Integration of a function in Schwartz space

How to prove the following: If $f(x)$ belongs to Schwartz space then integral from $x$ to infinity of $f(x)$ also belongs to Schwartz space?
2
votes
2answers
67 views

Doubts about infinite nested root

Find $f(a)=\sqrt{a-\sqrt{a^2-\sqrt{a^4-\cdots}}}$ where $a\in\mathbb{R}$. My Attempt : I consider $\frac{f(a)}{a}=\sqrt{1-\sqrt{1-\sqrt{1-\cdots}}}$. Now to finding this limit is easy but I cannot ...
1
vote
2answers
27 views

Prove $a_n$ is within a specific range for all $n\in\mathbb{N}$

Let $a_n = \cos (a_{n-1})$ and $a_1 = {\pi\over 4}$. How to prove the range of $a_n$ is within the closed interval: $\left[ {{1 \over {\sqrt 2 }},{\pi \over 4}} \right]$? I thought about ...
0
votes
1answer
110 views

Heaviside function in an integral

I having problems understanding how this integral is evaluated. Let $c,\beta$ be constants. $\begin{align*}u(x,t)&=\frac{c}{\beta}\int_{0}^{t}H(s-x/c)f(t-s)ds \\ ...
1
vote
1answer
49 views

How to interperet calculus thing

I have $\nabla \times (f\mathbb{F})$ where $f$ is a twice continuously differentiable scalar field and $\mathbb{F}$ is a twice continuously differentiable vector field. Is it right to interpret $f$ ...
1
vote
0answers
66 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
1
vote
0answers
42 views

Question on Drichlet Function?

Definition of Dirichlet function $$D(x)=\begin{cases}1&x\in \mathbb Q\\0&x\in \mathbb Q^c\end{cases}$$ Function is known as nowhere continuous function, since the limiting value does not ...
1
vote
0answers
33 views

What vector theorem should be used?

I have to integrate this line integral $$\int_C \mathbf F \cdot d\mathbf r$$ Where $\mathbf F = (\frac{y}{x^2 + y^2},\frac{-x}{x^2 + y^2})$ and $C$ is the curve $x^2 + 2y^2 = 1$ oriented ...
3
votes
1answer
61 views

Calculate integral using the method of parameter derivation or integration [duplicate]

$$F( \alpha ) = \int^{ \pi }_{ 0} \ln ( 1 - 2 \alpha \cos x + \alpha^{2} ) dx$$ Should I derive the inner function? But I can't process the derived outcome.
0
votes
4answers
61 views

Proving a function is periodic!

I am having trouble assimilating periodic function. Let me tell you, I have had a semester of fourier analysis already but reviewing the first chapter got me confused on a trivial equation. A ...
0
votes
3answers
290 views

geometric interpretation of Taylor's theorem

Deriving Taylor's theorem is not a problem. But I am curious if there is any nice geometric interpretation of the theorem.
5
votes
2answers
80 views

a question about how to prove mutivariable integral, I am struggling about it!

If $f(x)$ is Riemann integrable in $[a,b]$, and then how to prove $$\int_{a}^{b} f(x_1) \, dx_1 \int_{a}^{x_1}f(x_2) \, dx_2 \cdots \int_{a}^{x_{n-1}}f(x_n) \, dx_n={1\over n!} \left[\int_a^b f(x) \, ...
1
vote
1answer
52 views

Solve for $f(x)$ and $b$ in Riemann sum problem

The following sum $$ \sqrt{49-\frac{7}{n}^2}\cdot\frac{7}{n} + \sqrt{49-\frac{14}{n}^2}\cdot\frac{7}{n} + \cdots + \sqrt{49-\frac{7n}{n}^2}\cdot\frac{7}{n} $$ is a right Riemann sum for the ...
5
votes
3answers
80 views

How to calculate the range of $x\sin\frac{1}{x}$?

I want to find the range of $f(x)=x\sin\frac{1}{x}$. It is clearly that its upper boundary is $$\lim_{x\to\infty}x\sin\frac{1}{x}=1$$ but what is its lower boundary? I used software to obtain the ...
2
votes
1answer
42 views

Is all integration that are computable able to be compute by the substitution method and/or integration by parts method?

Is all single-variable integration that are computable by mathematician theoretically able to be compute by the substitution method and/or integration by parts method? If not, does that mean much ...
0
votes
1answer
57 views

Using derivative to find acceleration

Any help with the below question would be appreciated - I have a mental blank at the moment! A car, initially travelling at the speed $100 \text{ km/hr } (A)$, slows down according to the formula ...
2
votes
1answer
40 views

A Subadditive Function Satisfying an Integrability Property

Is there a function $f: [0,\infty) \rightarrow [0,\infty)$ that is Subadditive $\left(f(x + y) \leq f(x) + f(y), \forall x,y \in [0,\infty)\right)$, Satisfies $\lim_{r \rightarrow \infty} f(r) = ...
0
votes
2answers
43 views

How to evaluate this integral involving with a variable and exponential number?

How to evaluate this with integration by parts if necessary? $\displaystyle \int \frac {x^3e^{x^2}}{(x^2+1)^2} dx$
4
votes
3answers
96 views

Explanation of $\int_0^{2\pi}\sin^{100}x\,dx$.

I was browsing this thread when I came across this answer. I can neither make heads nor tail of it. Can someone help me understand it? This I understand: ...
0
votes
1answer
2k views

Use the midpoint rule to estimate distance traveled

The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Use the Midpoint Rule with n = 6 to estimate the distance (in km) traveled during this ...
2
votes
1answer
58 views

CW complex adjunction map

In topology we defined a quotient topology for glueing in the following way: Let $(X,O)$ and $(Y,O)$ be topological spaces and $f:A \subseteq X \rightarrow Y$ a continuous map, then we have that $X ...
4
votes
2answers
102 views

Solve multivariable limit

$$\lim_{(x,y) \to (0,0)} \frac{x^3 + y^4}{x^2 + y^2}$$ I am almost sure it is equal to $0$ but I can't prove it. Please give me some hint.
10
votes
1answer
116 views

Topologists glueing, cutting and so on. When is this rigorous?

I often see that things in topology are explained very non-rigorously recently. Thereby I mean that it is said that we can cut something and glue something together and so on in order to identify two ...
0
votes
1answer
56 views

Calculte new width and height of the video based on the original width, height and ratio

after searching through - I wasn't able to find the answer so I'll give it a shot here. I need to resize desktop video in order to feet it on mobile screen, let's say original width of the video was ...
0
votes
1answer
39 views

Recursive sequence problem

$$U(n+1) = (6+U(n))^{1/3},\text{ and } U(0) = 1.$$ Prove by induction that for all positive integers $n, U(n)$ is increasing. Prove by induction that for all positive integers $n, U(n) \leq 2$ ...
0
votes
1answer
23 views

composition of two functions problem

Let $\mathbb{R}^{n}$ and for $n=0$ we define $\mathbb{R}^{0}$ to be just a point. Consider two functions $$i:\mathbb{R}^{0}\to \mathbb{R}^{n}$$ ''that sends $\mathbb{R}^{0}$ to the origin''(do ...
3
votes
2answers
259 views

How to integrate this formula with secant, exponential, and tangent?

How to integrate this? $$\int \sec^2(3x)\ e^{\large\tan (3x)}\ dx$$
1
vote
0answers
32 views

How to establish a lower bound on this difference operator?

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
0
votes
1answer
65 views

convergence of a series (involving factorial, power and sum)

For all $n\in\mathbb{N}$ consider the positive number $$\alpha_n := \frac{2^{-n}}{n!}\,\sum_{k=0}^n\frac{k^n}{k!} \,.$$ Is the series $\sum_{n=0}^\infty\alpha_n$ convergent or divergent?
0
votes
2answers
82 views

Why does the derivative and integral of the funcion exist (doesn't exist)?

Given $ f(x) = \dfrac {2\sec(x)(\cos^2(x)+3x^4\cos(3x^2))}{3x^3(1+3x^2)} $, I know that $f(x)$ is not defined in $x=0$. And it is not defined in $ {k\fracπ2:k∈Z}$ either (thanks Git Gud) And i ...
2
votes
1answer
61 views

How to evaluate this integral for this type?

What is the skill to integrate this type of integral $\displaystyle\int \frac{4x^n}{x^2+9} dx$ for $n$ is constant How to use your general method to work out an example?
0
votes
2answers
68 views

An object is travelling in a straight line.

An object is travelling in a straight line. Its distance, $s$ meters, from a fixed point at time $t$ seconds is given by the expression: $$s=t^6-t^2-6t.$$ a) Find $\frac{ds}{dt}$ when $t = 3$ ...
0
votes
1answer
40 views

Boundedness of relative risk aversion function

Let $f\colon [0,\infty)\mapsto \mathbb{R}$ be a strictly increasing and strictly concave function. Let $$RRA_f(x)=-\frac{xf''(x)}{f'(x)}.$$ Is it possible that $RRA_f$ isn't bounded?
0
votes
0answers
79 views

Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0 $ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
2
votes
4answers
102 views

Prove: for $\forall x\ne 0, \cos x < 1 - {x^2\over 2} + {x^4\over 24}$

Prove: for $\forall x\ne 0, \cos x < 1 - {x^2\over 2} + {x^4\over 24}$ What I did: We can prove: $${\cos x -1 + {x^2\over 2} \over {x^4\over 24}} < 1$$ Lets define: $f(x) = \cos x -1 + ...
0
votes
1answer
95 views

The correct term for $y$ in $y=f(x)$

Given: $y=f(x)$ than y is: a) range b) domain c c) variable d) co do-main I saw this question on an fb page and I couldn't get the right answer. a,b,d cannot be the answers since these are ...
3
votes
2answers
99 views

Integral involving a logarithm and a linear rational function

$$\int_{0}^{1} \frac{\log x}{x-1}dx$$ I was wondering: is it possible to evaluate this integral with real methods? Playing around with a series expansion I was able to recognize that the integral is ...
2
votes
3answers
125 views

How do I evaluate integrals that involve the signum ($\text{sgn}$) function?

For example, I want to evaluate $$ \displaystyle \int_{0}^{2\pi} \left| \sin x \right| \text{ d}x $$ and I already know that: $$ \displaystyle \begin{aligned} \int \left| \sin x \right| \text{ d}x ...
1
vote
1answer
52 views

Calculating the mass flux through the curve $AB$

Flux through a flat curve We want to calculate the mass flux through the curve $AB$ $$\Delta m= \delta \cdot \Delta s \cdot \Delta t \cdot \overrightarrow{v} \cdot \hat{n}$$ ...