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

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2
votes
3answers
216 views

a question about double integral

Let $a,b$ be positive real numbers, and let $R$ be the region in $\Bbb R^2$ bounded by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Calculate the integral $$ ...
2
votes
1answer
400 views

Find the derivative of $F(x) = \int_0^x xf(t) dt$

this was given as an exercise in my first year honours math class. I can't seem to wrap my head around why this is not equal to $xf(t)$. Any help is appreciated! heres the question: Find the ...
0
votes
0answers
82 views

Prove a function is uniformly continuous (might be Lipschitz Condition)

Let $f$, a continuous function defined on the interval $[0,\infty)$. It is given there are $a,b$ such that: $$\mathop {\lim }\limits_{x \to \infty } \left[ {f(x) - (ax + b)} \right] = 0$$ ...
0
votes
1answer
45 views

Let $f:\mathbb{R} \to \mathbb{R}$ and $|f(x)| \le |\sin^3(x)|$ for $|x| \le 1$. Prove statements.

Let $f:\mathbb{R} \to \mathbb{R}$ and $|f(x)| \le |\sin^3(x)|$ for $|x| \le 1$. Prove that f is differentiable in x=0. I think that I can show that $0 \le |f(0)| \le 0$, thus $f(0)=0$ and then I ...
0
votes
1answer
33 views

Prove that $\exists x_1,x_2 \in \mathbb{R}$ with $|x_1-x_2|=\frac{T}{2}$ such that $f(x_1)=f(x_2)$

Let $f:\mathbb{R} \to \mathbb{R}$ be continuous periodic function with T>0. Prove that $\exists x_1,x_2 \in \mathbb{R}$ with $|x_1-x_2|=\frac{T}{2}$ such that $f(x_1)=f(x_2)$. I don't even know ...
0
votes
2answers
34 views

When to consider an approximation as a Good approximation?

What is the criteria for the Good approximation ? e.g. we can approxiamte $\sin(x)$ to $x$ for $x<0.16 rad $ why 0.16 ? why not 0.23 enother e.g. $\tanh(x/2s)=x$ for $x<s$ and so on .. how ...
2
votes
0answers
46 views

Integral containing inverse $\tanh$

How can I solve this integral containing inverse $\tanh$? Does it have any antiderivative? $$ \int \cfrac{t^2}{\sqrt{r^2-t^2}} \cdot \operatorname{arctanh}\sqrt{1-t^2}\; \mathrm{dt} $$
0
votes
1answer
61 views

Abscissa is the tangent line for the function…

I have such a problem: Find the parameter $a$ for which the abscissa is the tangent line for the following function: $$f(x)=a+9x-\frac{x^3}{3}$$ Thank you in advance!
3
votes
2answers
225 views

Two hyperreal numbers infinitely close to each other; $100$ and $100+\epsilon$

$100$ is a real number or we could call it a hyperreal number as every element of $\mathbb R$ is also an element of $\mathbb R^*$. If we add an infinitesimal say $\epsilon$ to $100$, the new number ...
11
votes
3answers
613 views

What's wrong with this definition of continuity?

Consider this definitions: A function $f:X \to Y$ is continuous at $x\in X$ iff for any open neighborhood $V_{f(x)}$ of $f(x)$ there is an open neighborhood $U_{x}$ of $x$ that gets mapped by $f$ ...
1
vote
2answers
746 views

How can I find the average y value of a function on a given domain?

Lets say that $f(x) = (10 - x)\ln x$. Over the domain: $1 ≤ x ≤ 10$. How can I find the average value of $y$ over this domain and what is that value?
0
votes
3answers
174 views

Finding velocity in optimization problem

Given $s=-16t^2+192t+144$, what is the velocity when $s=0$? This is part of a larger optimization problem which I solved, except for this last part. The critical point occurs at $t=6$, so after ...
4
votes
1answer
56 views

Continuous function calculus

Are there any examples satisfy the continuous function $f:\Bbb R\rightarrow \Bbb R$ such that the image of the closed interval $[0,\infty)$ under $f$ is the open interval $(-1,1)$..like ...
0
votes
3answers
49 views

Why is the domain of $F(t) = t^t$ for $ t \gt 0$ for all real numbers?

If you test values negatives values and even $0$ for $t$, you still get values for them.
2
votes
1answer
58 views

How come complex numbers represent coordinates?

I'm wondering why complex numbers represent coordinates without being on the form of a tuple (a,b). The complex numbers come in the form: $a+bi$ where $a$ denotes the real part and $bi$ denotes the ...
0
votes
2answers
134 views

Solving this equation with 3 variables.

I came across this equation. $m = (m + 1/2)x + (b-1)$ The goal is to solve for m and b. I want to make sure I understand the very last algebraic step required. Since the two sides are equal, you ...
1
vote
1answer
66 views

Algebraic Equation vs Algebraic Function

By definition, a function given by $y=f(x)$ is algebraic if it can be expressed in the form $$p_n(x)y^n+p_{n-1}(x)y^{n-1}+⋯+p_1(x)y+p_0(x)=0$$ where $p_0(x), p_1(x),...,p_n(x)$ are polynomials and ...
1
vote
1answer
76 views

Boundary of Set: $0<x<1, y = \sin(1/(1-x))$

So as stated in the title I need the boundary of the set in $\mathbb{R}^2$ of $0<x<1, y=\sin(1/(1-x))$. I understand what S looks like (the curve) and therefore the curve between $0$ and $1$ is ...
4
votes
2answers
42 views

If f is a function of $x$ and $t$ where $x$ itself is a function of time does this mean $\frac{\partial f}{\partial x}=0$?

$\frac{\partial f(x(t),t)}{\partial x}=0$? I suspect it probably doesn't but I can't justify it to myself.
2
votes
2answers
97 views

Find greatest value of $y(x) = (0.9^x)(300x + 650)$

Question and attempt $y(x) = (0.9^x)(300x + 650)$ Estimate at what x value that y reaches its maximum value The only way I could think of would be to use derivatives, so I tried it: $y'(x) = ...
2
votes
1answer
107 views

finding the value of the sum $\sum _{n=1}^{\infty} \frac{7^{n}}{8^{n}+2^{n}}$

I am concerned with finding the value of the sum $\sum _{n=1}^{\infty} \frac{7^{n}}{8^{n}+2^{n}}$. There may be some easy way to do this, but I have not found a way to compute this sum, and ones like ...
0
votes
1answer
37 views

Tangent and Taylor polynomials

We know that this series $x+ \frac{x^3}{3}+\frac{2x^5}{15}+\ldots$ is convergent in $|x|\lt \pi/2$, furthermore it converges to $\tan(x)$. I would like to know if we restrict to finite terms of this ...
1
vote
1answer
56 views

Am I understanding induction correctly?

Here is an induction proof that I have written for my homework and I want to know if I am understanding this correctly: Prove that for: $ \sum\limits_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ My proof: ...
0
votes
2answers
59 views

Bernoulli's inequality by induction

I'm proving Bernoulli's inequality by induction but I noticed something strange. See wikipedia proof: http://en.wikipedia.org/wiki/Bernoulli's_inequality Notice how they multiply both sides of the ...
0
votes
1answer
19 views

Integral convergence of (-1) floor

Question: does this integral converge? $\int _0^\infty (-1)^{\lfloor x^2 \rfloor}dx$ Thoughts: We tried using Cauchy to prove divergence. We don't know if that's true or not.
1
vote
1answer
52 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
10
votes
3answers
283 views

Should $f(x) \equiv 0$?

Assume $f(x)$ is a real-function defined on $[0,+\infty)$ and satisfies the followings: $f'(x) \geq 0$ $f(0)=0$ $f'(x) \leq f(x)$ Should we always have $f(x) \equiv 0$ ? Thanks for any solution.
0
votes
3answers
129 views

How to solve $\lim_{x\rightarrow1} \frac{\ln (x)}{x-1}$ using pre-derivative calculus?

How can I solve: $$\lim_{x\rightarrow1} \frac{\ln x}{x-1}$$using pre-derivative Calculus (no logarithm series or L'Hôpital's rule)?
0
votes
2answers
61 views

Optimization of electricity costs

I have to solve this exercise for the school and I do not really understand why the teacher solved it like this. Here is the exercise: I want to replace the 60 watt bulbs with 8 watt LED lamps. The ...
3
votes
3answers
245 views

Inflection point of a function

For what values of $a$ does the function $f(x)=e^x+ax^3$ have an inflection point? It is an old question in my mind and wanted to bring it out.
6
votes
3answers
374 views

Integrating $\int \cos^3(x)\cos(2x)$

How would Integrate the following. $\int \cos^3(x)\cos(2x)$ I did $\int \cos^3(x)(1-2\sin^2(x))$ $2\int \cos^3(x)-\cos^3x\sin^2x$ But I find myself stuck....
1
vote
0answers
40 views

Multiple equations with relations.

I'm very bad at math so please correct me or let me know if I'm posting in the wrong section. I have a set of equations (in reality any number > 2) but if its solvable for 3 it would be good enough. ...
8
votes
1answer
258 views

Darboux's Integral vs. the “High School” Integral

The definition of the integral below is what I usually call the "High School definition," because that's usually where I've seen it in use. Take a partition $\Delta = \{ x_0, x_1, x_2, \ldots, ...
4
votes
1answer
108 views

Inequality $ \left|x\sin\frac{1}{x}-y\sin\frac{1}{y}\right|\leq\sqrt{2|x-y|} $

$ \forall x,y>0 $, then $$ |x\sin\frac{1}{x}-y\sin\frac{1}{y}|\leq\sqrt 2\sqrt{|x-y|} $$ Is $\sqrt 2$ the minimum positive real number such that the inequality holds ? I try to apply Mean value ...
5
votes
2answers
172 views

if $f\left(x\right)=\int\limits _{0}^{x}f\left(t\right)dt$ then $f \equiv 0$?

$f:[0,\infty)\rightarrow\mathbb R$ is a function that for all $0\leq a<b\in \mathbb R$, $f_{|[a,b]}:[a,b]\rightarrow \mathbb R$ is integrable. assuming that for all $x\in[0,\infty)$ , ...
0
votes
1answer
37 views

Question about limit.

My question is that in my practical sheet I have been given a question which says show that limit doesn't exist and question is $f(x,y)= \frac{x^2}{x^2+y^2-x}$ s.t $(x,y)\ne(0,0)$ My question is: ...
3
votes
2answers
123 views

Absolute convergence of $ \sum \ln\left(1+\frac{ (-1)^n }{n}\right) $?

If we take: $$ \sum_2 ^\infty \ln\left(1+\frac{ (-1)^n }{n}\right) $$ I know it converges, because this is actually a kind of telescopic series . But, does it absolutely converge? I have no idea ...
2
votes
1answer
29 views

Two definitions of Taylor polynomials

I'm studying a book which states Given a function $f:I\to \mathbb R$, $n$ times derivable in the point $a\in I$, the Taylor polynomial of order $n$ of $f$ in the point $a$ is the polynomial: ...
0
votes
2answers
39 views

Limit of $\frac{1}{x}\ln{\frac{\exp{x}-1}{x}}$

How can I find the limit of $\dfrac{1}{x}\ln{\frac{\exp{x}-1}{x}}$ at $+\infty$ using equivalence ($\widetilde { +\infty }$ Little o)?
0
votes
2answers
42 views

Relation between two integrals

Let $f(x)$ be a monotonically decreasing continuous function for non-negative $x$. Let us define $g(x)=\sqrt{f(x)},\forall x\geq0$. For any two positive numbers $a$ and $b$ (such that $a\leq b$), is ...
1
vote
0answers
30 views

Fourier transform with a scaled variable

Can somebody explain me the following: Discrete Fourier transforms and their inverse are usually outlined as follows: \begin{equation} F(n)=\sum_{m=1}^N f(m) e^{-\frac{2\pi i}{N} mn} \end{equation} ...
1
vote
1answer
53 views

How prove this $\sum_{n=1}^{\infty}\frac{x^n(1-x)^n}{n!}f^{(n)}(x)=-\frac{1}{2}xf(x)$

let $$f(x)=\dfrac{x}{\ln{(1-x)}}$$ prove that for $0<x<1$, $$\sum_{n=1}^{\infty}\dfrac{x^n(1-x)^n}{n!}f^{(n)}(x)=-\dfrac{1}{2}xf(x)$$ I think use this Taylor series ...
0
votes
1answer
72 views

Integrating a Taylor series term-by-term

Why is $$\int_{0}^{z} \frac{\sin x}{x} \ dx =\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} \int_{0}^{z} x^{2n} \ dx$$ not valid for $z= \infty$? Well, at least I'm assuming it's not valid since ...
1
vote
2answers
2k views

Cross section with equilateral triangles and integration

Hello guys so I needed help with a problem which is: Let $S$ be the solid with flat base, whose base is the region in the $xy$-plane defined by the curves $y=e^x$, $y=−2$, $x=1$ and $x=3$, and ...
2
votes
2answers
122 views

Derivative and Integral of $\pi(x)$ (primes before $x$)?

What's the derivative and indefinite integral of $\pi(x)$, which gives the number of primes before $x$? I know there's no answer in terms of elementary functions, but is there an answer in terms of ...
0
votes
2answers
65 views

Build the function by its values. Only combination of +, -, *, abs() allowed for this function.

I've decided to open a new, more common question about the simplest function f(1)=-1; f(2)=0; f(3)=1; f(4)=0.. So, here is the question. Let's say we have some function $y=f(x)$ we'd like to find by ...
0
votes
1answer
40 views

Use the disc method to find the volume of solid of revolution.

Use the disc method to find the volume of solid of revolution obtained by revolving the the given region around the x-axis. $R\quad is\quad bounded\quad by\quad the\quad graphs\quad of\quad k(x)\quad ...
2
votes
3answers
85 views

Why is the value of this line integral constant

Consider the line integral given by $$\int_C \frac{(x+y)\,dx-(x-y)\,dy}{x^2+y^2}$$ where $C$ is any simple closed curve around the origin. Can someone explain, without using complex analysis, why this ...
1
vote
1answer
67 views

Area between two functions?

Find the area between the functions $x+y = 2$ and $x + 4 = y^2$. The question is relatively simple: The area between the functions is: $$\int^{2}_{-3} 2-y-y^2+4 \text{ }dy$$ But can the above ...
0
votes
1answer
30 views

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ [duplicate]

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle \int_1^2\int_0^1\frac ...