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

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0
votes
0answers
34 views

If a Taylor series about $x_{0}$ has a radius of converges $R = \infty$

If a Taylor series about $x_{0}$ has a radius of converges $R = \infty$, what does it say about the Taylor remainder about $x_{0}$? Does it say anything about the Taylor polynomial about $x_{0}$? ...
0
votes
3answers
34 views

Continuous functions and limits inequalities

Is it true that if $f(x)$ and $g(x)$ are both continuous functions and $f(x) \leq g(x)$ for all $x \in \mathbb{R}$, then $$\lim_{x\to\infty}f(x) \leq \lim_{x\to\infty}g(x)?$$ It makes sense to me ...
0
votes
6answers
112 views

There exists $c \in [0,1]$ for which $\int_{0}^{1}\sin(x^3) = \int_{0}^{c}\sin(x^2)$

T/F: There exists $c \in [0,1]$ for which $\int_{0}^{1}\sin(x^3) = \int_{0}^{c}\sin(x^2)$ I know the answer it true, and I already saw the proof. What I don't get is this: $\sin$ is monotonically ...
4
votes
5answers
128 views

$\lim_{x\to \infty} \ln x=\infty$

I'm reading the following reasoning: Since $\underset{n\to \infty}{\lim}\ln 2^n=\underset{n \to \infty}{\lim}n\cdot(\ln 2)=\infty$ then necessarily $\underset{x\to \infty}{\lim}\ln x =\infty$. ...
0
votes
2answers
28 views

Is my thought process in finding the tangent parallel to x & y axis correct?

I want to find the coordinates where the tangent to the curve is parallel to the x and y axis. The curve is $$2x^2 +xy - y^2 +18 = 0 $$ $$ Dy/dx = (-4x-y)/(x-2y) $$ Am I correct in saying that to ...
4
votes
6answers
146 views

Why is $\lim_{x\to \infty} x(\sqrt{x^2+1} - x) = 1/2$

I've been doing some calculus problems lately out of an old Russian book, and I came across something I didn't fully understand: One of the problems said that $$\lim_{x\to \infty} x(\sqrt{x^2+1} - x) =...
1
vote
1answer
55 views

If a multiple integral is zero over some region, can I say the integrand is zero?

Consider the following problem Let the integral of a real function of 3 real variables $F(x,y,z)$ over some volume $V$ of $R^{3}$ vanish, $\int$$\int$$\int$$dxdydz$ $F(x,y,z)$$=0$ Now assume this ...
2
votes
0answers
70 views

Any reason not to define a derivative as the average of the derivatives on all sides?

We all know $\operatorname{abs}$ is not differentiable in a classical sense, but one question that's always bothered me is, why not define the derivative as the average derivative in each direction? i....
3
votes
4answers
149 views

Different ways of evaluating $\int_{0}^{\pi/2}\frac{\cos(x)}{\sin(x)+\cos(x)}dx$

My friend showed me this integral (and a neat way he evaluated it) and I am interested in seeing a few ways of evaluating it, especially if they are "often" used tricks. I can't quite recall his way,...
0
votes
2answers
39 views

True or False - Taylor expansions

True or false: If $f$ is a Taylor expansion about $x_{0} = 0$ with radius $R$, then $$g(x) = f\left(\frac{x-1}{2}\right)$$ has a Taylor expansion about $x_{0} = 1$ with radius $2R$ I think this is ...
1
vote
0answers
46 views

Questions about a function that is not Riemann integrable

Consider the function: $$f(x) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \in \mathbb{Q} \\ -1 & \mbox{if } x \notin \mathbb{Q} \end{array} \right.$$ I want to check if this function ...
1
vote
1answer
44 views

Differentiation under integral sign- Multivariable case problem

Let $f_{\theta}(x,y)=f(x\cos \theta-y\sin \theta,x\sin\theta+y\cos\theta)$, where $f\in C^2(\Bbb{R}^2)$(Is the range necessarily $\Bbb{R}^2$? This is quite ambiguous.) a function with a bounded ...
9
votes
2answers
534 views

Prove that $ f(x) = e^x + \ln x $ attains every real number as its value exactly once

Prove that the function $$ f(x) = e^x + \ln x $$ attains every real number as its value exactly once. First, I thought to prove that this function is a monotonic continuous function. But then I ...
-1
votes
2answers
61 views

How to show is function surjective? [duplicate]

$f\left(x,y\right)=\left(\frac{x}{x^{2}+y^{2} } , \frac{y}{x^{2}+y^{2} } \right) $ $R^{2}-(0,0)$ Can someone help me with shwoing that this function is surjective?
0
votes
1answer
29 views

Prove that $\lim_{x \to \infty}\alpha(x) = -\lim_{x \to \infty^-}\alpha(x)$

Let $\alpha(x) = \displaystyle \int_{0}^x(1+t^2)^{-1}dt$. Prove that $\displaystyle \lim_{x \to \infty}\alpha(x) = -\lim_{x \to \infty^-}\alpha(x)$. I don't understand what is the difference between ...
0
votes
4answers
73 views

How to find inverse of function?

$f\left(x,y\right)=\left(\frac{x}{x^{2}+y^{2} } , \frac{y}{x^{2}+y^{2} } \right) $ How do I find inverse function? I have trouble here, because there are 2 variables and I am not sure how to do it?...
4
votes
3answers
106 views
+100

Position of Object Suspended on a String (Need Another Answer)

I'm going to try to make as few errors in typing this as possible, so please bear with me and ask me to clarify/correct whatever needed. Q: If an object is suspended on a string hung between two ...
1
vote
1answer
30 views

Taylor series expansion of function

I have the following statement - If $f$ has a Taylor series expansion about zero with radius $R$ , then $g(x) = \displaystyle f\left(\frac{x-1}{2}\right)$ has a Taylor expansion about $X = 1$ of ...
1
vote
3answers
29 views

Taylor series, Finding the sum of a series

I was given the following function: The area of convergence is [-5/3,7/3), I want to represent the series as a function, what is the framework? what are the steps for such a problem?
0
votes
4answers
54 views

How to prove this series converges ?

I have this series $\sum _{n=0}^{\infty }\:\left(\sqrt[n]{n}-1\right)^n$ Im having truoble to prove that this converges, I've tryind to use the ratio test but it didnt seem to get me to something ...
0
votes
2answers
64 views

T/F: if $\int_{1}^{\infty}f(x)dx$ converges then $\lim_{x\to\infty}f(x) = 0$.

I was asked to prove or disprove the following: If $\int_{1}^{\infty}f(x)dx$ converges then $\lim_{x\to\infty}f(x) = 0$. I said that this is false and gave this example: $f(x) = \left\{ \begin{...
-2
votes
2answers
29 views

Additive Property of Integrals of Step Functions [duplicate]

In Apostol's Calculus Volume-1 the proof of Additive Property for Integrals of Step Functions is given as an exercise that is: $$\int_a^b[u(x)+g(x)]dx=\int_a^b u(x)dx+\int_a^b g(x)dx$$ And Integrals ...
-2
votes
3answers
49 views

Help with Trigonometric Functions

so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the sin, cos and tan functions, I am not posting the formula for obvious reasons. My ...
0
votes
2answers
55 views

Convergence/divergence of $ \sum_{n=1}^{\infty} \sin \left(\frac{1}{n^2} \right) \cdot \sqrt{n^2+n+1} $

I tried proving $$\sum_{n=1}^{\infty} \sin\left(\frac{1}{n^2} \right) \cdot \sqrt{n^2+n+1}$$ diverges/converges with the comparison test but no matter what I can't prove it. I first tried with $\sin(...
1
vote
0answers
39 views

Integrals of the form $\int x^m(a+bx^n)^Pdx$

I was reading a book on Integral Calculus, and in one chapter, the author dealt with methods of solving Integrals of the form $$\int x^m(a+bx^n)^Pdx$$ The author broke it down into 4 cases:$$$$ $...
0
votes
0answers
19 views

A question based on finding range of a function.

$\mathbf{Question:}$ If $[5\sin(x))] + [\cos(x)] + 6=0$ then what is the range of $f(x)=\sin(x) + \sqrt{3}\cdot \cos(x)$, corresponding to the solution set of the given equation? (Where [.] denotes ...
0
votes
2answers
60 views

Am I doing this epsilon-delta algebra correctly?

I've done a handful of really basic epsilon/delta algebraic evaluations, but I'm not sure if my answer to this one is correct. Is it okay for delta to be a function rather than a constant? Are my ...
1
vote
1answer
58 views

evaluate if integral converge & determine antiderivative

The problem is i need to study the convergence of A and B and find the antiderivative of C $$A=\int_0^\infty \frac{\sin(x) +x}{\sqrt x + x^3}dx$$ $$B=\int_0^\infty \frac{1}{\sqrt {e^x-1}(x^2+x^{1/...
1
vote
1answer
30 views

Prove/Disprove - Integral converges so the lim is 0

I recieved the following question and solution: 1) I didn't understand the professor's reasoning, why is the integral clearly equals to the sigma. 2) Why is it obvious that the lim dosen't exist? ...
3
votes
2answers
51 views

Proof that the function is uniformly continuous

Proof that the function $f: [0, \infty) \ni x \mapsto \frac{x^{2}}{x + 1} \in \mathbb{R}$ is uniformly continuous. On the internet I found out that a function is uniformly continuous when $\forall ...
0
votes
2answers
90 views

How to prove $(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$

Q: How to prove $$(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint.
2
votes
2answers
123 views

Prove or disprove: if $\int_1^\infty f(x)\,dx$ converges, then $\lim\limits_{x\to\infty}f(x) =0$. [duplicate]

I received the following question: If $\int_1^\infty f(x)\,dx$ converges, then $\lim\limits_{x\to\infty}f(x) =0$. Image. I know that it is false, but i can't come up with a counter ...
1
vote
1answer
34 views

Show there exists $C\in\Bbb{R}^n$ such that $|C-A_i|=|B-A_i|+u_i$, with $A_i,B\in \Bbb{R}^n$ and $u_i$ close enough to $0$

Let $A_1,...,A_n,B$ be vectors in the $n$-dimensional Euclidean Space, such that they are never on the same affine $(n-1)$-dimensional subspace. (What? Is that a way to say they span $\Bbb{R}^n$?). ...
0
votes
2answers
44 views

How to find minimum and maximum of function?

Given: the function $$f\left(x,y\right)=4x^{2} +3y^{2} -5x$$ Find the $x$ values of the minimum and of the maximum on the set: $$ \left\{ \left(x,y\right)\in \mathbb R ^{2}: x^{2}+y^{2}=9 \right\}...
1
vote
2answers
63 views

True or False: Do Two Anti-Derivatives Necessarily Differ by a Constant?

Is it true or false that if $$F'(x) = G'(x)$$ then $$F(x) = G(x) + C.$$ No more details are given (continuity, etc...) I think this is false but I can't think of an example.
1
vote
1answer
52 views

A differentiation with first principles question for two variables

I know this question is probably quite easy but it's been some time since I've done any sort of calculus and since a google search failed to turn up anything relevant to this specific question I ...
4
votes
1answer
44 views

Finding extreme value

It is given: $f\left(x,y\right)=-7x^{2}-5xy+4y^{2} $ and I should find x coordinate of the extreme value with condition $x-4y=4$. I think I know how to do this, but my solution is not the correct....
0
votes
1answer
30 views

Prove using taylor series $\lim\limits_{n\to \infty}\max\limits_{0<x\leq 1}\frac{d^n }{dx^n}\exp\left(\frac{-1}{x^2}\right) = \infty$ [closed]

Prove this using taylor series : $$\lim_{n\to \infty}\max_{0<x\leq 1}\frac{d^n }{dx^n}\exp\left(\frac{-1}{x^2}\right) = \infty$$ That is we need to provethat the derivative of $\exp\left(\...
2
votes
2answers
112 views

I want to know if the way I derived the surface area of a sphere by integration is correct?

I am using the alias of Sillysack Buttowski and this is my first question. I searched on other links on stack exchange regarding "how to find the surface area of a sphere by integration". They seemed ...
1
vote
2answers
22 views

Supremum $\sup_{x\in (0,+\infty)}|F(x,t)|=\sup_{x\in (0,+\infty)}\dfrac{e^{-t^2}}{x+|t|}$

Let : \begin{aligned}F \colon (0,+\infty)\times \mathbb{R} &\longrightarrow \mathbb{R} \\(x,t) &\longmapsto F(x,t)=\dfrac{e^{-t^2}}{x+|t|}.\end{aligned} How they do to find $\...
1
vote
2answers
69 views

Determine $f(x)$ which satisfies given condition

Suppose $f(x)$ is real valued function of degree $6$ satisfying the following conditions: $1.$ $f(x)$ has minimum at $x=0$ and $x=2$ $2.$ $f(x)$ has maximum at x=1 $3.$ $lim(x \to 0)$ $\frac{ln(\...
0
votes
0answers
37 views

How to approximate the largest eigenvalue of a monodromy matrix [closed]

Would you happen to know of a method to calculate the largest eigenvalue of a monodromy matrix? For my case the fundamental matrix cannot be calculated explicitly but it exists!
-3
votes
2answers
72 views

proving $\frac{1}{n+3}+\frac{1}{n+4}+…+\frac{1}{2n+4}>\frac{1}{2}$

how can one prove that: $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ For all natural $n$, without using induction? thank you.
0
votes
2answers
79 views

$\left| x \right| \le 3\left[ {\sqrt x } \right]$ [closed]

Let $\left| x \right| \le 3\lfloor {\sqrt x } \rfloor$. What is the answer to this inequality?
1
vote
1answer
38 views

Role of the absolute value in $\int \frac{dx}{\sqrt{1-x^2}}$

In the derivation of the value of the indefinite integral \begin{equation} \int \frac{dx}{\sqrt{1-x^2}}, \end{equation} I can substitute $x = \sin(u)$, $dx = \cos(u)du$ to get this: \begin{equation} \...
0
votes
0answers
26 views

Change the integral variable

$\int _{ x=-\infty }^{ x=+\infty }{ exp\left[ +i\xi \left( { \mu }^{ '' }\left[ { x }_{ 0 } \right] \frac { { \left( x-{ x }_{ 0 } \right) }^{ 2 } }{ 2 } \right) \right] dx=\left( \sqrt { \frac {...
0
votes
1answer
30 views

relation between a curve's function and it's tangent (from graph)

The figure attached represents the curve of the function $y = f(x) $, if the equation of the tangent to the curve at any point $(x,y)$ on it is $y=g(x)$ so the following statement is correct ... ...
1
vote
1answer
40 views

Can $A^2\preceq \gamma^2 B^2$ lead to that $A\preceq \gamma B$?

In the question, $A$ and $B$ are positive semi-definite matrices, $\gamma\geq 0$ is a constant, and $A\preceq \gamma B$ means that $\gamma B-A$ is positive semi-definite. We have known another fact ...
1
vote
1answer
180 views

Is $n^x-x^n$ increasing?

For integers $n \geq3$, I want to prove that $f(x)=n^x-x^n$ is increasing on the interval $x\geq n$. Also I want to prove that for integers $x\geq 3$, $f(x)=x^{x+1}-(x+1)^x$ is increasing. I ...