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

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0
votes
1answer
70 views

Number of real roots of polynomial derivative

Let $W(x)$ be a polynomial with n real roots and $P(x) = \alpha W(x) + W'(x)$. Prove that for any $\alpha \in \mathbb{R}$: $P(x)$ have at least $n-1$ real roots. I know that the degree of the ...
0
votes
2answers
47 views

Problem with understanding the application of the Intermediate Value Theorem in the proof of the Mean Value Theorem for Integrals

I am struggling to understand the last parts of this proof because I know that the IVT states that on the interval $[a,b]$ of $f$, where it is continuous, there exists a value $L$ between $f(a)$ and ...
0
votes
1answer
38 views

$f(x)=2-|x-3|, 1\le x\le 5$ and for other values, $f(x)$ is obtained using the relation $f(5x)=kf(x)$ for $x\in R$. then…

Question: The maximum value of f(x) in $[5^4,5^5]$ for $k=2$ is? Also, if $$\lim_{x\to \infty}\int_1^xf(x)dx$$ is a finite number, find the exhaustive set of $k$. Attempt : For first part, ...
-1
votes
1answer
36 views

Collinearity and complex numbers

How to show that $z_1, z_2, z_3 \in \Bbb C$ are collinear if and only if the quotient ${z_2-z_1}\over{z_3 - z_1}$ is a real number?
1
vote
3answers
66 views

Prove inequality $x^sy^{1-s} \leq sx + (1-s)y$ [duplicate]

Given $s \in (0,1)$, prove $$x^sy^{1-s} \leq sx + (1-s)y$$ for $x,y > 0$ Tried some algebraic manipulations but I'm guessing I need to use some trick. Any suggestions, hints?
4
votes
2answers
104 views

“Mean value like” problem.

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be differentiable, take $a<a'<b<b'$. Prove that there exists $c<c'$ such that $$\frac{f(b)-f(a)}{b-a}=f'(c) \quad and \quad ...
1
vote
1answer
59 views

Matrix integration by parts

It seems to me that the integration by parts rule carries over simply to the matrix case. This can be seen by applying: $(AB)' = A'B + AB'$ and then integrating for square (time dependent) complex ...
1
vote
2answers
95 views

A problem on the chapter: Work, Power and Energy

An engine of $150$ Kilowatt is drawing a train of total mass $1.5\times 10^5$ Kg up an incline of $1$ in $50$. The frictional resistance is $4$ Kg-wt per ton. Show that the maximum speed of the ...
0
votes
3answers
57 views

Find x for which $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges.

For what values of $x$ the infinite series $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges? Please help me out. Using ratio test $\large{\lim_{n\to ...
0
votes
2answers
47 views

Finding conditional extrema with trig functions

Find the conditional extrema of $$f(x,y)=\cos^2x+\cos^2y,\quad g(x,y)=x-y+\frac{\pi}{4}=0.$$ I have a problem with finding a solution to this problem. Using Lagrange multipliers i come up with a ...
0
votes
0answers
46 views

Cesaro sum of a series [duplicate]

$\sum_{n=0}^{\infty}a_n$ diverges in the regular term but is Cesaro summable Prove $a_n/n\to 0$ when $n\to \infty$ We used the definition of the Cesaro sum and obtained: $\lim_{N\to ...
0
votes
2answers
76 views

How many methods are available for finding this volume?

I wonder how many methods are available for finding the volume required by the question. Two spheres (of radii $r$ and $a$, with $r \lt 2a$) meet in such a way that the centre of the one of radius ...
10
votes
4answers
2k views

10th derivative of a function

I want to find $f^{(10)}(0)$ where $f(x)=\ln(2+x^2)$. I know that it can be done "by hand", but I believe there is a smarter way. I think I should use Taylor series and the fact that ...
11
votes
1answer
95 views

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

Let denote $K$ and $E$ the complete elliptic integral of the first and second kind. The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and ...
0
votes
4answers
135 views

Which number is bigger? [duplicate]

Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$? Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those ...
1
vote
0answers
39 views

$e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$

I have to show that $e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$ holds for all positive integers k and m whenever $m>4\cdot k\cdot log(k)$. I replaced $(1-\frac{1}{k})^m$ with ...
1
vote
7answers
86 views

Find min & max of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$

Problem: Find the maximum and minimal value of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$. Since $x^2 > x$ (edit $x^2 \geq x$) for all $x \in \mathbb{R}$, $f$ is bowl-ish with a minimal ...
1
vote
2answers
53 views

Calculating limit-sequences

So i have this limit to calculate: $\lim_{n\to\infty}\frac{[x] + [2x]+ ... +[nx]}{n^2}$ And i tried to make some boundaries and got this two limits: $\lim_{n\to\infty}\frac{[x] + [x]+ ... ...
1
vote
3answers
36 views

How to verify my solution to an separable differential equation?

I have this question: Find the general solution to the separable differential equation $$ \frac{dy}{dx} = y(1-y). $$ My attempt is : $$ \frac{dy}{y(1-y)} = dx $$ $$ \frac{1}{y(1-y)} = ...
0
votes
3answers
102 views

Calculate the sum of this series

$$ \sum_{n=1}^\infty \frac{1}{n^2 3^n} $$ I tried to use the regular way to calculate the sum of a power series $(x=1/3)$ to solve it but in the end I get to an integral I can't calculate. Thanks
-4
votes
1answer
81 views

An example of discontinuous function on $\mathbb{R}$ [duplicate]

Is there any example of a function which is discontinuous on $\mathbb{R}$
6
votes
2answers
559 views

A problem in integration.

As you know from basic trigonometry that $\sin(2x) = 2\sin(x)\cos(x)$. If you integrate both sides with respect to x, one finds $$\int \sin(2x) \ dx = -\frac{1}{2}\cos(2x)+c$$ on the left hand side ...
0
votes
1answer
41 views

Prove $\unicode{x222F}_S{f(x,y,z)d\vec{S}}=\iiint_E{}\nabla f(x,y,z)dV$.

Let the closed surface $S$ be delimit volume $E$ and let the scalar function $f$. I need to demonstrate that : $$\unicode{x222F}_S{f(x,y,z)d\vec{S}}=\iiint_E{}\nabla f(x,y,z)dV$$ I do have some ...
1
vote
0answers
52 views

Proof of additivity of domain for definite integrals

I would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ...
0
votes
3answers
51 views

The derivative function is not continuous

(Sorry about the bad title, couldn't think of a way to word it concisely.) Let $C[0, 1]$ be the metric space whose points are all continuous functions from $[0, 1] \rightarrow \mathbb{R}$ with the ...
-3
votes
2answers
64 views

Analytic integration of this function [closed]

Integrate \begin{equation} \int{\frac{1}{(1-\frac{a}{r}-b r^2)}} \, \mathrm{d}r \end{equation} where $a$ and $b$ are constants.
-2
votes
1answer
44 views

Convergence of $\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$ [closed]

Test the convergence of the following series $$\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$$
4
votes
3answers
472 views

Angle between two parabolas

I'm a little confused about a problem that asks me to find the angle between the two parabolas $$y^2=2px-p^2$$ and $$y^2=p^2-2px$$ at their intersection. I used implicit differentiation to find the ...
5
votes
2answers
240 views

Rearranging Pokemon Experience Formula to make Level the Subject

As the title suggests, I am trying to rearrange some of the formulas for calculating experience based on level to be the other way around (calculating level based on experience). I am having trouble ...
1
vote
2answers
53 views

Differentiate vector function wrt vector

I have a function $\frac{df(\mathbf{y})}{d\mathbf{y}}=\mathbf{y}g(\kappa)$ where $\kappa=||\mathbf{y}||_2$ and $g(\cdot)$ is a scalar function. Thing is when I differentiate this function I get a ...
2
votes
1answer
203 views

Equilibrium Points Second Order Differential

Attempt: I get the system of the two first order equations (first order in $w$) by considering the different signs the first derivative takes. Problem is by equilibrium points: do I just set the ...
6
votes
2answers
231 views

Is it possible to develop Analysis solely from Peano's axioms

...and a few definitions on the way? When I studied Calculus using Spivak's book It was clearly shown that, in order to prove some fundamental theorems (intermediate value theorem being one of them), ...
3
votes
4answers
104 views

Expressing the integral in terms of the original variable

In evaluating the integral: $$ \int{dx\over(a^2-x^2)^{3/2}} $$ or $$ \int{dx\over(a^2-x^2)^{1/2}\ (a^2-x^2)}$$ Let $ x=a\sin\theta $ and $ dx=a\cos\theta\ d\theta $. Then $$ \int{{a\cos\theta\ ...
1
vote
3answers
101 views

The simplification of divided difference of cosine function

What is the following limit? $$\lim_{h \to 0}\frac{\cos(\pi/2+h)-\cos(\pi/2)}{h}$$ Why when simplified do you get $(-\sin(h))/h)$?
-1
votes
1answer
65 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [closed]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
0
votes
2answers
32 views

Continuous Compound

You own an antique that is currently worth 1500, and whose value increases linearly at a rate of 175 per year. If the prevailing interest rate remains constant at 5%, per year compounded continuously, ...
0
votes
1answer
37 views

LN word problem

measurement of a child's ability to learn is given by the function $$L(t)=\frac{ln(t+1)}{t+1}$$ where t is the child's age in years, for $0 ≤ t ≤ 5$ At what age does a child have the greatest ...
8
votes
5answers
2k views

Math Subject GRE 1268 Question 55

If $a$ and $b$ are positive numbers, what is the value of $\displaystyle \int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$. A: $0$ B: $1$ C: $a-b$ D: $(a-b)\log 2$ E: ...
0
votes
1answer
50 views

Question on continuity and differentiability of min() and max() functions.

Question: $f(x)=x^2-2|x|$. Test the continuity of $g(x)$ in the interval $[-2,3]$ if $g(x)$ is defined as: attempt: $f(x)$ is defined as: But i am finding it difficult to understand $g(x)$. ...
1
vote
3answers
68 views

What does it mean for a function to “preserve the limits of sequences”?

I've translated the English Wikipedia page "Limit of a sequence". What does the following statement mean? In fact, any real-valued function f is continuous if and only if it preserves the limits ...
2
votes
1answer
46 views

Calculus stay to Real Analysis as $x$ stay to Functional Analysis

Hi guys i had a look to book which treat the subject of Calculus (of course...) Analysis and Functional Analysis. Is that correct to state that Calculus is more focused on "computing" while ...
2
votes
0answers
46 views

Problem with a step involving a type of Riemann integration

I am reading this text, , and I find it unclear how the ratio of the considered rectangle's area to its length tends to become the derivative of a function $S$ as the lenght of the considered ...
1
vote
2answers
29 views

Factoring out quotient of an expression.

I am following the MIT Open Courseware on Single Variable Calculus and in the first lesson when taking the derivative of a simple function I found myself confused because of my knowledge gap in ...
1
vote
1answer
86 views

Convergence of $\sum \frac{2n+1}{(n^2+n)^n}$

I have to choose the right option: The series $$\sum_{n\geq 1} \frac{2n+1}{(n^2+n)^n}$$ a. Converges to 1. b. Converges to a number >1. Using ...
2
votes
0answers
64 views

An integral that I cannot simplify.

Good day, esteemed students of mathematics! I have been trying to prove that the convolution of $2q$ Gaussian probability distributions is another $q$ Gaussian probability distribution with the same ...
2
votes
1answer
68 views

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ ...
1
vote
2answers
60 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ ...
2
votes
2answers
654 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
3
votes
3answers
95 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
2
votes
1answer
101 views

Integrating functions with $x^3$

After learning the integration of various functions with $x^2$ involved, I was given the following integration, as a challenge: $$\sqrt{1+x^3}$$ I tried various methods - too long to even try and ...