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

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17
votes
10answers
1k views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
1
vote
4answers
72 views

$\lim_{x \to 0}(x^2(1+2+3+\cdots+[\frac {1} {|x|}]))$ where [a] is largest integer not greater than a and |x| is absolute value of x

As x tends to 0, the first term $x^2$ tends to 0 while the second term tends to infinity. So is the limit undefined
0
votes
1answer
46 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
0
votes
2answers
97 views

Convergence of $\sum_{n=1}^\infty\frac {n^{n}}{e^nn!}$

Check the convergence of: $\displaystyle\sum_{n=1}^\infty\frac {n^{n}}{e^nn!}$ Using the root test I get: $\displaystyle\lim_{n \to\infty} \dfrac {n}{e\sqrt[n]{n!}}$ now I'm left with showing ...
1
vote
3answers
46 views

Why is it required to change variable to get the right answer for this question?

The question is this : $$\lim_{x\to-\infty} {\sqrt{x^2+x}+\cos x\over x+\sin x}$$ The solution is $-1$ and this seems to be only obtained from the change variable strategy, such as $t=-x$. However, ...
2
votes
1answer
41 views

Advanced Calculus – (Real Analysis) function f

Def. The statement that $f$ is continuous means that $f$ is continuous at each point in its domain. Def. if $D$ is a subset of $\mathbb{R}$ and $f$ is real valued function with domain $D$ then the ...
19
votes
11answers
753 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
1
vote
4answers
84 views

A limit tend to infinite! Need a little help…

Can someone help me solve this limit? $$\lim_{t \to \infty} \frac{t-t\sqrt{t}}{2t^{3/2}+3t-5} $$
1
vote
3answers
94 views

A Horrible looking limit

I have the following limit question: $$\lim_{x \rightarrow 1 }\frac {({\rm log} (1+x)-{\rm log}\space 2)(3\times4^{x-1}-3x)}{[(7+x)^{1/3}-(1+3x)^{1/2}]{\rm sin}\space \pi x}$$ This has the form ...
1
vote
2answers
49 views

$\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ If $\sum_{n=1}^\infty b_n$ converges then $\sum_{n=1}^\infty a_n$ converges as well [duplicate]

We have two positive series: $\displaystyle\sum_{n=1}^\infty a_n$, $\displaystyle\sum_{n=1}^\infty b_n$ and we know that: $\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ (from a certain index). ...
2
votes
0answers
90 views

Convergence of $\sum_{n=3}^{\infty}\frac{1}{n\log n(\log\log n)^\alpha} $

Does the following series converge: $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n\log n(\log\log n)^\alpha} $ and $\alpha>0$ ? Using Cauchy condensation test twice: $\begin{align} ...
8
votes
1answer
99 views

A question of rationality

This problem was asked to me by a friend and I simply have no idea about it. So I have not progressed a single bit. The problem is this: If $f :\mathbb{R}\to \mathbb{R}$ is an infinitely ...
2
votes
1answer
46 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
1
vote
2answers
108 views

Convergence of $\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}}$

Does the following series converges ? $$\displaystyle\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}} \ \text{for} \ \ k,m\in \mathbb N$$ I tried the ratio test: $ ...
9
votes
5answers
588 views

Evaluating $\displaystyle\lim_{x\to 0}\left(\frac{1}{\sin x} - \frac{1}{\tan x}\right)$

How to solve this limit $$ \displaystyle\lim_{x\to 0}\left(\frac{1}{\sin x} - \frac{1}{\tan x}\right) $$ without using L'hospital's rule?
0
votes
1answer
78 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
2
votes
1answer
73 views

$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
1
vote
1answer
338 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
3
votes
1answer
169 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
4
votes
1answer
118 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
2
votes
6answers
211 views

General solution for squared trigonometry questions: $\cos^2 x = 1$

$\cos^2 x = 1$ How do you solve trig equations with a power? Unsure what to do with the square? I get this $\frac{1+\cos2x}2 =1$ $\cos2x =1$ $2x=2n\pi\pm0$ $x=n\pi$ but the answer says $\pm ...
0
votes
1answer
38 views

How does a sequence's convergency change finite sums?

What has been troubling me lately is that I cannot grasp how a finite series could ever diverge if a finite sequence that is divergent can only imply to a finite sum every time. Perhaps my main ...
0
votes
0answers
32 views

smooth extensions with unique critical point

Let $f:B^{2}\rightarrow\mathbb{R}$ be a continuous function on the unit disk $B^{2}$ which is smooth in $B^{2}\backslash\{0\}$ and has no critical points there. May we find a smooth function ...
3
votes
1answer
49 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
1
vote
0answers
57 views

The Flat Function

I have to write an essay on the flat function $$\text{flat}(x) = \begin{cases} e^{-\frac{1}{x^2}} & \text{for } x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$$ and I want to prove ...
0
votes
1answer
26 views

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is?

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is? I know the condition for minima but here there are two simultaneous variables , how and with respect to what do I ...
2
votes
0answers
94 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
0
votes
1answer
29 views

Evaluate the definite integral and substitute

solving for definite integral. I substitute $4x+6$ with $U$ and solved for $du=4dx$ $$\int_0^1 x^2 \sqrt{4x+6}\ dx$$
2
votes
1answer
543 views

Proving uniform convergence

Prove uniform convergence of function series: $$ \sum_{n=0}^\infty \frac{1}{n^2 + x} \sin \frac{1}{n^2 + x}$$ on $ \Bbb R $ I'm stuck with a problem, because I've proven that is uniformly ...
3
votes
0answers
47 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
0
votes
3answers
65 views

Finding the limit of $\lim_{ x\rightarrow 3^-} ((\sqrt{3x+7}-4)/(\sqrt{3-x}))$

How do I evaluate $\lim\limits_{ x\rightarrow 3^-}\dfrac{\sqrt{3x+7}-4}{\sqrt{3-x}}$? Can someone explain the steps by steps solution to this problem?
3
votes
2answers
122 views

Limit of $f(x)=\frac{x}{1+\sin^2x}$ as $x\to0$ and proof

I computed the limit using the limit theorems and the answer is obviously $0$. So now I am attempting to prove it using the $\epsilon,\delta$ definition. $f(x)=\frac{x}{1+\sin^2x}$ ...
3
votes
2answers
76 views

Limit of a Cosine Sequence

This seems trivial, and yet after a bit of thinking, I couldn't supply a simple proof. Is the following true? The series $$\underset{n=1}{\overset{\infty}\sum}\cos(nx)$$ is divergent for almost ...
0
votes
3answers
36 views

Functional equation regarding differentiability

How do you solve this problem ? I'm more interested in the method than the result . Find all the differentiable functions that satisfy the following condition : f(x+y)=f(x)+f(y).
0
votes
0answers
72 views

The fundamental theorem of calculus: $\frac{\mathrm d}{\mathrm{d}x} \int_a^x f(t)\:\mathrm{d}t = f(x)$ and nothing more

My textbook writes, just as wikipedia, but I'd like to write it shorter. Can the fundamental theorem of calculus be written as $$\frac{\mathrm d}{\mathrm{d}x} \int_a^x f(t)\: \mathrm{d}t = f(x)$$
1
vote
1answer
22 views

Find the function $y(x)$ with the property that $y(x)$ has a horizontal tangent line

This is a calculus problem I've been struggling on: "Find the function $y(x)$ with the property that $y(x)$ has a horizontal tangent line at the point $(1,-2)$ and $\frac{d^2y}{dx^2}=2x+5$. So, I ...
1
vote
1answer
36 views

How to parameterized implicit curve.

How could I parameterize: $$\frac{1}{2}\left(x^2+y^2\right)-\frac{1}{3}x^3=\frac{1}{6}$$ as $x(t)$ and $y(t)$?
2
votes
2answers
36 views

Basic doubt on Taylor's polynomial

I have a doubt about a general situation in where I am asked to calculate $f(x)$ with a certain precision. How can I compute the number of terms of the Taylor polynomial needed for that? For example ...
1
vote
3answers
264 views

Finding the limit of $\lim_{x\to 1} (x^2-\sqrt x)/(1-\sqrt x)$

How do I evaluate $$\lim_{x\to 1} \frac{(x^2-\sqrt x)}{(1-\sqrt x)}$$ Can someone explain the steps by steps solution to this problem?
10
votes
2answers
211 views

Convergence of a series (quite frustrating)

The sum is $$\sum_{k=1}^{\infty}\frac{1}{k}\left( \frac{\sin{k} + 2}{3} \right) ^k$$ Here's what I've tried so far: Root test (in its stronger limsup form), gives nothing, so I didn't bother with ...
0
votes
2answers
2k views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
1
vote
0answers
14 views

Rearranging 2 discriminant function to solve for 1 parameter (to derive a decision boundary)

I have a task where I want to classify patterns from 2 classes where the samples are drawn from a bivariate Gaussian distribution. I use the 2 discriminant functions ($g_1$ and $g_2$) to classify the ...
2
votes
1answer
35 views

Determine if the following series converges or diverges using the alternating series test.

$$\sum_{n=1}^\infty\frac{(-1)^n}{n^{\frac{4}{3}}}$$ Using the $p$-series test I know that this converges. But how would I use the alternating series to solve this?
0
votes
1answer
53 views

Use alternating series test to determine convergence/divergence.

I'm positive this series converges, but I just wanted to double check: $$\sum_{n}\frac{(-1)^n}{4n^5 +7}$$
0
votes
1answer
85 views

Stokes' Theorem and Surface Independence Failure

As we know, if $\vec{F}=\nabla\times\vec{A}$ then from Stokes' Theorem, $\iint_{S_1} \vec{F}\dot \,d\vec{S}=\iint_{S_2}\vec{F}\dot \,d\vec{S}$ where $S_1$ and $S_2$ have the same boundary. Does ...
0
votes
1answer
22 views

Determining whether a point with unknown values is positive

I have 2 roots for a function: $\Large x=\frac{-(1+b-m)\pm\sqrt{(1+b-m)^2-4dn}}{2d}$ Is it possible to determine whether a root is positive or negative given only that $b,d,m,n > 0$
3
votes
0answers
161 views

Differentiation under integral sign

There is this integral that I used a lot in my research: $$\int_{-\infty}^{\infty}\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x = ...
0
votes
0answers
36 views

Can operations on arbitrarily differentiable functions form a group?

Consider the the set $S=${All uniary operations (all operations that perform on one function and give you another function, e.g $(x^2)'=2x$) on all arbitrarily differentiable functions}, does it form ...
4
votes
1answer
81 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...