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

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-6
votes
1answer
34 views

$\frac{\partial^2 x}{\partial y^2}=\frac{1}{2}\frac{\partial}{\partial y}(\frac{\partial x}{\partial y})^2$ [on hold]

This is just a curiosity question, but how do you prove: $\frac{\partial^2 x}{\partial y^2}=\frac{1}{2}\frac{\partial}{\partial y}(\frac{\partial x}{\partial y})^2$? $x=x(y,t)$ and the above is ...
0
votes
1answer
46 views

Find constant $c < 1$, such that Fibonacci number $F(n) \le 2^{cn}$ for every $n \ge 0$

I have an outline for solution, but I am afraid that it's not mathematically rigorous at all. Would you be so kind to point problems with this solution if any? 1) I looked up in Wikipedia that $F(n)$ ...
1
vote
1answer
54 views

Why is $f(x,y)$ said to be discontinuous at $(0,0)$?

Why is $f(x,y)=\begin{cases} \frac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq (0,0)$}\\[2ex] 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ said to be discontinuous at $(0,0)$? I am supposed to show ...
1
vote
1answer
45 views

Proof of a theorem about limits

The following is the introduction part of the proof of the theorem which says limit of the sum is equal to sum of limits. Here I could not understand why it is sufficient to show that theorem holds ...
1
vote
1answer
28 views

Prove that $\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$

Prove that $$\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$$ I started out with the following identity: $$ \frac{1}{s}\mathcal{L}(f)=\frac{1}{s}\int_{0}^\infty e^{-st}f(t)dt ...
-4
votes
0answers
25 views

I'm trying to calculate DTFT, but I'm stuck [on hold]

Help me solve this, Its DTFT and I don't know how to continue. $$\sum_{n=-\infty}^\infty 2^n(e^{jn(\pi/4-w)}-e^{-jn(\pi/4+w)})$$ Thanks
3
votes
5answers
613 views

Find the value of this series

what is the value of this series $$\sum_{n=1}^\infty \frac{n^2}{2^n} = \frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots$$ I really tried, but I couldn't, help guys?
2
votes
2answers
53 views

find the following integral

I really dont know why i got confused by this but i need assistance, it is school break and i am reading ahead. How would i get this? I got the asswer to be -7 is this right? I have tried simplifing ...
2
votes
3answers
98 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
2
votes
1answer
49 views

Did I do this implicit differentation right? [on hold]

I have just solved an implicit differentiation question and feel that I have made a mistake after checking some online calculators. The questions states to use implicit differentiation to find dy/dx ...
-1
votes
1answer
48 views

Proving that the integrals of two functions are the same if they are equal everywhere except a point [on hold]

Let $f(x)$ and $g(x)$ be integrable functions over $[a,b]$ and let $∂$ be a point on $[a,b]$. If $f(x) = g(x)$ for all $x≠∂$, then $$\int_a^b f(x)dx=\int_a^b g(x)dx$$
1
vote
1answer
23 views

Differentiating composite function

Can anyone say the basic formula for the differentiation of the composite functions? Is it similar to chain rule?
3
votes
7answers
132 views

Evaluating the indefinite integral $\int e^{-x^2}\,\mathrm{d}x$ [on hold]

In my book, it is said that $$\int e^{-x^2} \, \mathrm{d}x$$ cannot be solved by the method of inspection. It then turned to method of substitution as a new topic. I am not able to solve this ...
1
vote
3answers
64 views

About the sum of $\sum_{n=1}^{\infty} \frac 1 {n(n+1)}$ [on hold]

Find the sum of $\displaystyle \sum_{n=1}^{\infty} \frac 1 {n(n+1)}$ So I can see that it's a telescopic sum: $\displaystyle \sum_{n=1}^{\infty} \frac 1 {n}-\frac 1 {n+1}$, but since the sum ...
1
vote
4answers
74 views

Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$

Let we have the following sequence $(a_n)$ such that $$a_n=\sum_{k=1}^n\frac{1}{n+k}$$ How can I prove that $(a_n)$ is increasing bounded sequence, then prove it is convergent and find its limit?
0
votes
1answer
62 views

Calculus 2 - $\int(\sqrt{72+36x^2}dx$

I have done this problem several times and this is the only answer i ever come to. My schools webwork gives me incorrect for my answer (answer is not simplified but it should be accepted in this ...
-1
votes
0answers
46 views

Find indefinite integral $dx/(x^6+1)$. [closed]

Help to find indefinite integral $$ \int \frac{\mathrm{d}x}{x^6+1} $$
0
votes
1answer
41 views

Derivative of a trigonometric function

What is the derivative of $$\cos^2 a (\tan a - \tan b)$$ Please anyone explain in detail. The differentiation is with respect to $a$. I tried to obtain the answer using chain rule, but didn't get it. ...
1
vote
1answer
30 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
-1
votes
0answers
31 views

Solve the following related rate problem. [on hold]

A gravity water tank in the shape of a cone is being drained at a rate of 8 gallons per minute. The tank has a depth of 8 feet and a diameter at the mouth of 6 feet. How fast is the water level ...
2
votes
1answer
47 views

Limits in two dimensions

First some context: I was trying to find the limit of $\frac{e^x - 1}{x}$ as x approaches zero without using L'Hopital's rule to avoid circular reasoning. Then, I was told that I could use the ...
6
votes
2answers
94 views

$ \lim_{n\to+\infty} \frac{1\times 3\times \ldots \times (2n+1)}{2\times 4\times \ldots\times 2n}\times\frac{1}{\sqrt{n}}$

Knowing that : $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$$ ...
0
votes
2answers
58 views

Find the area of the entire region that lies between $r=1+\sin\theta; r=1+\cos\theta$

I have to find the area of the region that lies between the curves $r=1+\sin\theta; r=1+\cos\theta$ . The answer the book gave was $\frac {3\pi}{2}-2\sqrt{2}$ . I tried generating the curve for ...
-4
votes
1answer
40 views

Chain rule differentiation [on hold]

Can any one show me the steps to differentiate $v^2$ according to chain rule? Why is derivative of $v^2$ found out by chain rule and not by exponent formula?
0
votes
1answer
29 views

Uniform convergence of f(x/n) to f(o)

I am trying the following problem: the solution feels wrong to me since I don't think it deals with a situation in which f can drive to infinity around some point in [a,b]. Thanks for your help
0
votes
1answer
25 views

proving convergence of this sequence and calculating limit

So i have this sequence: $a_{n}=1+ \frac{1}{3}\cos1 +\frac{1}{3^2}\cos2+ ... +\frac{1}{3^n}\cos(n) $ I have to prove it is convergent, and then calculate the limit. I'm not totally sure how to find ...
2
votes
1answer
64 views

How to solve this seemingly easy problem?

In short, I need to prove that: $\sin 2nx\not\to-\sin 2x\quad x\ne\frac{k\pi}2,k\in\Bbb Z,n\in\Bbb N\quad \text{as}\quad n\to\infty$ The biggest trouble is that I know little about $x$, not even ...
0
votes
6answers
328 views

calculating 2 sums of series

So I have these two series given. 1: $\displaystyle\sum_{n=1}^{\infty}\frac{\sin{(2n)}}{n(n+1)} $ And I have to show that this sum is $\leq$ 1. 2: ...
1
vote
1answer
40 views

How to calculate this limit of a function defined in pieces?

Calculate $$\lim_{x\rightarrow~ 0} f(x)$$ where $$f(x)=\left\{\begin{matrix} x~\textrm{if}~x\in \mathbb{Q}\\x^2 ~\textrm{if}~x\notin \mathbb{Q} \end{matrix}\right.$$ To me it seems like the ...
4
votes
1answer
68 views

Area under tangent to a curve.

The tangent to the graph of the function $y=f(x)$ at the point with abscissa $x=a$ forms with the line $x$-axis an angle $\frac{\pi}{6}$ and at the point with abscissa $x=b$ an angle of ...
1
vote
2answers
68 views

Integral of $x^2\sqrt{5+x}\ dx$

I have the following integral to solve, with my working out below. This is a bit more complicated than I am used to, so I'm hoping for some feedback as I'm not sure if my process & solution are ...
-3
votes
2answers
36 views

Prove this equality about improper integral [closed]

Prove that $$\lim_{ x \rightarrow\infty} \int_x^{x^2} e^{t^2} dt= \infty$$
0
votes
3answers
36 views

Limit of integral of difference between f(x) & f(x+h)

If $f(x)$ is defined on $[0,1]$, Given that $f(x)$ is integrable on $[0,1]$, is the following limit true for all $c$ with $0 ≤ c < 1$? $$ \lim_{h\to0^+}\int_0^c\bigl(f(x+h)-f(x)\bigr)\,dx=0 $$ ...
1
vote
2answers
68 views

find $\lim_n\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+…+\frac{a^\frac nn}{n+\frac 1n}$ using Riemann integral

Here is the question: prove that $S_n=\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+...+\frac{a^\frac nn}{n+\frac 1n}$ is convergent for $a>0$ then find its limit. My attempt: If we accept ...
2
votes
3answers
59 views

derivative integral $\int_0^{x^2} \sin(t^2)dt$

I want to know how I derivative this integral: $$\int_0^{x^2} \sin(t^2)dt$$ what are the steps to derivative it?
4
votes
1answer
67 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
0
votes
2answers
51 views

Differentiating $f(x)=\sum_{i=1}^{N}|x-y_i|^2$ where $y_1,…,y_N\in \Bbb{R}^n$.

Let $y_1,...,y_N\in \Bbb{R}^n$ and let $f(x)=\sum_{i=1}^{N}|x-y_i|^2$. I need to show that $f$ has a minimum. I try to differentiate but I am having troubles doing so. First of all, does $|x-y_i|$ ...
1
vote
1answer
58 views

Alternative Proof of the Extreme Value Theorem

I have proven the Boundedness Theorem for continuous functions and would now like to prove the Extreme Value Theorem; that is, show that the upper bound is indeed attained for continuous functions. I ...
0
votes
3answers
67 views

n-th derivative test.

Let $f(x)$ be a function such that it is $n$ times differentiable and $f^{'}(a)=f^{''}=(a)f^{'''}=(a)....=f^{n-1}(a)=0$ and $f^{n}\ne0.$ The $n^{th}$ derivative test tells us about the concavity of ...
1
vote
2answers
33 views

What is the cosine of angle of intersection of following functions?

1st Function: $\displaystyle 3^{x-1}\log x$ 2nd Function: $\displaystyle x^x-1$ How to find the cosine of angle of intersection of these two curves? Their $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ are not ...
-4
votes
2answers
39 views

Find the derivative of the function at the indicated point [closed]

ln x y+5x=30 at (1,e^25) Not sure where to go with this problem
1
vote
1answer
28 views

Continuity of composite functions

The continuity theorem for composite functions states that if $f(x)$ is continuous at $x = a$ and $g(x)$ is continuous at $x = a$ , then the composite function $f\circ g$ and $g\circ f$ are also ...
13
votes
1answer
207 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion [duplicate]

(Note: I apreciate very much who marked this as a duplicate but I would like an answer for why my proof is wrong) This is my solution, I have no clue why it failed. Let's start: define $$I_n(m) = ...
3
votes
1answer
42 views

Probability of True Positive of a random variable defined by an integral expression

$\newcommand{\Prob}{\operatorname{Prob}}$Let's assume that we have a random variable with the following pdf: \begin{equation} f_T(x) = \int_0^\infty f_T(x,g) \cdot f_{g}(g) \, dg = \int_0^\infty ...
2
votes
3answers
68 views

$I_{2n}=\dfrac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$

let $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ show that $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq ...
3
votes
4answers
125 views

Does this series converge, and if so to what value?: $\sum_{n=0}^\infty \left\{\frac{1}{(n+1)^2} \right\}\ln(2n+1)$

I've arrived at this series from a given sequence of terms, but now I'm at a loss as to how to proceed... How does one know which convergence test to use? This isn't a geometric series, so I don't ...
1
vote
1answer
17 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
0
votes
4answers
64 views

Can you prove the simplification of this sum?

I'm still learning Calculus (in parallel) and I'm stuck on this sum simplification. It is the 2nd part of the Tail to expectation formula from statistics #1. From here : $$ \sum_{k=a+1}^{b} \frac{ ...
-1
votes
2answers
33 views

Continuous compound word problem using ordinary differential equation

I have a problem with one of my homework questions. (b) A certain bank compounds interest continuously at an annualized interest rate $0<r<1$ (measured in inverse-years), meaning that ...
12
votes
1answer
204 views

Calculating 2 integrals in polylogarithmic functions

Are we aware of any nice way of calculating these $2$ integrals? $$i) \space \int_0^1 \frac{\text{Li}_2\left(x-x^2\right)}{x^2-x+1} \, dx$$ $$ii)\space \int_0^1 ...