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

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0
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0answers
8 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}$. I've understood that it can be seen as a set of $n$ functions; $f = (f_1,f_2,\ldots ,f_n)$. What are $f_i$ exactly? I'd be glad if you could ...
0
votes
3answers
50 views

What is wrong in my $f'(x)$?

We have $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=\frac{x^2-x+1}{x^2+x+1}$ and we need to find $f'(x)$. Here is all my steps: ...
3
votes
2answers
32 views

An integral using a trig. substitution

I came across the following integral, and I don't know how to solve it. $$ \int\log\left(x+\sqrt{x^2-1}\right)dx $$ I tried the "obvious" substitution of $x=\tan\theta$, which gives you: $$ ...
2
votes
4answers
48 views

Differentiate the Function $f(x)= \sqrt{x} \ln x$

Differentiate the Function $f(x)= \sqrt{x} \ln x$
1
vote
0answers
19 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
1
vote
2answers
55 views

Integral of $x/(2x-1)$

I'm not sure how to do this, I'm also new to math.stackexchange so please excuse any novice mistakes. So anyways, here is a question I have on a summer assignment for Calculus BC (this is review from ...
4
votes
1answer
39 views

Prove a sequence converges using sub-sequences

Let there be a sequence $a_n$ The following sub-sequences converge: $a_{n^3},a^3_{2n+3}-a^3_{2n+4},a^2_{2n+3}-a^2_{2n+4},a_{2n+15}$ Prove: $a_n$ converges I think it has something to ...
0
votes
1answer
34 views

Determine the roots of equation if possible

How to determine the roots of equation using numerical methods? I have this particular equation: $$\arctan(e^x)=\ln \left(\sqrt{\frac{e^{2x}}{e^{2x}+1}}\right)$$ In my solution I have that this ...
0
votes
4answers
38 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
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1answer
29 views

How to calculate this derivative $D^{\alpha}f(x)$?

Let $v\in\mathbb{R}^n$ be a fixed vector, and $f$ a function given by $f(x)=\cos(x\bullet v)$, where $x\bullet y$ is the dot product. What is the derivative $D^{\alpha}f(x)$ for $x\in\mathbb{R}^n$ ...
0
votes
2answers
50 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...
2
votes
1answer
28 views

The value of an integral of a piecewise defined function

I have been given $$f(x)=\begin{cases} 4 & 2\le x<5\\ 3 & x=5\end{cases}$$ and I want to find the value of $$\int_2^5 f(x) dx.$$ I proceeded as follows: $$\int_2^5 f(x) dx = 4(5-2) = ...
0
votes
2answers
29 views

Calculus: simpler way of showing that derivative is negative?

I want to show that $\frac{1-(1-\beta)^N}{\beta}$ is strictly decreasing in $\beta$ for $\beta \in (0,1)$ and $N \geq 2$. My approach so far is as follows: I take the derivative with respect to ...
0
votes
2answers
18 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ I get an answer of $\frac ...
0
votes
2answers
54 views

Integral of cos(1/x) dx

Is the following integral expression correct (neglecting the constant of integration)? $$ \int\cos\left(\frac{1}{x}\right)dx = x^2\sin\left(2x\right) $$ When I take the derivative, it returns to the ...
2
votes
2answers
41 views

Why is the integral starts from $0$?

Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$ It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem: $$\lim_{x\to 1^-} f(x) = ...
12
votes
1answer
144 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
0
votes
1answer
44 views

Prove the limit is $e^\alpha$

prove that $\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=e^\alpha$ $$\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=\lim_{n \to \infty} \left(\left(1+{\alpha\over n}\right)^{n\cdot ...
1
vote
2answers
39 views

Find maximum of a function

I want to find the maximum of a function. $$ d = \frac{35}{3} + \frac{7}{3}\sin( \frac{2\pi}{365}t ) $$ I don't know if I applied the chain rule correctly. $$ w = \frac{2\pi}{365}t $$ $$ w' = ...
2
votes
3answers
71 views

Integrate $\displaystyle \int \sin(\sqrt{at})dt$

Integrate $\displaystyle \int \sin(\sqrt{at})dt$ Here is what I tried. Let $u=\sqrt{at}$, then $\displaystyle\ du=\frac{a}{2\sqrt{at}}dt=\frac{a}{2u}dt\implies \frac{2udu}{a}=dt.$ So by subsitution, ...
3
votes
2answers
53 views

$\sum_{n=1}^{\infty} \frac{1}{n+1!} \prod_{k=1}^{n} f(k)$ Prove the divergence of a series [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
1
vote
3answers
49 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
1
vote
3answers
54 views

$x^2+y^2<1, x+y<3$ is open or closed?

I'm trying to figure out if $$\{x^2+y^2<1, x+y<3|(x,y)\in \mathbb R^2\}$$ is open or closed. I tried to imagine this set. It looks, for me, as a 'pizza', or a circular sector, which have two ...
1
vote
3answers
24 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
0
votes
3answers
72 views

Evaluate the following integration below [on hold]

Evaluate the following integration $$\int_{0}^{\infty }\frac{x^2}{x^6 +1}dx$$ help guys please, I tried but I got nothing.
1
vote
1answer
33 views

Intermediate value theorem

Suppose $f$ is a continuous function on $[a,b]$ and $\lambda$ is a value between $f(a)$ and $f(b)$. Prove that $\exists c \in [a,b]$ s.t. $f( c) = \lambda$ Let, $$g(x) = f(x) - \lambda$$ $g$ is ...
1
vote
1answer
34 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,R))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,R)^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le \frac{4}{R}$ for almost all ...
-6
votes
1answer
31 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
50 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
40 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
24 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
592 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
50 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 ...
1
vote
3answers
93 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 ...
1
vote
1answer
45 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
45 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
20 views

Differentiating composite function

Can anyone say the basic formula for the differentiation of the composite functions? Is it similar to chain rule?
2
votes
7answers
122 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
62 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
56 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
59 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
40 views

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

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
25 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
18 views

Solve the following related rate problem.

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
46 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
92 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
57 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 ...