For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.
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1answer
8 views
A question on functional equations.
Question:
If it is given that
$$ e^xf(x) = 2 + \int_0^x\sqrt{1+x^4}\,dx $$
then what is the value of $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $
Where I am stuck:
Now, since we are to ...
0
votes
1answer
18 views
how to prove an identity related to $\int_0^\infty\sin(x^{1-a})dx$?
i have made some experiments in maple evaluating the integral $$\int_0^\infty\sin(x^{1+a})dx$$
and the computer give me the following result
...
1
vote
0answers
10 views
Find a function $u(x,y)$ such that a line integral $I=u(B) - u(A)$ where B and A are limits of the integral
As the title, where function $u(x,y)$ can satisfy $I=u(B)-u(A)$
the line integral $I$ is already shown to be path independent and is defined as
...
6
votes
3answers
120 views
What does $\frac{dy}{du}$ mean?
I understand that $dy/dx$ is the rate of change, and that it means "the rate of change of $y$ with respect to $x$", but when I see people use $dy/du$ I get confused ($u$ of course being any variable). ...
2
votes
1answer
29 views
Calculate the ROI when the rate of return has its own rate of change.
This is a real-world problem, not a home-work assignment or anything of the sort.
Where I live, the current cost of electricity (Tariff) is $T_0$. Historically, I've noticed that this rate changes at ...
4
votes
5answers
468 views
How can I solve $\cos^2 x + \sin x +1 = 0$?
The solution set of the equation $$\cos^2 x + \sin x +1 = 0$$ is? I haven't studied trigonometry, I'm kinda lost on this issue ...
4
votes
1answer
84 views
The limit of a sum
How to compute this limit:
$$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$
Please give me some hint. Thank you.
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3answers
73 views
How can I prove that a set of natural numbers always have a minimum?
Let's say I have a finite not-empty set named A, which is a set of natural numbers.
How do I prove it has a minimum?
(In Calculus)
0
votes
1answer
49 views
solve the following differentiation equations.
Please solve the following differential equation:
A)$$\frac{d y} {dx} + \frac{1}{t}y=3 \cos(2t)$$
here is my method: writing equation in form of the $y' + p(t)y=g(t)$ and I have difficulty with ...
1
vote
1answer
20 views
Maximum volume of rectangular prism with known surface area
I have a sheet of cardboard 64cm x 51cm. I need to make a cereal box that is a rectangular prism out of the cardboard. However I must use as much of the cardboard as possible to minimise waste. The ...
3
votes
1answer
72 views
What is the sum of the series?
How to find the sum of the series $$\sum_{n=0}^\infty \frac {x^{3^n}+(x^{3^n})^2} {1-x^{3^{n+1}}}$$ under the assumptions $x >0,\,x\neq 1,$ in a closed form?
1
vote
3answers
48 views
Prove that $\sum_{n=1}^{\infty} \frac{1}{2^n - 1}$ is convergent and $\sum_{n=1}^{\infty} \frac{1}{2^n - 1} < 2$.
Prove that $\sum_{n=1}^{\infty} \frac{1}{2^n - 1}$ is convergent and $\sum_{n=1}^{\infty} \frac{1}{2^n - 1} < 2$.
I'm not sure, but I suppose that $$\sum_{n=1}^{\infty} \frac{1}{2^n - 1} < ...
1
vote
2answers
19 views
Finding arc length and binormal vector for a given curve
Can somebody show me the arc length of a curve formula, and the binormal vector formula. The curve C with equation $r(t)=(\sqrt{3}\cos t,t,\sqrt{3}\sin t)$
How do you find the arc length from $t=0$ ...
0
votes
2answers
35 views
limit of square root of function and example of group
can somebody help in getting the value of
$\displaystyle\lim_{x\to 9}\frac{\sqrt{f(x)}- 3}{\sqrt{x}-3}$, if $f(9) = 9$ and $f(9) = 4$.
Though the problem seems to be very simple, when I tried the ...
0
votes
2answers
77 views
Finding the maximum value of $t + \frac{1}{t}$
Finding the maximum of $t + \frac{1}{t}$ by making the first derivative equal to zero and taking $t=-1$ (because second derivative < 0) gives -2 as the answer. But this is not correct. I guess I am ...
