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

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4
votes
2answers
45 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
0
votes
1answer
17 views

How to represent y as a function of w?

Assume : $F(y)=G(w)$ where $F,G$ are two real-valued functions from $R \to R$. We want to find the function $C(w)$ such that : $F'(y)=C(w)$ and C should be built based on F and G. Thanks so much.
0
votes
0answers
19 views

Recursively enumerable VS recursively sets [on hold]

How can I show that are many recursively enumerable sets than recursively sets?
0
votes
0answers
18 views

normal plane to a level curve [on hold]

$\ f(x,y,z)=(x^2 + y^2 - z^2, x + y + 2z)$ $\ C: f(x,y,z)=(1,0). $ Find the cartesian equation of the normal plane to C at $\ (1,1,-1)$ Where do I start here?
0
votes
1answer
36 views

Find $\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$

$$\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$$ I have multiply by $\frac{\sqrt{2x^2+2}+(x+1)}{\sqrt{2x^2+2}+(x+1)}$ and got: $$\lim_{x\to 1} ...
-1
votes
0answers
22 views

Function Limit & Continuity [on hold]

What is Function Limit & Continuity? I'm a little bit silly.Is there anyone to explain those terms precisely? Thanks in Advance...
-3
votes
2answers
60 views

Help with continuity [on hold]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
1
vote
3answers
63 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...
-3
votes
3answers
58 views

Find the following integrals [on hold]

I am really having a hard time trying to solve these integrals and I would be very thankful if you would help me solve them:
5
votes
1answer
36 views

$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$ - show that $a_n$ is convergent sequence

Problem: Show that $a_n$ is convergent sequence and find a limit of $a_n$. $$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$$ I tried to look at this as normal limit problem so I wrote ...
1
vote
4answers
75 views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [on hold]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
1
vote
1answer
23 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
3
votes
4answers
30 views

Finding the formula for acceleration from $v=2s^3+5s$, where $s$ is the displacement at time $t$

This is the question: I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't ...
1
vote
1answer
43 views

solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
1
vote
1answer
47 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
0
votes
0answers
27 views

Range of Derivative

Let $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$, such that $f(0)=4$ and $f(5)=-1$. What is the range of values $g'(c)$ for a $c$ belonging to $[0,5]$? Considering values of ...
3
votes
1answer
65 views

Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
1
vote
1answer
23 views

Finding a limit on multiple square roots in a row?

Here are basically my two problems, which I have the answer from WolframAlpha: $$ \lim_{n\to\infty}(1-\sqrt 2-\sqrt{n+1}+\sqrt{n+2})=1-\sqrt 2 $$ $$ \lim_{n\to\infty}(\sqrt n-2\sqrt{n+1}+\sqrt{n+2})=0 ...
-5
votes
0answers
42 views

Give a clear and lucid description of the function $f(x,y,z)=\sin(x)\sin(y)\sin(z)$. [on hold]

Give a clear and lucid description of the function $f(x,y,z)=\sin(x)\sin(y)\sin(z)$.
0
votes
3answers
41 views

$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is this done correctly?

$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is it allowed to split a multi-variable limit into its component variables as in the next step? $= ...
0
votes
1answer
38 views

Is the gap between successive real roots of $x(t) = \frac{1}{30000 e^t} + \frac{2 e^{t/2} \cos (\sqrt{3}t/2)}{30000} $ eventually less than $5$?

Consider the function $x : \mathbb{R} \to \mathbb{R}$ given by $$x(t) = \frac{1}{30000} \frac{1}{\mathrm{e}^t}+ \frac{2}{30000} \mathrm{e}^{\frac{t}{2}} \cos \left(\frac{\sqrt{3}}{2}t\right), \quad ...
-4
votes
1answer
48 views

Integration CALCULUS [on hold]

I need help on problem based on integration calculus. Q: how to integrate $$\int\frac{dx}{1+\sin(x)\tan(x)}$$ Wolfram and integrate calculator does not help me.
-1
votes
1answer
43 views

Evaluating the following definite integral calculus

Given the following definite integral $$\int_0^4 \left[\left(1/2x^2 - 2x +8\right)-\left(1/4x^2+x\right)\right]\;\mathrm dx$$ I have done in the following process. $$\int_0^4 \left[\left(1/2x^2 - 2x ...
0
votes
0answers
30 views

Find required degree of Maclaurin polynomial to estimate the cosine to two decimal places

I have a question where I am asked to find the amount of terms required in a Maclaurin polynomial to estimate $\cos(1)$ to be correct to two decimal places. So far what I have done is used Taylor's ...
1
vote
2answers
61 views

Memorizing Formulas for Differentiation

Once upon a time, I memorized the following formula out of laziness. Let $k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}$. Then $k'(x)$ is as follows. ...
0
votes
1answer
29 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
-5
votes
2answers
46 views

Why is $ \lim_{x\to 25}(25−x)/(\sqrt x−5)= -10$? [on hold]

I know the answer is $-10$ but I don't know where the negative sign is coming from. This is what I ended up with. $$\frac{(x-25)(\sqrt{x}+5)}{x-25} = (1)\sqrt x+5 = 10 $$ ...
-4
votes
0answers
37 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
0
votes
1answer
43 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
2
votes
2answers
55 views

Finding roots of an equation wich involves floor function

I'm trying to solve this equation $$ \left \lfloor{x +\frac{1}{100}}\right \rfloor + \left \lfloor{x +\frac{2}{100}}\right \rfloor + ... + \left \lfloor{x +\frac{223}{100}}\right \rfloor = 521 $$ I ...
1
vote
4answers
73 views

What is the difference between sum and integral?

I am a beginner in calculus and I want to know what is the difference between sum and integral. More specifically I came across this example: Compare $$\sum^\infty_1\frac1x\space \text{and} \space ...
0
votes
0answers
42 views

Finding the integral of a 1/variable*radical function

I'm trying to find the integral of $$\int\frac{1}{x* (\sqrt{4x^4 - 9})}$$ Attempt: I assumed that the integral would be some sort of inverse trigonometric function. Because of this, I did the ...
2
votes
1answer
37 views

Complex derivative numerically using real $h$ and imaginary $h i$?

I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like $$ \frac ...
0
votes
2answers
28 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
0
votes
2answers
32 views

How do I find the equation for a semicircle with a radius of 2 on the x axis?

I'm doing a calculus project where we have to make a model of some graph rotated about the y axis. I am doing a fish bowl and I have most of it understood and ready. The only thing I'm not sure of is ...
0
votes
1answer
79 views

Limit of a sequence with binomial coefficient. Can I use Stirling?

I was trying to solve this limit: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $ I solved it with Cesaro theorem: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $= $\lim_\limits{n\to \infty} ...
0
votes
1answer
18 views

Using Rodrigues' formula to show a result

use the formula $P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}((x^2-1)^n)$ to show that $P_{2n}(0) = \dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ and odd terms are 0. I first subbed in 2n to the formula and got ...
-3
votes
1answer
29 views

Notational problem [on hold]

Please, how do I write the following as a combination of a sum and a product: $$ (c-a_1)(c-a_2)(c-a_3)b_1 + (c-a_2)(c-a_3)b_2+(c-a_3)b_3 ?$$ Also, how can I generalize it?
-1
votes
1answer
17 views

How to find $x$ such that $f(x)$ takes a prescribed value [on hold]

Find $x$ such that \begin{equation} x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}} \end{equation}
0
votes
2answers
54 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
1
vote
0answers
23 views

limit of $\frac{(k+1)[x^k(a+sin(x^{-k-1}))-x^{-1}cos(x^{-k-1})]}{k[x^{k-1}(a+sin(x^{-k}))-x^{-1}cos(x^{-k})]}$ as $x \to 0$

Limit of equation: $\frac{(k+1)[x^k(a+sin(x^{-k-1}))-x^{-1}cos(x^{-k-1})]}{k[x^{k-1}(a+sin(x^{-k}))-x^{-1}cos(x^{-k})]}$ as $x \to 0$ and k=0,1,2,... My calculation steps: $=\frac{k+1}{k} ...
3
votes
5answers
149 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
0
votes
2answers
25 views

How do I use the ratio test to determine convergence or divergence in this problem?

I have the problem: $$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$ I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive). So I take the limit: ...
0
votes
0answers
9 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
-1
votes
2answers
46 views

$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$ iff $\lim_{n\to\infty}{(a_{n}-b_{n})=0}$ [on hold]

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
1
vote
0answers
34 views

$\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$

Let $P(x)$ be a real polynomial with degree $n$ such that $\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$. Prove that $|P(x)|\le n+1$ for all $x\in [-1,1]$. This question was posted some years ...
0
votes
0answers
32 views

Leibniz integral rule definition

https://en.wikipedia.org/wiki/Leibniz_integral_rule If we have an integral $$\int_{y_0}^{y_1} f(x, y) \,\mathrm{d}y$$ then for $x$ in $(x_0, x_1)$ the derivative of this integral is thus ...
0
votes
1answer
37 views

Simplify exponential equation

I really need your help to solve this exponential equation. It looks so simple, but I haven't been able to find a solution so far: $$ {A_1 + A_2 \over 2} = A_1 \exp\left({-x^2 \over c_1^2}\right) + ...
-1
votes
0answers
16 views

Express $\prod \frac{a_i}{x+b_i}$ in terms of known functions [on hold]

I am interested if there is a way to express the finite product $$ \prod_{i=1}^{S} \frac{a_i}{x+b_i} $$ in terms of known functions like Gamma, Beta, etc.
2
votes
0answers
25 views

Volumes by Cylindrical Shells - What am I doing wrong?

I am trying to solve this exercise from a textbook: $y = x^4, y = 0, x = 1;$ rotated about $x=2$ This is my attempt at solving the problem: Shell radius: $2 - x$ Shell height: $x^4$ $a = 1$ $b = 2$ ...