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

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0
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1answer
37 views

What is valid and what is not in limits.

I have been looking for $$ \lim_\limits{x\to 0}{\left({\sin x\over x}\right)}^{1\over x^2}. $$ So I took $$ \lim_\limits{x\to 0}\left({\left({1+{\sin x-x\over x}}\right)}^{({x\over \sin ...
2
votes
5answers
66 views

One integral, two solutions?

I ran into an interesting integral problem: the indefinite integral of $\int \frac{dx}{a^2 x^2 - b^2}$. I can do a hyperbolic trig substitution and get that the result is $- ...
1
vote
4answers
76 views

Using mathematical induction to show that for any $n\ge$ 2 then $\prod_{i=2}^n\bigl(1-\frac{1}{i^2}\bigr)=\binom{n+1}{2 \cdot n}$

I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ...
2
votes
1answer
21 views

Related Rates Problem Involving Airplanes

I took a test yesterday, and would like to know how to answer this specific question on the exam: One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane ...
2
votes
0answers
30 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
2
votes
1answer
51 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
1
vote
1answer
28 views

Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration

Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq ...
0
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0answers
9 views

find angular velocity for so that: $\exp(jt) = \exp( j(3t+\pi/3) )$

I have a fourier series in which there are two different arguments on the exponential function: $jt$ and $j(3t+\pi/3)$ and I have to "choose" a fitting angular velocity. It it probably easy yet it ...
2
votes
2answers
68 views

Solving functional equation $f(x)f(y) = f(x+y)$

I'm having some trouble solving the following equation for $f: A \rightarrow B$ where $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{C}$ such as: $$f(x)f(y) = f(x+y) \quad \forall x,y \in A$$ ...
1
vote
4answers
53 views

Find $y^{(n)}(0)$ for every $n$.

Let $y(x)$ fulfill $y''-xy=0$. Furthermore: $y(0)=0,y'(0)=1$. Find $y^{(n)}(0)$ for every $n$. I tried different forms of recurrence relations but I couldn't do much with it without it becoming a ...
0
votes
0answers
8 views

Polar and Spherical Coordinates Conversion

I learnt $dxdy = rdrd\theta$ from Calculus, but why the following doesn't work as expected? \begin{align} dxdy &= (drcos\theta)(drsin\theta)\\ &= ((dr)cos\theta-rsin\theta ...
0
votes
0answers
12 views

Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals

The problem is to find the volume of $y = 6\cdot \sqrt{\sin (x)}$ rotated around the $y$-axis when $0 \leq y \leq 6$. I know this can be done by the sv-calc method of volumes of revolution but I ...
13
votes
3answers
774 views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
1
vote
1answer
23 views

Evaluate an integral quickly

Evaluate the integral $$\int \sqrt{x} \ln(1+x)dx $$ so we should start with the substitution: $t=\sqrt{x}$ $$ \int t\ln(1+t)dt2t = 2\int t^2\ln(1+t)dt $$ From here, it seems reasonable to ...
6
votes
2answers
121 views

Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example(s) of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ ...
0
votes
0answers
16 views

Calculus - Vector Functions

Consider the following vector function. $r(t) = ({5\sqrt2}{t}, e^{5t}, e^{-5t})$: Find the unit tangent and unit normal vectors $T(t)$ and $N(t)$. Find the curvature $\kappa(t)$
0
votes
1answer
16 views

Find the derivative of the question mentioned

Question: find the derivative of $$ x^3\cdot y^7+\frac{x}{y}-x^2+y^4=3 $$ I got $$ \frac{dy}{dx}=\frac{2xy^2-y-3x^2y^9}{4y^5+7x^3y^8-x} $$
1
vote
3answers
441 views

how to compute the last 2 digits of 3^3^3^3 to n times?

input n, output the last 2 digits of the result. n=1 03 3=3 n=2 27 3^3=27 n=3 87 3^27=7625597484987 n=4 ?? 3^7625597484987=?? Sorry guys, the formula given ...
2
votes
2answers
254 views

Find $\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$

It's asked to solve this: $$\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$$ And I have no idea how to do it...
3
votes
2answers
112 views

Does this infimum tend to infinity?

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\lim _{x\to +\infty}f(x,y)=+\infty\quad\text{for each fixed }y\in\mathbb{R}.$$ Further, let $\mathcal{I}\subset\mathbb{R}$ be a ...
0
votes
0answers
9 views

Iteratively solve this equation

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
1
vote
3answers
22 views

Multivariable: Show the limit is $0$.

I already proved that for $\alpha =1$ the limit doesn't exists. Now I need to show that for $\alpha > 1$ the limit does exists and equals $0$. $$\lim_{(x,y)\to (0,0)} \frac{\left|x\right|^\alpha ...
0
votes
0answers
19 views

Courant. Real numbers determined by nested sequences of rational intervals.

In his book Introduction to Calculus and Analysis vol.1, page 95 Courant writes: Every nested sequence of intervals with real end points contains a real number. To prove this, consider closed ...
1
vote
1answer
39 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I used WolframAlpha to know that $\frac{\pi}{2} - x < ...
1
vote
2answers
56 views

Find all planes which are tangent to a surface

I'm given the surface $z=1-x^2-y^2$ and must find all planes tangent to the surface and contain the line passing through the points $(1, 0, 2)$ and $(0, 2, 2).$ I know how to calculate tangent planes ...
0
votes
0answers
14 views

When should we use $f'(c)=\lim_{h\to0}\frac{[f(c+h)-f(c)}{h}$ and $\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$?

When should we use $f'(c)=\lim\limits_{h\to0}\dfrac{[f(c+h)-f(c)}{h}$ and $\lim\limits_{x\to c}\dfrac{f(x)-f(c)}{x-c}$? I don't really understand when should I use them. If the question by ...
1
vote
2answers
20 views

What are the derivative, differentiable and differentiation? [on hold]

If the question is $f(x)=\frac{2x+1}{1-x}$? The derivative is $f'(x)=\frac{(1-x)(2)-(2x+1)(-1)}{(1-x)^2}$?
0
votes
3answers
100 views

Solving $\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$

Evaluate $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$$ This is my attempt: $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x} = ...
1
vote
3answers
19 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
-1
votes
2answers
38 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
1
vote
3answers
67 views

Help with $\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1})$ [duplicate]

$\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1}) = ?$ I don't know how to solve the indetermination there... is it possible to rearrange the expression in brackets in order to use ...
0
votes
1answer
14 views

Rate of change question involving velocity, displacement and acceleration

I have been having trouble understanding questions c)-e) and am in need of some help: An object is moving in a straight line from a fixed point. The displacement $s$ in metres is given by ...
1
vote
2answers
31 views

Prove that $f_n(x) = \left(\frac{x}{n}\right)^ne^{-x}$ converges uniformly

Prove that $f_n(x) = \left(\frac{x}{n}\right)^ne^{-x}$ converges uniformly at $[1,\infty)$. So for every $x$, there's $N\in\mathbb{N}$, such that for all $n>N$: $\frac{x}{n} < 1$. ...
0
votes
0answers
18 views

A tank of the shape of a right circular cylinder $5$ feet across the top and $9$ feet deep is full of water. [on hold]

A tank of the shape of a right circular cylinder $5$ feet across the top and $9$ feet deep is full of water. How much work is done by pumping the water out of the tank, over the top edge? I need your ...
1
vote
0answers
34 views

Volume of figure between $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ if $z\geq 0$

I have a problem where I have to find volume of figure formed, when $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ intersects if $z\geq 0$. Here is a graphic for clarity: So far I have transformed the problem to ...
2
votes
0answers
14 views

How do I integrate this master equation from a time-continuous Markov chain?

I hope the question is not too vague. My calculus courses are way in the past and I can't remember how to do it :-). I have this master equation for a time-continuous Markov chain I have a two ...
6
votes
3answers
170 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1(x)_n dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1)\: dx$$ to five significant digits. I've used ...
1
vote
0answers
27 views

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r$ is a smooth real-valued function?

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r: \mathbb R \rightarrow \mathbb R$ is a smooth real-valued function ? In Calculus I've seen linear (higher-order) ...
0
votes
1answer
25 views

Generalized angle sum identity for $\arctan$?

The angle sum identity for arctan is: $$\arctan (\alpha)+\arctan(\beta)=\arctan\left(\frac{\alpha+\beta}{1-\alpha\beta}\right)$$ I was wondering if there exists a relationship for any linear ...
0
votes
1answer
13 views

Diffeomorphism between open sets of half-space

Let $\mathbb{H}^{m}=\left\{(x_{1},...,x_{m})|x_{m}\geq0\right\}$. How can i prove that if $A$ and $B$ are respectively open set of $\mathbb{H}^{m}$ and of $\mathbb{H}^{n}$, with $n\ne m$, then they ...
1
vote
3answers
25 views

Identify the Differential Equations from the given problem [on hold]

Dear Math expert, Please solve the above problem. Thanks in advance for your support!
0
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0answers
15 views

Partial integration and substition rule.

Well in my day to day usage I now came upon an example of using the substition rule where I can't see how it works, and I wonder how one could handle such an equation with ease. The set of equations ...
2
votes
4answers
60 views

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$.

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$. I have been trying for hours using the continuity of $e$ and using L'Hopital rule but it gets really scattered and ugly. I am in despaire. ...
5
votes
5answers
57 views

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converge.

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converges. We cannot apply Abel's/Dirichliet's tests here (For example, Dirichliet's test demands that for $g(x)=\ln x$, $\int_0^1 g(x)dx < \infty$ ...
0
votes
3answers
36 views

Integral $ \int \frac{1}{x^{1+a} (1-x)^{1-a}} dx~,~a \gt 0$

The following integral is part of a large problem I'm trying to solve and I'm stuck. I'd appreciate some guidance. I would like to know how to compute integrals of the form $$ \int ...
2
votes
1answer
44 views

Can this special case happen when working with L'Hopitals rule?

I am using this version of L'Hopital's rule Assume that $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)=0$, and that the limit-value $\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$ exists (could ...
0
votes
1answer
46 views

continous function s.t $ \int_{0}^{\infty} f(x) dx$ exist [on hold]

Let $f:R \to R$ be a continuous function s.t. $ \int_0^\infty f(x) dx$ exists. Which of following is correct. If $\lim_{x \to \infty} f(x)$ exists then $\lim f(x) = 0$ The $\lim f(x)$ must exist and ...
0
votes
2answers
59 views

How to evaluate the integral $\int x^2/\sqrt{4-x^2}\,dx$?

How to compute this integral? $$\int \frac{x^2}{\sqrt{4-x^2}}dx$$ If there were $x$ instead of $x^2$ in the numerator I know how to do a substitution $y=4-x^2$. But this doesn't help with the $x^2$.
2
votes
2answers
26 views

Can every differentiable scalar function be written as a divergence of some vector field?

My question is simple: can every differentiable function $f$ defined on a bounded, connected subset of $\mathbb{R}^3$ be written as a divergence of some vector field ? That is, given the vector field ...
3
votes
2answers
285 views

Calculating derivative using definiton for $f(x)=\frac{x - \sin x}{x^2}$

Really stuck on this one.... $\displaystyle f(x) = \frac{x - \sin{x}}{x^{2}}$ for $x \neq 0$ and $0$ when $x = 0$ Using the definition of the derivative, find $f'(0)$ I know the definition is ...