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

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2
votes
0answers
61 views

Integration in Cylindrical Please help.

Being that some people just taught me something interesting i would like to apply it. Use cylindrical coordinates to find the volume of the region bounded above by the plane $z = 2x$ and below by the ...
0
votes
1answer
365 views

Multivariable Integration by parts (basic)

Use integration by parts twice on $ \int e^{xy} \cos(y) dy$ i wanna integrate $\cos(y)$ first and derivate $e^{xy}$ since x is a constant of the integration. we have $e^{xy} \sin(y)$ + $ \int ...
1
vote
2answers
61 views

Improper Integral

Question is as follows Evaluate the integral. $$\iint_S \cfrac{dxdy}{(1+x^{2}+y^{2})^2}$$ $S= (x,y)\in \mathbb{R}^{2}$ such that $ x=0$ or $x>0$ Little lost here $y=-2, x=-1$ and we divide by ...
1
vote
0answers
50 views

evaluation of $\int \ln \left(1+2m\cos x+2m^2\right)dx$

How can i calculate $\displaystyle \int \ln \left(1+2m\cos x+2m^2\right)dx$ My try:: Let $\displaystyle I(m) = \int \ln \left(1+2m\cos x+2m^2\right)dx$ dIfferentiate both side w.r.to $x$ ...
4
votes
3answers
145 views

evaluation of limit $\lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$

Calculate value of limit $\displaystyle \lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$ Can we solve without using Reinman sum Method. If yes then how can i calculate it Thanks ...
2
votes
3answers
178 views

Does this series converge or diverge?

I have a series here, and I'm supposed to determine whether it converges or diverges. I've tried the different tests, but I can't quite get the answer. ...
1
vote
1answer
39 views

Sequence of continued powers defined recursively

I have a series ${a_n}$ defined recursively by $a_1 = b$ and $a_{n+1} = b^{a_n}$, with $b \ge 1$ and $n \in \mathbb{Z} $. I am trying to show that ${a_n}$ is bounded above if $1 \le b < 3^{1/3}$, ...
1
vote
1answer
341 views

Minimizing mean squared error

I want to find a $d$ that minimizes the value of the expression below. I think the first step is to find the derivative w.r.t. $d$ (is that correct? If not, what is the first step?). If so, I'm having ...
4
votes
1answer
1k views

Finding the volume of a tetrahedron by given vertices.

Please help me with the problem below. Find the volume of a tetrahedron with vertices: $O(0,0,0)$, $A(1,2,3)$, $B(-2,1,5)$, $C(3,7,1)$ by using triple integral. Hint: First find the the equations of ...
7
votes
2answers
484 views

If series $\sum a_n$ is convergent with positive terms does $\sum \sin a_n$ also converge?

If $\{a_n\}$ is a sequence of positive terms such that the series $$\sum_{n=1}^\infty a_n$$ coverges, does the series $$\sum_{n=1}^\infty \sin a_n$$ also converge? I believe that limit comparison ...
0
votes
1answer
167 views

Finding local extrema of $f(x,y,z)$

$$f(x,y,z) = x\cdot y\cdot \left ( x-y-2 \right )$$ How can I find all the local maxima, local minima, and saddle points of the function?I tried but I only found one point.
0
votes
2answers
42 views

Show that the integrals are equivalent

Show that: $$\int_o^{\infty}\frac{\cos(x)}{1+x}dx=\int_o^{\infty}\frac{\sin(x)}{(1+x)^2}dx$$ I have no idea how to approach. The only thing I can think is substitution $y=\pi/2-x$ or integration by ...
4
votes
1answer
150 views

A problem with limit

How to attack this one? Does the following limit exists: $$\lim_{x\to +\infty}\dfrac {\cos^5x\sin^5x} {x^8\sin^2x-2x^7\sin x\cos^2x+x^6\cos^4x+x^2\cos^8x}$$
1
vote
2answers
362 views

Riemann Integral of $f(x)=1$ if $x=\frac{1}{n}$ where $ n\in N$ or $0$ otherwise

Simple question. I have the function: $$f(x)=\begin{cases}1&,\;\;\;x=\frac{1}{n}\;,\;\; n\in\mathbb{N}\\{}\\0 &,\;\;\; \text{otherwise}\end{cases}$$ I am being asked whether it is Riemann ...
2
votes
2answers
95 views

Evaluate $\int \dfrac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx$

Evaluate: $$ \space \space \int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx $$ Please give proper directions/hints to evaluate this.
1
vote
1answer
189 views

Probability hit “bullseye”

I have to finish en example, but don't know how to do it. We hahve a shooting target. Let's say that distribution in this target is bivariate normal distribution (x,y). . We have this formula for ...
1
vote
1answer
85 views

Polar coordinate

Let $f(x,y)$ be a differntiable function in $\mathbb{R}^2$ so that $f_x(x,y)y=f_y(x,y)x$ for all $(x,y)\in\mathbb{R}^2$. Find $g(r)$ so that $g(\sqrt{x^2+y^2})=f(x,y)$ and $g$ is differentiable in ...
0
votes
2answers
77 views

Finding the derivative of $a^{f(x)}$

I have a question on finding the derivative of $a^{f(x)}$ where $a$ is a constant and $x$ is the variable that we want to differentiate w.r.t. UPDATE I got it! $$ \begin{align*}a^{f(x)} &= ...
1
vote
0answers
29 views

Prove limit without using operator norm

prove $$\lim_{h \to 0}\frac{f(\mathbf{x}+h)-f(\mathbf{x})-\langle\nabla f(\mathbf{x})h,h\rangle-1/2\langle\nabla^{2}f(\mathbf{x})h,h\rangle}{\|h\|^{2}}=0$$ where function $f:\Bbb{R}^{n} \to \Bbb{R}$ ...
0
votes
3answers
106 views

Uniform convergence of sequence of functions

Given that $$\gamma_n\rightarrow\gamma$$ uniformly, can we conclude that $$\int^b_a\|\gamma_n'\|\rightarrow\int^b_a\|\gamma'\|$$ uniformly? I know that we even do not have ...
0
votes
3answers
140 views

$\int_{-\infty}^{\infty}e^{-at^2}\cos btdt=?$ [duplicate]

We know that the Gaussian integral is $$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.$$ Now, we want to compute the following integral $$\int_{-\infty}^{\infty}e^{-at^2}\cos btdt=?$$ where $a>0$, ...
0
votes
1answer
297 views

Change of Summation and Differentiation

I am looking explicitly for a proof that that infinite summation of sequence of functions, uniformly convergent and differentiation are interchangable. ...
4
votes
4answers
209 views

Solve a Seemingly Simple Limit

$$\lim_{n\to\infty}\left(\frac{n-2}n\right)^\left(n^2\right)$$ Why does this go to 0? Why can I not just divide each item in the fraction by n and assume it would go to 1?
2
votes
6answers
124 views

Calculate: $\lim_{x \rightarrow e}\left(\frac{x}{e} \right)^\frac{1}{x-e}$

How to calcylate following limit? $$\lim_{x \rightarrow e}\left(\frac{x}{e} \right)^\frac{1}{x-e}$$ Can we solve it without using L'Hospital?
2
votes
2answers
378 views

Integration using euler's formula

I need to find the following integral: $$\int\frac{1}{(1+x)\sqrt{1+x-x^2}}dx.$$ I tried using Euler's formula and put $xt+1=\sqrt{1+x-x^2}$ and after to do integration in parts but that goes nowhere. ...
0
votes
2answers
36 views

Finding Convergence of a Series

According to WolframAlpha this series converges, but I can't find out how to properly use the limit comparison test with it. Can anyone at least tell me what my $b_n$ might be? ...
5
votes
5answers
115 views

$\lim_{x \rightarrow 0}\left(\frac{(1+57x)^{67}-(1+67x)^{57}}{x^{2}} \right)$

Calculate: $$\lim_{x \rightarrow 0}\left(\frac{(1+57x)^{67}-(1+67x)^{57}}{x^{2}} \right)$$ Without using L'Hospital rule
1
vote
3answers
35 views

How to differentiate these type of problems?

This is an example of the problem type I am asking about Let $f\left(x\right) = \int_1^{2x-x^2} e^{1-t}\, \mathrm{d}t$ Find $f^{\prime}\left(x\right)$ $$\int_1^{2x-x^2} e^{1-t}\, \mathrm{d}t$$ ...
8
votes
3answers
623 views

Calculate: $\lim\limits_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$

How do I calculate the following limit without using l'Hôpital's rule? $$\lim_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$$
0
votes
1answer
36 views

Find $n$ for $\lim_{x \rightarrow 0}\left(\frac{(\cos x-1)(\cos x-e^{x})}{x^{n}}\right)$

If $$\lim_{x \rightarrow 0}\left(\frac{(\cos{x}-1)(\cos{x}-e^{x})}{x^{n}} \right)$$ is a non-zero finite number for an integer $n$, then find the value of $n$.
2
votes
1answer
60 views

Check my work for this calculus word problem.

The region bounded by the $x$-axis and the graph of $y=\sin x$ is divided by the vertical line $x = k$. If the area of the region $0 \leq x \leq k$ is three times the area of the region $k\leq ...
13
votes
3answers
391 views

Calculate:$y'$ for $y = x^{x^{x^{x^{x^{.^{.^{.^{\infty}}}}}}}}$ and $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+…\infty}}}}$

(1) If $y = x^{x^{x^{x^{x^{.^{.^{.^{\infty}}}}}}}}$ (2) If $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+....\infty}}}}$ then find $y'$ in both cases (3)If $ y= ...
1
vote
3answers
152 views

How should I calculate the Lebesgue integral of logarithm function from zero to infinity?

Does the area under the $\ln(x)$ in $(0,+\infty)$ is measurable? If yes, how can I calculate it?
2
votes
1answer
86 views

Critical Points

I'm slightly confused as to how to solve for critical points, for example: $$f(x) = x^{3} + 3x^{2}-24x$$ You take the derivative of the function, $$f'(x) = 3x^{2}+6x-24$$ And then solve for x, ...
4
votes
4answers
7k views

when is the particle speeding up and when is it slowing down

Based on this graph i have to figuere out when the particle is speeding up and when it is slowing down. My understanding is that when velocity and accelaration have the same sign then we are ...
4
votes
1answer
993 views

Point on Sphere Closest/Farthest to Another Point

"What points on the sphere centered at the origin with a radius of 3 are closest to and farthest from the point P = (6,6,-3)?" The approach I took was to make a vector v going from the origin to P ...
2
votes
2answers
81 views

Volume by rotation?

Find the volume of the solid that results when the regions bounded by $y=x^3, x=2$, and the x-axis is revolved around the line $x=2$? I don't really understand what it meant by "x-axis is revolved ...
0
votes
1answer
111 views

How to solve for the coefficients of this polynomial?

For what values of the constants $a$, $b$, $c$, and $d$ does the function $f(x) = ax^3 + bx^2 + cx + d$ satisfy both of the following conditions? a) $f''(0) = 0$ at the origin b) a ...
5
votes
3answers
154 views

Evaluate : $\int_0^1 \frac{dx}{\sqrt[3]{1-x^3}}$.

Evaluate : $$ \int_0^1 \frac{\mathrm{d} x}{\sqrt[3]{1-x^3}}.$$ if you feel that this integral is easy, just post hints.
0
votes
3answers
34 views

simplifying $e^\frac{-t}{4} (\frac{-1}{4})+(\frac{-1}{4}t+1) e^\frac{-t}{4}(\frac{-1}{4})$

Im stuck yet once again simplifying a derivative i get so close to finishing a problem then i spend an hour trying to do something that should otherwise be simple. simplifying $$e^\frac{-t}{4} ...
2
votes
3answers
65 views

$e^{\large\frac{-t}{5}}(1-\frac{t}{5})=0$ solve for t

I'm given the following equation an i have to solve for $"t"$ This is actually the derivative of a function, set equal to zero: $$f'(t) = e^\frac{-t}{5}(1-\frac{t}{5})=0$$ I will admit im just stuck ...
0
votes
1answer
51 views

Finding differentials of functions $p,q,v$

I am given the following 2 equations, where $p$ and $q$ are implicit functions of $v$. $$p^2 + vpq+q^2-1=0\\p^2+q^2-v^2+3=0$$ I need to find the values of $\large{\frac{dp}{dv}}$ and ...
0
votes
1answer
140 views

If $f(x)$ is a continuous function on $(0,1]$, show that g(x) is continuous on [0,1].

If $f(x)$ is a continuous function on $(0,1]$, show that g(x) is continuous on [0,1] where g is equal to f on (0,1] and g(0)= $\lim_{x\to 0^+} f(x)$. Assume limit exists. It is obvious that g(x) is ...
1
vote
3answers
2k views

Isotope Decay / half-life

Suppose you start with 100g of an isotope that has a half life of 17 years. How long before 20g of the isotope are left? What's the general formula for these problems?
1
vote
3answers
111 views

Calculate:$\lim_{x \rightarrow\infty}\left(\frac{(2+x)^{40}(4+x)^{5}}{(2-x)^{45}} \right)$

How to calculate following limit without using L'Hospital rule? $$\lim_{x \rightarrow\infty}\left(\frac{(2+x)^{40}(4+x)^{5}}{(2-x)^{45}} \right)$$
0
votes
3answers
55 views

Why does this integral give a different answer? a water tank depletes at 10000g per day, the tank starts at 150,000 gallons

How long before the tank runs dry? So, this is basic division. take the $\frac{150,000}{10000}$ to get the days it takes for the water tank to deplete right? To set this up using integral form, ...
2
votes
2answers
145 views

Related Rates Word Problem Help?

If two resistors with resistances $R_1$ and $R_2$ are connected in parallel then the total resistance $R$, measured in ohms, is given by $\frac{1}{R}=\frac{1}{R1}+\frac{1}{R2}$. If $R_1$ and $R_2$ are ...
2
votes
1answer
558 views

2D Partial Integration

I have a (probably very simple problem): I try to find the variational form of a PDE, at one time we have to partially integrate: $\int_{\Omega_j} v \frac{\partial}{\partial x}E d(x,y)$ where v is ...
0
votes
2answers
591 views

Boyle's Law Related Rates question.

Boyle's law states that when a sample of gas is compressed at a constant temperature P and volume V are related by the equation PV=C, where C is a constant. Suppose that at a certain instant the ...
2
votes
1answer
47 views

How to expand this matrix formula?

$L$ is a matrix, is it possible to expand this formula in terms of $\operatorname{Tr}L^n$? $$\log\operatorname{Tr}e^L$$