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

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1answer
48 views

series convergence test with parameter

As part of a bigger proof I reached the series $\sum {1 \over n^\alpha}$. $\alpha \in \mathbb{R}$ Obviously, the convergence depends on the value of $\alpha$. I already know the harmonic series ...
3
votes
1answer
56 views

Constructing a new function

I've got two function $f(x)=\sin(x)$ and $g(x)=\sin\left(\frac{x}{4}\right)$. I'd like to construct a function $h$, which would equal $0$ when $\sin(x)=0$ except when ...
1
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1answer
22 views

minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$ \min_{r\geq0} r(e^r-w) $$ Is there any way to find it. Thanks.
5
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3answers
173 views

Evaluate $\displaystyle\int_1^e\left(x^x+\ln x^{\large x^x}\right)\ dx$

I just took calculus test & I cannot find $$\int_1^e\left(x^x+\ln x^{x^x}\right)\ dx.$$ I don't know how to answer this integral. Can you help me? Thanks in advance!
3
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2answers
56 views

How Do I Integrate? $\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}$

How do I integrate this one? $$\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}\,dx$$ Is my answer correct: $$-3\ln\left \| x+2 \right \|+\ln\left \| x \right \|+\frac{4}{x}+C$$
0
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1answer
22 views

Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
1
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1answer
79 views

Confused with Leibniz notation of a derivative

If $f$ is a function and $x$ is function of $t$, how do you find the derivative of $f(x)$ in terms of the derivative of $f(t)$? With Leibniz' notation this is shown as (using the chain rule) ...
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1answer
55 views

Sigmoid derivative using quotient rule

Sigmoid function defined by $f(x)=\frac{1}{1+e^{-x}}$ can be derived easily with derivative of a composed function like here: Derivative of sigmoid function. However I was asking myself how this could ...
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3answers
78 views

Partial fraction decomposition of a rational function

The form of the partial fraction decomposition of a rational function is given below. $$\frac{x−3x^2−26}{(x+1)(x^2+9)} = \frac{A}{x+1}+ \frac{Bx+C}{x^2+9}$$ What are the values of $A,B$ and $C$? ...
2
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0answers
73 views

Can someone clear up my question on an area under the curve.

Find the area under the curve $y=2x^{−3}$ from $x=9$ to $x=t$ and evaluate it for $t=10$, $t=100$. Then find the total area under this curve for $x \geq 9$. I found the first two answers, but I ...
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2answers
85 views

Painful? Moment Generating Function

Part 1 Let $X$ be a random variable with the p.d.f. $f(x)=\frac{1}{4\pi}e^{\frac{-x^2}{4}}$, compute the MGF of $X$. So I know I want ...
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1answer
67 views

What method should I use when integrating this problem?

What method should I use when integrating this problem? $$\int \frac{dx}{x^{\frac12}+x^{\frac13}}$$
1
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2answers
69 views

Use Logarithmic Differentiation to find $\frac{d}{dx} (x^{{x}^{x}})$ at $x=1$

How do do this? Use Logarithmic Differentiation to find $\frac{d}{dx} (x^{{x}^{x}})$ at $x=1$.
3
votes
4answers
135 views

Calculation of $\int_{0}^{1}\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}dx$

Calculation of $\displaystyle \int_{0}^{1}\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}dx$ $\bf{My\; Try::}$ Let $x=\tan \psi\;,$ Then $\displaystyle dx = \sec^2 \psi$ So Integral convert into ...
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2answers
115 views

Using limits to find the range

Say a function is continuous everywhere. Could one determine the range of this continuous function by taking the limit as $x$ approaches $0$, $\infty$ and $-\infty$? My guess is yes. I would, however, ...
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2answers
320 views

Find the equation of the tangent line to the curve y=3x^2-x^3 at the point (3,0)

So far I have this, not sure what I am doing wrong.
6
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5answers
2k views

Solving limit without L'Hôpital

I need to solve this limit without L'Hôpital's rule. These questions always seem to have some algebraic trick which I just can't see this time. $$ \lim_{x\to0} \frac{5-\sqrt{x+25}}{x}$$ Could ...
0
votes
1answer
64 views

Integration Question

If we know the integral $$\int \frac{\mathrm{d}x}{f(x)+1}$$ can we find the integral of $$\int\frac{\mathrm{d}x}{f(x)+c}$$ for arbitrary $c\in\mathbb{R}$ (where defined)? Does it make a difference if ...
0
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2answers
44 views

Evaluating $\int \frac{1}{9+4x^2} dx$

Evaluate $$\int \dfrac{1}{9+4x^2} dx$$ I let $u = 9 + 4x^2$ so $du$ would be $8x$. But I don't know any way to make the numerator $1$ become $8x$. I could multiply by $1/8$ but then I'd still ...
0
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0answers
57 views

Riemman sum to get surface area of cone, and cone height

I have this itch to think about math ideas in my free time. Like when I play a game or something, I kind of "work in parallel" on math ideas. Back in one of my calculus classes we worked on Riemann ...
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2answers
25 views

Find radius of convergence, and then test the endpoints to determine the interval of convergence.

Consider $$\sum_{k=1}^{\infty}\left(\dfrac{x}{5}\right)^k$$ In class we had an extremely brief discussion on this topic, and so I still have many questions on how to start these problems. It ...
3
votes
2answers
98 views

How do I evaluate this integral by hand?

TL;DR how do I evaluate $\int_0^{2 \pi } \frac{1}{\cos ^2(\theta )+1} \, d\theta$ by hand? I'm trying to solve this problem: Find the volume of the region defined by $x^2+xy+y^2+yz+z^2\le1$. ...
1
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1answer
59 views

Finding the area between two curves?

I was trying to "discover" a way of finding the length of a curve, and I can across something: Let's say I want to find the length of $y=x^2+1$ between $0$ and $1$, and let's say the length is $L$. ...
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4answers
54 views

Evaluate the integral of sec(2x + 1) dx

I got $\ln|\sec(2x +1) + \tan(2x+1)| + \text C$ as an answer. I saw that the integral of $\sec x$ is $\ln|\sec x + \tan x| + \text C$. But I feel I may have left something out because that was too ...
0
votes
5answers
241 views

How to create alternating series with happening every two terms

I'm looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -... etc. I have every other part of the ...
0
votes
1answer
27 views

Evaluate the integral $(3x-2)/(x+1)$

The answer I have which I'm sure is wrong is $(3x-2)\ln(\left \vert x+1 \right \vert + \text{constant}$ I let $x + 1$ be $u$ and $du$ would be $1$. But I didn't know how to get $1$ on top so I just ...
0
votes
2answers
374 views

Given the graph of F prime (x) on [-5,2], answer the following questions about the graph f(x). Answer to the nearest integer.

Where is the graph of f(x) simultaneously increasing and concave down? Ok, so I know that the answer is (-3,-2)U(1,2) but I don't know how you're supposed to get that answer. I've attached a picture ...
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2answers
79 views

Evaluate the integral $\int\frac{x^2 + 1}{x^3 + 3x + 1} dx$

Evaluate the integral $$\int\frac{x^2 + 1}{x^3 + 3x + 1} dx$$ I've looked at similar examples online and I can't find one like the one above. In class we did one where we had to do long division and ...
0
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3answers
58 views

Find the $\lim_{x\to \infty} \frac{\ln(A + e^{Cx})}{x}$

Okay so I know that the answer is just C, but I don't understand how to get there. I figured out that I have to use L Hospitals rule because I have an indeterminate form of infinity over infinity. I ...
0
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1answer
18 views

Find B such that $\lim_{x\rightarrow-3^-}\frac{x^2-4}{x^2+7x+B}=-\infty$

Okay so I know that the answer is 12 but I don't understand how to get there. Am I right in thinking that we're going to use l hospitals rule for this? Can someone please guide me and explain to me ...
0
votes
1answer
64 views

Application of double integrals problem?

The centroid of a uniform plane region is at $(0,0)$ and the region has total mass $m$. Show that its moment of inertia about an axis perpendicular to the $xy$-plane at the point ($x_0$,$y_0$) is ...
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2answers
44 views

Given $y=(x+1)^{6x}$ use logarithmic differentiation to find $\frac{dy}{dx}$

The final answer I got after working this is $\frac{6(x+1)^{6x}}{x+1}$. But I am not confident in my answer. Is this correct?
0
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1answer
55 views

Converting from Cartesian polar coordinates

I am wondering whether I converted the following correctly. In Cartesian coordinates: $$\int_{-2}^{0}\int_{0}^{(4-x^2)^{1/2}}x^2(y) dydx$$ In polar coordinates would this be ...
0
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1answer
44 views

Curl of a function with only angular dependence

Let a function in spherical coordinates $$\vec F(\vec r) = \int{ d^3\vec r\,' \vec j(\vec r\,') } \,e^{-ik\hat r \cdot \vec r \,'}$$ Where $\vec j$ is a vector function. So $\vec F$ only depends on ...
0
votes
2answers
25 views

Find the derivative with respect to $t$ of $y=2^{\left(t^2\right)}$

I got $\left(2^{t^2}\right)(\ln2)(2t)$ but I feel it isn't right, or should be simplified more or something. Can someone explain if this is wrong?
4
votes
2answers
86 views

How to find $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}$?

find the integral of $f(x)=\frac1{(x-1)^2\sqrt{x^2+6x}}$ my attempt = $(x-1)=a$, $a=x+1$ so the integral'd be $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}=\int\frac{da}{a^2\sqrt{a^2+8a+7}} $ lets ...
0
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1answer
54 views

Integration by parts (fluids question)

I can't quite follow the solution for the part highlighted. Writing $U(x,t)=\phi(x)e^{\lambda t}$ casts $(2)$ into $$\lambda\phi=\phi^{\prime\prime}-(\overline u\phi)^\prime,\tag3 \\ ...
4
votes
0answers
115 views

Integral of product spherical harmonics

I'm trying to calculate this integral: $$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$ We could compute this integral ...
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votes
5answers
232 views

$99$th derivative of $\sin x$

Can someone help me calculate the $99$th derivative of $\sin(x)$? Calculate $f^{(99)}(x) $ for the function $f(x) = \sin(x) $
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1answer
42 views

Function continuous Uryson's lemma?

when we proved Uryson's lemma we checked that the function $f:X \rightarrow [0,1]$, where $X$ is a $T_4$ space, i continuous by checking whether $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open. $f$ is ...
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1answer
61 views

Differentiation/ find the derivative

Can anybody please help me with my work? I have to find the differentiate/ find the derivative of these two question: Please HELP!!! $sin^2(cos3x^3)^5 $ $cot^2(x)((x^2)(3cos^3(3x)))^2$
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0answers
46 views

First order equation to a transfer function

A process can be represented by the first order equation $$4 \frac{dy(t)}{dt} + y(t) = 3\,u(t).$$ Assume the initial state is steady ($y = 0$ at $t = –0$). Determine the transfer function of this ...
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4answers
103 views

Antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$

What is the antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$ ? $\displaystyle\int t^2e^{-\frac{1}{2}t^2}\,dt=\displaystyle\int{t}_{}te^{-\frac{1}{2}t^2}\,dt=-te^{-\frac{1}{2}t^2}\Big|_{?}^?+\int ...
0
votes
1answer
75 views

Reduced row echelon form with variables

I'm new to this, but if I have the matrix \begin{equation} A= \begin{bmatrix}1&2&3&1\\2&1&1&x^2+x \\ 3&6&x&x-6\end{bmatrix}\end{equation} and if I want to use the ...
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3answers
144 views

Why does $\lim\limits_{t\to-\infty}te^t=0$?

Intuitively, $\lim\limits_{t\to -\infty}te^t=0$, since $e^t\to 0$ much faster than $t\to -\infty$. Is there a way to more rigorously compute this? Writing it as $e^t/(1/t)$ seems resistant to ...
1
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0answers
23 views

Differential of a function

In simple words, the differential of a function is no more than the value of the function (for a given value of x), calculated on the tangent on the curve instead. But calculating the value of the ...
0
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1answer
122 views

Matrices word problem?

Julie and Bill are waiters at the Ogling Ogre Convention Center, which is well-known for serving the most deliciously disgusting meals to its guests. One of their tasks was to count the 2-eyed and ...
1
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1answer
46 views

Find velocity at the given point

I'm fighting with a problem of a sad sort. It's when you face a task with allusion to physics in the pure mathematical book. Here it goes: We have a boat which moves in the still water. We have one ...
0
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1answer
33 views

Convergence of $\int_{1}^{\infty}\frac{t^q}{1+t^q}dt; q\in{\mathbb{R}}$

How do i proceed to check the convergence\absolute convergence of the integral $$\int_{1}^{\infty}\frac{t^q}{1+t^q}dt; q\in{\mathbb{R}}.$$ Can anyone help me with this please.
3
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4answers
122 views

Why is $ \int \frac{\sin x (b-a\cos x)}{(b^2+a^2-2ab \cos x)^{3/2}}\,dx = \frac{a-b\cos x}{b^2 \sqrt{a^2-2ab\cos x + b^2}}$?

Why is $$ \int \frac{\sin x (b-a\cos x)}{(b^2+a^2-2ab \cos x)^{3/2}}\,dx = \frac{a-b\cos x}{b^2 \sqrt{a^2-2ab\cos x + b^2}}\text{ ?}$$ Constant of integration omitted.