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

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2answers
60 views

Limits and Integration Problem

I have no idea as to how to go about this. Could somebody please help? Let $$\displaystyle ...
1
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0answers
25 views

Need help with integrating the function $\frac{-cb}{b(ax+Pbe^{bx})+a}$ w.r.t. $x$

Can someone give me some hints on how to solve the following: $$exp\left(-\int{\frac{cb}{b(ax+Pbe^{bx})+a}\,dx}\right),$$ where $exp(t)=e^{\,t}\,\forall t\in \mathbb{R},e \approx2.71$, and $a,b,c,P$ ...
1
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2answers
67 views

How do I split this into partial sums?

Given the series, $\sum(s_n - s_{n+1})$, can someone please explain to me how to split this into partial sums and the basic concept behind it? I know that I can split it into $\sum s_n - \sum ...
0
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3answers
159 views

explain why ${\left(\frac{{1}}{{2}}\right)}^{\infty}=0$

Mathematica shows ${\left(\frac{{1}}{{2}}\right)}^{\infty}=0$, anyone can explain why ? I know we can get $\lim\limits_{{{x}\to\infty}}{\left(\frac{{1}}{{2}}\right)}^{{x}}={0}$ by taking limit , ...
2
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3answers
279 views

How to create very hard problems on Lagrange Multipliers

This is a rather odd request. I only recently started studying the Lagrange Multipliers, and was given a task to create some challenging (as much as possible) problems on them and also provide ...
1
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1answer
104 views

Evaluating an integral: $\int \frac{1}{(x+\sqrt{x+x^2})^2} dx$

$$\int \frac{1}{(x+\sqrt{x+x^2})^2} dx$$ I don't know how to approach this integral. I tried a few substitutions, but none of them got me to a desirable point.
3
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5answers
156 views

How do you solve for the limit of this series?

I need to take the limit of this summation so that I kind find out whether it converges or diverges. The equation is: $$\sum_{k=1}^\infty \frac{4}{k+4}$$ What I have tried so far is the following: ...
3
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2answers
88 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
1
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2answers
78 views

Dual tensor for partial derivative, if it has any meaning

I'm trying to find out some details about tensors, so my question maybe isn't quite correct. What if $\omega$ is volume form in $(x,y,z)$ coordinates, then how to understand that ...
1
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1answer
92 views

Find the 28th derivative of $f(x) = x^3 \sin(x^2)$

I am trying to do this problem using the Maclaurin series of sine but I get that the exponent on $x$ is $4n+5$. Therefore, there is no integer value for which $28 = 4n+5$. Does this imply $c_{28}=0$ ...
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1answer
47 views

Calculus 2 Series convergence - For which positive integers k is the series convergent?

For which positive integers k is the series given below convergent? $$ \sum _{n=1}^{\infty }\:\frac{\left(n!\right)^6}{\left(kn\right)!} $$ I tried using Root/Ratio tests but that didn't work out. Not ...
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0answers
124 views

Handling complex arguments of elliptic integrals in Maple

I want to solve the integral \begin{equation} I(y)=\int_0^y\sqrt{\dfrac{x(1+ax)}{(1+ax)^2-b^2}}\,\mathrm{d}x,\qquad a<0,\; b\in(0,1),\; y>0,\; (1+ax)^2-b^2>0, \end{equation} using Maple 18. ...
0
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2answers
264 views

Example of a Power Series Given Interval of Convergence

This was a thought question assigned to our calc II class, and I wasn't sure how to approach it. Give an example of a power series whose interval of convergence is $(0, \frac{4}{3}]$. Show ...
2
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1answer
45 views

Where did I got wrong with this surface integral

It appears that I don't quite have surface integrals like I thought I did. The following is a problem from the back of the book (not homework because it wasn't prescribed but I'm working it to ...
1
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1answer
103 views

Integral of $ \int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$

How to solve: $$\int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$$ This question is from a list and it's in the category of problems that involving $\sqrt{x^2\pm a^2}$ and $\sqrt{a^2\pm x^2}$ (triangle ...
3
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3answers
1k views

Is 1/x, integrated from 1 to infinity, considered “integrable”?

$\large \int\limits_1^\infty \frac{dx}{x} $ diverges by comparison with the infinite sum $\sum\limits^\infty\frac{1}{n}$. But $\frac{1}{x}$ is continuous at all non-zero values of $x$. So is ...
2
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3answers
38 views

How to find the convergence/divergence of the sequence $a_n = \left(2n+3 \over n\right)^n$

$$a_n = \left(2n+3 \over n\right)^n = \left(2 + {3 \over n}\right)^n$$ $$\lim_{n \to \infty} \left(2 + {3 \over n}\right)^n$$ $$\lim_{n \to \infty} n*ln\left(2 + {3 \over n}\right) $$ $$\lim_{n \to ...
3
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4answers
132 views

How can I integrate $\int {dx \over x^2\sqrt{x^2 -1}}$?

Let $x=\sec\theta$, so $\mathrm{d}x=\tan\theta \sec\theta\, \mathrm{d}\theta$. Then $$\int {\mathrm{d}x \over x^2\sqrt{x^2 -1}}=\int {{\tan\theta \sec\theta} \over {\sec^2\theta \sqrt{\sec^2\theta ...
2
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1answer
76 views

TI-84 gives 100 for $\frac{d}{dx}\sqrt[3]{x}\,\big|_{x=0}$

My TI-84 Silver Edition is doing something strange. If $f(x)=\sqrt[3]{x}$, $\frac{d}{dx}\sqrt[3]{x}=\frac{1}{3\sqrt[3]{{x^2}}}$ At $x=0$, $\frac{d}{dx}f(0)$ is undefined. When I type ...
3
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5answers
156 views

How to calculate the integral $I=\int\limits_0^1\frac{x^n-1}{\ln(x)} \,\mathrm dx$ [duplicate]

How can we calculate this integral: $$I=\int\limits_0^1\frac{x^n-1}{\ln(x)}\,\mathrm dx$$ I believe that integral is equal to $\ln(n+1)$, but I don't lnow how to prove it.
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2answers
513 views

What do convergence and divergence mean? And why do they matter?

I understand that when a series diverges, y doesn't approach 0 when x approaches infinity, and converging series do. But what does this say? I just want to understand some applications. NOTE: I'm ...
3
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7answers
942 views

How to evaluate the limit $\lim_{h \to 0} \frac{e^{2+h}-e^2}h$?

$$ {\lim \limits_{h \to 0}} { {e^{2+h}-e^2 } \over {h} } $$ Due to time constraints, evaluating limits with e in them wasn't covered and I have this on the AP exam review. How do I proceed?
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2answers
43 views

How can I prove that $a_{n}$ and $a_{3n+2}$ converge to the same value?

I am given that $a_{n}\to L$, how can I prove that $a_{3n+2} \to L$ also? It makes sense since $3n + 2$ is still in $\mathbb{N}$, but I don't know how to say that in proof form.
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5answers
273 views

How to factorize $x^5 -1=0$?

how can I factorize $x^5 -1 =0 $ so I get a quartic equation separately. Is there any specific way of factorizing such equations to get something of a lesser power?
10
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1answer
126 views

Simple true/false statement about function composition

Given the functions $f,g$ from $\mathbb{R}$ to $\mathbb{R}$ is it true that If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is monotonic I believe this is false but I can't ...
2
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7answers
61 views

How can I solve the improper integral $\int_{1}^\infty {dx \over {(x+1)(x+2)}}$

$$\int_{1}^\infty {dx \over {(x+1)(x+2)}}$$ I have the indefinite integral solved for: $$\ln(x+1)-\ln(x+2) + C$$ But I don't know how to finish with $[1, \infty]$.
2
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0answers
57 views

Is there a way to follow the curve of a polynomial at a fixed speed?

I'm trying to follow the curve of a polynomial between $x=0$ and $x=1$ at a fixed speed using the arc length formula: $\int_0^1\sqrt{1+f'(x)^2}dx$ I've gotten around the square root problem by ...
3
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2answers
60 views

How to draw the graph of $f(x)=\int_0^x\left(\frac{t^3-2t^2-4}{t^2+1}\right)\ dt$ using only your calculator?

$$f(x)=\int_0^x\left(\frac{t^3-2t^2-4}{t^2+1}\right)\ dt$$ I need to find the $x$ and $y$ intercepts, and the inflection points of the function $f(x)$ (with both $x$ and $y$ coordinates). I need to ...
1
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1answer
42 views

Show that $\sum_{m \in \mathbb{Z}} e^{-(x-m)^4}$ converges for all $x \in \mathbb{R}$ converges

how can I rigorously show that $$\sum_{m \in \mathbb{Z}} e^{-(x-m)^4}$$ converges for all real $x$? I am aware of convergence criteria for ordinary series, but not for $\sum_{m \in \mathbb{Z}}$. ...
1
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0answers
29 views

Is the set of all points $(x,y)$ outside of an ellipse, or on the ellipse?

If $\vec{r} = \langle x,y \rangle$, $\vec{r}_{1} = \langle x_{1},y_{1} \rangle$, and $\vec{r}_{2} = \langle x_{2},y_{2} \rangle$, describe the set of all points $(x,y)$ such that ...
1
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2answers
61 views

Power series properties. The sum of two series.

Let $\sum a_nz^n$ and $\sum b_nz^n$ power series with radiuses of convergences $R_1,R_2$ respectively. Suppose the radius of convergence of $(\sum a_n+b_n)z^n$ is $R$. Find an example in which ...
4
votes
3answers
59 views

How can I solve the integral $ \int {1 \over {x(x+1)(x-2)}}dx$ using partial fractions?

$$ \int {1 \over {x(x+1)(x-2)}}dx$$ $$ \int {A \over x}+{B \over x+1}+{C \over x-2}dx $$ I then simplified out and got: $$1= x^2(A+B+C) +x(C-2B-A) -2A$$ $$A+B+C=0$$ $$C-2B-A=0$$ $$A=-{1 \over 2}$$ ...
1
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1answer
43 views

Partial limit of a sequence

In my assignment I have to find all the partial limits of the following sequence: $$\lim_{n\rightarrow\infty}\frac{(-5)^n-2^n+2}{3^n+(-2)^n-2}$$ I wanted to seperate this question to $n$ elements ...
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3answers
63 views

How do I know when to use partial fractions or long divison with this integral? $ \int {{x^4+1} \over {x(x^2+1)^2}} dx$

$$ \int {{x^4+1} \over {x(x^2+1)^2}} dx$$ Is there a method to determine which way is better?
2
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1answer
43 views

How to evaluate the integral $\int_{0}^{2}{g(x) dx}$, where $g(a)$ is a solution of the equation $x^{5}+x=a$

We consider an integral $\int_{0}^{2}{g(x)dx}$, where $g(a)$ is a solution of $x^{5}+x=a$. Actually, it means that $g^{5}(a)+g(a)-a=0$. Moreover, it somehow possible to reestablish $g(x)$ on $[0, 2]$ ...
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2answers
202 views

If $f(x)=\sin^2(3-x)$, then what is $f'(0)?$

I've been doing the math myself and my answer happened to be $-\sin(6)$, am I just being really stupid here and unable to convert it to any of the answers or my answer is wrong (or the answers are ...
1
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2answers
42 views

Is it true that $\frac{d}{dt}f(g(t),h(t))=f'(g(t),h(t))g'(t)+f'(g(t),h(t))h'(t)$

I want to solve the following question: We want to find $\frac{du}{dt}$ where $u(x,y)=x^2y^3$ and $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$. I know we can just plug $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$ in ...
26
votes
4answers
1k views

Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$

Can we possibly compute the following integral in terms of known constants? $$\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$$ Some progress was already done here ...
3
votes
2answers
109 views

If $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim \limits_ {x\rightarrow \infty} f(x) = 0$ [duplicate]

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function and $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim\limits_{x\rightarrow \infty} f(x) = 0$. I ...
3
votes
2answers
90 views

Check the convergence of $a_{n+1}=\sqrt{a_n+\frac{4}{a_n}}$ where $a_1=4$

Check the convergence of $a_{n+1}=\sqrt{a_n+\frac{4}{a_n}}$ where $a_1=4$. If it converges, find its limit. I tried to prove that the sequence is monotonically decreasing and bounded by 2, but I ...
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1answer
59 views

Show that $\tan (nx) \neq 0$

I would appreciate if somebody could help me with the following problem Show that $$\tan (nx) \neq 0$$ where $n$ is a positive integer and $\tan x=\frac{3}{2}, (0<x<\frac{π}{2})$ ...
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1answer
30 views

convergence test

show that a function of cos(n) diverges and prove.
3
votes
2answers
54 views

Find derivative of integrate square function [closed]

I am finding a solution of that function. Could you have me to resolve it $$F=\left( \int {(ax+b-c)}^2 dx \right) +\lambda_1(a-m)^2+\lambda_2(b-n)^2$$ where $c,m,n ,\lambda_1,\lambda_2$ are constant ...
0
votes
1answer
128 views

What is the relationship between sequences and series?

If I am given a sequence made up of positive real numbers, and I know the series $\sum{M_n}$ converges, what can I say about Mn? I am trying to prove that Mn converge. I know it has something to do ...
2
votes
1answer
77 views

Prove that the sequence is increasing.

I want to prove that the sequence $M_n$ is increasing and bounded.
1
vote
1answer
427 views

How do I find the second derivative from a differential equation? [closed]

How do I find the second derivative from a differential equation? From the 2012 AP BC Calc, ...
1
vote
2answers
272 views

Centre of mass of region using density (multivariable calculus)

A lamina (two–dimensional plate) occupies the region inside the circle $x^2 + y^2 = 2y$, but outside the circle $x^2 +y^2 = 1$. Find the centre of mass if the density = $\frac{k}{r}$ (inversely ...
3
votes
3answers
104 views

user friendly proof of fundamental theorem of calculus

Silly question. Can someone show me a nice easy to follow proof on the fundamental theorem of calculus. More specifically, $\displaystyle\int_{a}^{b}f(x)dx = F(b) - F(a)$ I know that by just ...
5
votes
1answer
41 views

Simplifying the method of solving a problem

One of my peers from high school asked me how can this problem be solved: $$\begin{cases} x^2+y^2=z \\ x+y+z=m \end{cases}$$ Considering the mentioned equations, find $m$ such that the system has ...
3
votes
1answer
110 views

A possible dilogarithm identity?

I'm curious to find out if the sum can be expressed in some known constants. What do you think about that? Is it feasible? Have you met it before? $$2 ...