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

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3answers
54 views
5
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1answer
103 views

Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$

Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$ Abel's/Dirichlet's tests cannot be applied here. I guess it's something more tricky involving integration maybe (?)
3
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2answers
64 views

convergence of $ \sum_{n=1}^{\infty} (-1)^n \frac{2^n \sin ^{2n}x }{n } $

Find values of $x$ for which the following series converges $$ \sum_{n=1}^{\infty} (-1)^n \dfrac{2^n \sin ^{2n}x }{n } $$ Attempt: (a) Check for Absolute Convergence If we consider $ ...
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1answer
220 views

Derivative of bilinear forms

I want to solve the following problems: Let $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form. Prove that it's differential is $$ Df_{(x,y)}(a,b) = f(x,b) + f(a,y).$$ ...
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1answer
34 views

Plane equation x units from point

I'm trying to find the equation of a plane normal to a certain vector $<x_1, y_1, z_1>$, and x units from a given point, $(a,b,c)$. Normally this question would be trivial, and I would simply ...
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2answers
30 views

Convergence of $\int_0^1 \frac{dx}{(\cos(x)-1)\sqrt{1-x^2}}$

Study the convergence of $$\int_0^1 \frac{dx}{(\cos(x)-1)\sqrt{1-x^2}}$$ Well, we can observe the $$\left| \frac{1}{(\cos(x)-1)\sqrt{1-x^2}} \right| \le \left| \frac{1}{(\cos(x)-1)} \right|$$ ...
3
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2answers
253 views

Deriving formula for derivative

I have a formula in my book for differentiating numerically. $$f'(x_0)=\frac{1}{12h}[-25f(x_0)+48f(x_0+h)-36f(x_0+2h)+16f(x_0+3h)-3f(x_0+4h)]+\frac{4}{5}f^{(5)}(\xi)$$ I was wondering if anyone ...
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1answer
187 views

Does there exist continuously differentiable function $f:\mathbb{R}\longrightarrow\mathbb{R}$?

Does there exist continuously differentiable function $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that for all $x\in \mathbb{R},\,\,f(x)>0$ and, $f'(x)=(f\circ f)(x)$? I see this question in ...
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0answers
62 views

Problem about limit of an integral

I came across this question while doing some exercises on integrals, and I was wondering if I could get some help. a) Show that for $n < -1$, $\int_1^N x^n dx$ converges as $N \to\infty$, and for ...
2
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3answers
65 views

Radius of convergence of $\sum_{n=0}^\infty a_n z^{n^2}$

Find radius of convergence of power series $\sum_{n=0}^\infty a_n z^{n^2}$ where $a_0=1, a_n=3^{-n}a_{n-1}$ for n $ \in $N. I tried to get expression for $ a_n $ first which comes to be $ a_n$ ...
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1answer
32 views

Using the tangent to the curve to find when it matches another equation

Find the point at which the tangent to the curve $y=x^2$ has the equation $2x+y+1=0$? So I find the tangent to the curve of the equation but then to find the point what do I do next like sub it in?
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1answer
32 views

$x$-values for tangent to the curve then to a perpendicular lines

Find any $x$-values for which the tangent to the curve $y=5x-x^2$ is perpendicular to the line $4x-2y-1=0$. So do I use the $m_1 \times m_2= -1$ to show it's perpendicular or what? I'm really ...
2
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0answers
34 views

Extremal points relative to origin for an ellipsoid

Suppose I have an ellipsoid of the form $ax^2 + by^2 + az^2 - cxy -cyz = d$ How would I find the points nearest to, and furthest from, the origin?
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0answers
32 views

How do i solve for $v_0$ in the equation $s(t)=-4.9t^2+v_0t+2.032$?

I was given the equation $s(t)=-4.9t^2+v_0t+s_0$, for the initial height we used our own height as if we were throwing the ball and converted it to meters that is where the $2.032$ is from but now I ...
2
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1answer
44 views

Integral of $f$ and $g$ equal for all intervals containing positive and negative numbers sufficient condition for $f=g$?

Is there a way to get from $$\int_a^b \delta(t)dt=-\int_a^bt \frac{d (\delta (t))} {dt} dt$$ for all $$(a,b):(a<0) \land (b>0) \land a,b\in\Re $$ to $$\delta (x)=t \frac {d (\delta (x))} ...
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2answers
81 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
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1answer
78 views

Coming Up With A Neutral Fixed Points Theorem

Question: If $f(x_0)=x_0,f'(x_0)=1$ and $f''(x_0)>0$, is $x_0$ weakly attracting, weakly repelling, or neither? (weakly attracting meaning $\exists\delta,\forall x\in ...
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1answer
22 views

Does the equality $\partial^\alpha(x^\alpha)(0)=\alpha!$ hold?

Do we have $\partial^\alpha(x^\beta)(0)=\alpha!=\beta!$ if $\alpha=\beta$ and $0$ else? I tried to proof it on induction, can include my attempts if needed, but they seem to have failed anyway...
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2answers
70 views

What is the integral of $1/(1+x)$

The integral of $1/(1+x)$ with respect to $x$ can either be $\arctan(\sqrt{x})$ or $\ln(1+x)$. Which one is it? Because I am sure these two functions are not the same.
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1answer
41 views

Find the derivative of an integral.

Find the derivative of the following integral $$ F(x)=\int_x^{x^2}e^{t^7}dt $$ Find F′(x) given F(x). Normally I would show my attempt in working out the problem: however, I don't even know where ...
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1answer
52 views

How to integrate $(x^2 - y^2) / (x^2 + y^2)^2$

How do I integrate $$\int \int \frac{(x^2 - y^2)}{(x^2 + y^2)^2} dx dy?$$ The WolframAlpha page gives $$ c_1 + c_2 + \tan^{-1}(x/y). $$ And I kind of specifically need $$ \int_{0}^{x} \frac{(x^2 - ...
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2answers
43 views

Can someone help check if I evaluated my integral right?

The integral given to be evaluated is $\int 6x\cdot\mathrm{arctanh}(x)\,dx$. I tried to evaluated and got the following answer: $$3x^2\operatorname{arctanh}x + 3x ...
0
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1answer
79 views

Angles and tangents finding specific points

Find any points on the graph $y=x^3$ where the tangent makes an angle of $45$ degrees with the $x$ axis in the positive direction I don't understand the $45$ degree bit, I differentiated but I'm ...
4
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3answers
173 views

Conceptual question on differentiation in calculus?

In Calc class, my teacher told us that the only solution to $y' = y$ is $y = ce^x$, with $c$ being a real number. I am having difficulty understanding the only part. Is there a proof of this? Or am I ...
5
votes
3answers
126 views

Convergence of $ \sum_{n=1} ^\infty \frac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$

Convergence of $$ \sum_{n=1} ^\infty \dfrac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$$ Attempt: I believe not a nice attempt: $ n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} ...
3
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2answers
55 views

Limit of non-linear multi-variable function

I'm trying to prove the limit of the following function is $0$: $\lim_{(x,y) \to (1,-1)} {x^3} - {2xy^2} + 1$ I know that I'm trying to find a $\delta$ s.t $ 0 < \sqrt{(x - 1)^2 + (y + 1)^2} < ...
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1answer
83 views

How to integrate $1/\sqrt{(1+x^2)^3}$?

Normally I use WolframAlpha pro to help me with problems I don't know however wolfram wont/cant show me the steps only the final solution to this integration problem. Is anyone able to assist me with ...
1
vote
1answer
44 views

Simple Trig Integration. Why is my answer wrong?

$$\int \dfrac{\cos x+\sin 2x}{\sin x}dx=\int \dfrac{\cos x+2\cos x\sin x}{\sin x}dx=\int \dfrac{\cos x\left(1+2\sin x\right)}{\sin x}dx$$ Substitute $u=\sin x$ and $du=\cos x\ dx$: ...
0
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3answers
1k views

Finding a point on a curve where the tangent is parallel to another line noted

Find the point on the curve $y=x^2+2$ where the tangent is parallel to the line $2x+y-1=0$ I understand the answer is $(-1,3)$ but I can't find a way to get there... Thanks
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2answers
56 views

Is this identify valid?

$$\sin(t) \dot{}e^{if(t)} = \sin(t+f(t))$$ I've never seen this identify before but it follows directly from the relation between complex exponentials and the trigonometric functions.
0
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2answers
74 views

Find $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$

Find $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$ My professor showed us a few ways to compute the limit 1) Factor the numerator $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3} = \lim_{x ...
0
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1answer
40 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
0
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1answer
28 views

taylor series approximation of e function

in the equation $$e^{y(x)}=1+2x-\frac{y(x)}{1-x}$$ $y(0)=0$ because using the taylor series and by comparing the coefficients we obtain $$1+y(0)=1-y(0)$$But why is using the taylor series allowed. ...
2
votes
5answers
95 views

Convergence of $\sum_{n=1}^{\infty} \log~ ( n ~\sin \frac {1 }{ n })$

Convergence of $$\sum_{n=1}^{\infty} \log~ ( n ~\sin \dfrac {1 }{ n })$$ Attempt: Initial Check : $\lim_{n \rightarrow \infty } \log~ ( n ~\sin \dfrac {1 }{ n }) = 0$ $\log~ ( n ~\sin \dfrac ...
0
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3answers
58 views

Does $f_n(x) = \frac{x^n}{1+x^n}$ converges uniformly on $[0,1]$?

Does $f_n(x) = \frac{x^n}{1+x^n}$ converges uniformly on $[0,1]$? My answer is: No because obviously $f_n(0) = 0$ and $f_n(1)=\frac{1}{2}$, so for every $n\in\mathbb{N}$ it's true that for ...
0
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0answers
629 views

The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles.

Find the volume of the following solid S: The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles. So far I got ...
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1answer
54 views

Is it true that $\lim_{x \to 0} \frac{f(x)}{g(x)} = 0 \implies \lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0$?

Is it true that $$\lim_{x \to 0} \frac{f(x)}{g(x)} = 0 \implies \lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0$$ ? Assume that $f$ and $g$ admit derivatives $f'$ and $g'$ at every point in an open interval ...
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3answers
74 views

convergence of $ \sum_{n=1}^\infty \frac {1}{\log (1 +\frac {1}{n})}$

Test convergence of $$ \sum_{n=2}^\infty \dfrac {1}{\log (1 +\frac {1}{n})}$$ I am not really sure how to move forward. Could anyone give me a direction to proceed please. EDIT" The only part I ...
1
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2answers
47 views

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$ So first we have two potentially problematic points which are $1,\infty$ We split the integral to $$\int_1^2 \frac{x\ln x}{x^4-1} ...
2
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3answers
73 views

Convergence of $\sum_{n=1}^{\infty} \frac {1}{n\log^2(n+1)}$

Convergence of $$\sum_{n=1}^{\infty} \dfrac {1}{n\log^2(n+1)}$$ Attempt: We note that $\lim_{n\rightarrow \infty} \dfrac {n}{ \log^2(n+1)} = \infty$ Hence, for a sufficiently large $n: \dfrac {n}{ ...
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3answers
66 views

convergence of $\int_a^b \frac{1}{x^2} dx$

Why is it true that $\int_0^a \frac{dx}{x^2} = \infty$ but $\int_a^\infty \frac{dx}{x^2} < \infty$? Shouldn't it be symmetric?
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2answers
67 views

convergence of $\sum_{n=1}^{\infty} \frac {1}{\log(e^n+e^{-n})}$?

Test convergence of $\sum_{n=1}^{\infty} \dfrac {1}{\log(e^n+e^{-n})}$ Attempt: I have tried the integral test, the comparison test ( for which I couldn't find a suitable comparator). However, I ...
0
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1answer
25 views

Find the change of electromotive-force per degree, at 15 degrees, 20 degrees, and 25 degrees.

I started to read "Calculus Made Easy", by Silvanus P. Thompson, and i can't figure out how to solve one problem E = 1.4340[1 - 0.000814(t-15)+0.0000007(t-15)^2] volts Find the change of ...
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3answers
140 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
0
votes
2answers
29 views

Show using inequality of means that $a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$

Show using inequality of means that for $a>0$ and $n\in\mathbb{N}$: $$a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$$ I'm sure it's not that complicated, but I'm probably missing ...
1
vote
1answer
68 views

Prove $\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}$

In the book "Lehrbuch der Analysis Teil I" of Heuser page 303, there was a task: Prove $$\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}.$$ When I tried, I ended up with ...
0
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1answer
54 views

How to calculate the multiplication result in two different ways?

First I want to apologize for a possibly ambiguous title, but don't know how to expose clearly the problem through the title because the speech is much more structured. I'm working on a system that ...
1
vote
1answer
40 views

Area between two functions

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 16. Let $f(x)=x-x^2$, $g(x)=ax$. Determine $a$ so that the region above ...
0
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2answers
30 views

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
2
votes
2answers
62 views

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$ Why is it wrong to... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$