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

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0answers
6 views

Radius and Interval of Convergence of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{2n-1}$

This is my first time finding the radius and interval of convergence of a series, so please bear with me. I would like to find the radius and interval of convergence of ...
0
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2answers
12 views

Given f(x) =$ a/x^2 + x $

Use the second-derivative test to show that if a is positive then the graph has a local minimum, and if a is negative then the graph has a local maximum. So part 1 of this asked to find the critical ...
0
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0answers
14 views

A inverse Trigonometric multiple Integrals

How to calculate the closed form of the integral $$\int\limits_0^1 {\frac{{\int\limits_0^x {{{\left( {\arctan t} \right)}^2}dt} }}{{x\left( {1 + {x^2}} \right)}}} dx$$
2
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1answer
16 views

$cos(x)$ and $sin(x)$ span the solution space $V$ of $f''(x)=-f(x)$

Show that for $g(x)\in V$, $(g(x))^2+(g'(x))^2$ is constant. Show that for $g(x)\in V$ with $g(0)=g'(0)=0$, $g(x)=0$ for all $x$. Show that for $f(x)\in V$, $f(x)=f(x)-f(0)\cos(x)-f'(0)\sin(x)$. ...
2
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2answers
33 views

Tedious undefined limit without L'Hospital $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$

When I try to calculate this limit: $$\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$$ I find this: $$\begin{array}{l} L = \mathop {\lim }\limits_{x \to ...
0
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1answer
17 views

Calculate the volume of intersection of $x^2+y^2+z^2=4$ and $r=2\cos \theta$ by using cylindrical coordinates.

Calculate the volume of intersection of $x^2+y^2+z^2=4$ and $r=2\cos \theta$ by using cylindrical coordinates. My try:Intersection will be a cylinder $x^2+y^2+z^2=4\implies r^2+z^2=4$ Then ...
3
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2answers
26 views

Integrating algebraic functions

The function $y = f(x)$, restricted on the domain $ 0 < x < 1$ and satisfying $$y^{5}+y^{4} + x = 0,$$ seems to be well-defined and smooth. So how does one integrate this thing? That is, what ...
1
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1answer
20 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
1
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0answers
21 views

2 variable limit

So, I understand why these bigger limit above does not exist (I'll name it 1), but I can understand why the other (2) is $0$. It seems to me that the $y^4/(x^6+y^8)$ is a non limited function and so ...
0
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1answer
13 views

Mathematica Cubic Polynomial

I am interested to find the roots of a cubic polynomial f(r)=r^2 - 2 r + Q^2 - [Alpha] r^3. \Alpha is positive real number hare, Q is real. I am interested in 2 complex roots and third root to be ...
1
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2answers
17 views

Limits with squares

I have problem with finding such limits: a) $\displaystyle \lim_{n \to \infty}\frac{n^2}{7^{\sqrt{n}}}$ b) $\displaystyle \lim_{n \to \infty}\frac{3^{\sqrt{n}}}{2^n} $ I think the method of ...
-1
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2answers
27 views

Find an equation of the straight line

Find an equation of the straight line passes through the origin and is tangent to the curve $y=x^3+2$ I got stuck because I have no points.
1
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0answers
17 views

Closed form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
0
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0answers
17 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty ...
2
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1answer
24 views

Finding extremes of a function in some domain

Find extreme values of the function $$f(x,y)=e^{xy}$$ subject to constraints $x^2 + y^2 = 2$ What I have so far: I know the gradient of f is $$\nabla f(x,y) = \langle e^{xy}y,e^{xy}x\rangle$$ ...
6
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0answers
40 views

Closed form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
0
votes
1answer
28 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
0
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0answers
11 views

Finding and classifying critical points

Find all critical points of the function $g(x,y) = x^2-xy+y^3/3$ Determine whether each point is a local minimum, maximum or saddle. What I did: I found the gradient of g to be gradient of ...
2
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1answer
51 views

Are all calculus textbooks “the same”?

I'm not satisfied with my calculus textbook,[1] and because of that I have searched for books by other authors. The problem is: all the books I have taken a look at are almost the same, even the ...
0
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0answers
13 views

Linear Approximation confusions

The problem: Find the linear approximation of the function $f(x,y)=ln(e+x+y)$ at point $(0, 0)$. Use it to approximate the value of the function at $(0.1, 0.2)$ What I have so far: I found the ...
1
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0answers
12 views

Equation of tangent plane at a point

The problem: Consider the cone given by equation $x^2 = z^2 + 3y^2$ Sketch a picture of the cone, and identify in which direction the cone is opening. Find an equation of the tangent plane to the ...
2
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3answers
22 views

Find the tangen to $\cos(\pi \cdot x)$

I have the following assignment. Find the tangent to $y=f(x)=\cos(\pi \cdot x)$ at $x=\displaystyle\frac{1}{6}$. First step would be to take the derivative of $f(x)$ $f'(x)= -\pi \sin(\pi \cdot x)$ ...
0
votes
2answers
40 views

How to find the limit without L'Hospital rule

Find the limit $$\lim_{x\to\infty} x\left[\left(1+\frac{1}{2x}+\frac{1}{4x^2}+\frac{1}{8x^3}\right)^{1/3}-1\right].$$ I assume that L'Hospital rule here is useless and something else must be done.
0
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0answers
11 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
0
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0answers
16 views

Find the minimum number of terms in the series it would take before the error would be guaranteed to be less than 10^-9

Consider the function $f(t)=\ln t$ about the point $t_0=1$. Find the minimum number of terms in the series it would take before the error would be guaranteed to be less than $10^{-9}$.
-2
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0answers
11 views

Inscribed and circumscribed polygons [on hold]

Given a circle, prove (with basic geometric methods: no trigonometry) that the area of any inscribed irregular polygon is strictly smaller than the area of any circumscribed polygon. Extra ...
0
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1answer
21 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
4
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1answer
21 views

Volumes and surfaces of revolution?

Please can someone explain to me why we use $dx$ in a volume of revolution i.e. $$\pi \int{f(x)^2 dx}$$ but $ds$ (an elementary bit of arc) in a surface of revolution i.e. $$2\pi \int{f(x)ds}$$ does ...
3
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5answers
200 views

Calculus Proof of inequality

I asked this question yesterday and have been working on it. I have to prove that $2015^{2013}<2014^{2014}<2013^{2015}$. I set $x=2014$, so now I have $$(x+1)^{x-1}<x^x<(x-1)^{x+1}.$$ ...
-1
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0answers
12 views

if \Sigma a_n is convergent then what we can say about \Sigma a^{\frac{n}{n+1}} [on hold]

if Sigma a_n is convergent then what we can say about Sigma {a_n}^{n/n+1}
-5
votes
0answers
26 views

trust fund maturity [on hold]

My grandfather started a trust for me when I was born. He opened this account Oct. of 1991 placing $\$200.00$ as the initial deposit. He then deposited $\$200.00$ every Christmas and birthday ...
0
votes
1answer
47 views

Limit Delta-Epsilon proof

Prove $\lim_{x \to a} 2x = 2a$ Using the formal proof, not informal. So we know $2|x - a| < \epsilon$ We need to find some $\delta$ We only need to prove there IS SOME $\delta$ right? Only ...
3
votes
1answer
30 views

Implications of some sort of $l^2$/uniform convergence

Sorry about the title, but I couldn't really figure out how to describe my problem in one sentence... I'm having some problems with real limits: For $f,g : \mathbb{N} \to \mathbb{R}$ let ...
1
vote
2answers
26 views

derivative of $\frac{2}{3}x^{3-e}$

Find the derivative:$\;\;\;\;\;\;\dfrac{2}{3}x^{3-e}$ I am not sure how to solve this problem. My try: $\ln y=\dfrac{2}{3}(3-e)\ln x$ $\dfrac{1}{y}\times y\;'=\dfrac{2}{3}(3-e)\dfrac{1}{x}$ ...
1
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2answers
107 views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
0
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0answers
48 views

Can this be expressed by a contour integral?

Let $f(z)$ be a real entire function of the form $f(z) = a_1 z + a_2 z^2 + ...$ such that $0 < a_{n+1} < a_n$. Consider $g(x) = f^{-1}(f(x)-f(x-1))$ where $x$ is a positive real and $f^{-1}$ ...
5
votes
1answer
87 views

Why limit $\sqrt{\frac{\sin(x)}{x}}$ as $x \rightarrow \infty$ is not a real number?

Let $f(x)=\sqrt{\frac{\sin(x)}{x}}$. Why isn't the $\lim\limits_{x\rightarrow \infty} f(x)$ equals to some $l \in \mathbb{R}$? The definition of a finite limit at inifinity is: $$\forall ...
7
votes
3answers
69 views

Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$

How to evaluate the following improper integral : $$\int_0^{+\infty}\frac{x\sin x}{x^2+1}\,dx$$ I have tried integration by parts and variable substitution, but it didn't work.
2
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4answers
108 views

Inverse Trigonometric Integrals

How to calculate the value of the integrals $$\int_0^1\left(\frac{\arctan x}{x}\right)^2\,dx,$$ $$\int_0^1\left(\frac{\arctan x}{x}\right)^3\,dx $$ and $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
4
votes
2answers
29 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...
3
votes
3answers
73 views

Evaluate integral: $\int_0^{+\infty}\frac{\cos{bx}-\cos{ax}}{x}dx$

It seems that $\displaystyle\int_0^{+\infty}\frac{\cos x}{x}$ is divergent, so how to solve this problem? $$\int_0^\infty \frac{\cos bx -\cos ax}{x}\, dx\quad,\quad\mbox{where}\,a,b>0$$ It's ...
1
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2answers
13 views

Critical points for undefined fraction on closed interval

I am told to find the absolute extrema of $$h(x) = \frac{8+x}{8-x},[4,6]$$ So I obtain the derivative of $$\frac{16}{(8-x)^2}$$ The trouble I am having is trying to determine the critical points. ...
0
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1answer
52 views

calculus first impressions

I just did the first lecture on differentiation what the derivative is, and how is it calculated I didn't find it to be difficult at all it's just some forms of algebraic calculations when will ...
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0answers
14 views

Calculating volumes using integral.

Given $y=x,y=0,x=2$ and $x=7$. Calculate the volume6 about $x=1$. I just need to get the concept right. Please tell me what mistake I did here. The region looks like a trapezium right? From $y=0$ ...
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0answers
18 views

with wich values of the parameter “a” serie converges [on hold]

To know for each value of parameter a the convergence of the serie
2
votes
5answers
320 views

How calculate limit without using L'hospital theorem

$$\lim_{x \to 0}\left(\frac{\sin{x}-\ln({\text{e}^{x}}\cos{x})}{x\sin{x}}\right)$$ My question is: This limit can be calculated without using the L'Hospital theorem? any ideas?
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0answers
23 views

Will this method find the taylor expansion of ANY function $f(x)$?

Polynomials are themselves Taylor expansions, correct? ex. $4x+5x^2+3 = 3+4x+5x^2 +0x^3 +0x^4 + \dots$ I'm assuming has no closed form besides $\sum_{n=0}^{2}(3+n)x^n + \sum_{n=3}^{\infty}0x^n$ but ...
0
votes
1answer
13 views

Integral of a normal function multiplied by heaviside and delta functions

$\int_{-\infty}^{\infty} e^{2t}u(\tau - t)t^{2}\delta(t)dt$ Hi! How would I go about computing this integral? I understand I can change one of the integration limits and eliminate the heaviside ...
0
votes
1answer
21 views

Studying the convergence of a series with logarithm [on hold]

I would like to know if this sum is convergent, and why: $$\sum_1^\infty\ln\bigg(1+\dfrac1{n^a}\bigg)$$
2
votes
1answer
15 views

checking slope = $0$ at a point for a function using $\epsilon $, $\delta $ definition

From the continuity definition, a function is continuous at a point $a$ if : $$\forall \epsilon \gt 0 \exists \delta \gt 0 : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$$ If I change the ...