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

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1answer
50 views

Definition of improper Riemann integral

My lecture notes have defined the improper Riemann integral as follows Let $f : (a,b] \rightarrow \mathbb{R}$ be a function such that $f|_{[\tilde a,b]}$ is Riemann integrable on every closed ...
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2answers
27 views

How to show path wise connectedness?

How do I show that the set $A=\{(x,y)\in\mathbb{R}^2:x≥0,y≥0\}\cup\{(x,y)\in\mathbb{R}^2:x≤0,y≤0\}$ is path connected. I know that I need to construct a continuous function $f:[0,1]\rightarrow A$ such ...
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1answer
122 views

Griffiths Electrodynamics Example 1.8 - Calculating Volume Integral

In Example 1.8 in the electrodynamics textbook by Griffiths, he calculates the volume integral over a prism. The prism is formed of two triangles in the xy plane, with sides $x=0$ to $x=1$, $y=0$ to ...
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1answer
72 views

Compute the integral using Riemann sums.

Let $f(x)= 1 , \;-1\leqslant x\gt0,\; f(x)= 2,\; x=0,\; f(x)= 1 ,\; 0\lt x\leqslant1.$ Compute this integral using Riemann sums: $$\int_{-1}^1 f(x)\,\mathrm dx.$$ Any tips/solutions? I don't even ...
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0answers
47 views

Some basic Topological proof help please.

Basically im really bad at proofs and i havent done math in almost a year and decided id like to learn topology on my own... just want someone to be really critical on my solutions please also i would ...
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0answers
25 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
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0answers
109 views

A trigonometric integral with sin(cos(x)) in exponent

Evaluate: $$\int_0^{\pi} x\csc^{\sin(\cos x)}(x)\,dx$$ I honestly don't know how to deal with this case. If I apply the property $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$, I get: $$\int_0^{\pi} ...
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1answer
66 views

indefinite integral of inverse trigonometric function

How can we integrate the following I am unable to find a suitable method or formula by which i can get the value of this integral. $$ \int {\sqrt{\cot^{-1} x}} + {\sqrt{\tan^{-1} x}} \, dx$$
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4answers
204 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
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1answer
37 views

Equation of the tangent line, derivative denominator is 0

I was wondering what the equation of the tangent line is at $x=0$ for the following function: $$\frac{(3x^2+1)^4}{(x^5-4x^2)^2}$$. The derivative works out to be: ...
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2answers
47 views

Integrate the function $w=x+y^2$

I have the following exercise: We want to integrate the function $w=x+y^2$ and we have a path that begins from $A(0,0)$ and reaches at $B(1,1)$. $$$$ Could you give me some hint what I am supposed ...
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2answers
37 views

Is the series $\frac{1}{(n+1)^p}-\frac{1}{(n-1)^p}$ where 0<p<1 convergent or divergent?

Sorry for my bad English. I really suspect it is convergent. But I can't prove it. Since ${x^p}$ is not derivable at x=0, I can't using taylor expansion to find the order of infinitesimal, thus ...
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1answer
53 views

Finding solution to a unidirectional nonlinear wave equation

I can do the parts a), b) and c) and find that in part c) that the condition in which the solution will break down is when $1+tf'(x-tu)=0$ However I am unable to part d) I tried ...
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1answer
168 views

Evaluating dirac deltas in the following integrals

I am able to do parts a) and b). I think that the other three parts are similiar. For part c) I get that $\displaystyle u(x,t)=\frac{1}{2}\int_0^{t} ...
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3answers
62 views

How to solve limits when x approaches minus value?

How to solve the following limit? I can't think of a way to find $\delta$ values because x approaches a minus value . $$\lim_{x\to -3} x^2+5x+6=0$$
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4answers
112 views

How to find $\lim_{n\rightarrow 0} \cos(\frac{\pi}{n} \sin n \cos n)$ and $\lim_{n\rightarrow 0} \frac{n}{\sin(2n) - \cos(\frac{n}{2}) +1}$

How can I determine these limits: $$ a) \lim_{n\rightarrow 0} \cos(\frac{\pi}{n} \sin n \cos n)$$ $$ b) \lim_{n\rightarrow 0} \frac{n}{\sin(2n) - \cos(\frac{n}{2}) +1}$$ Note I cannot use ...
2
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2answers
52 views

How to solve this first order non-linear ODE

I am struggling to solve this first order ODE. $$ u'\,^2 = a u^2 + b u^3$$ Mathematica gives me, $$ u(v) = \dfrac{a}{b} \mathrm{sech} \left( \dfrac{1}{2} \sqrt{a} \ (v + c) \right)^2 $$ So I ...
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1answer
57 views

$\int_{-\pi}^{\pi} \cos nx \, dx$ is always 0 when actually integrated.

It's pretty apparent $\int_{-\pi}^{\pi} \cos nx \, dx$ should be 2$\pi$ when n = 0. However it seems $\int_{-\pi}^{\pi} \cos nx \, dx$ is always 0 when actually integrated given that n is integer. I ...
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1answer
45 views

Second order ordinary differential equation

I have to solve the following ODE: $$ \frac{f''(x)}{f(x)}=\frac{a''(x)}{a(x)}, $$ where $a(x)$ is a well behaved function. It is easy to check that the solution is given by $$ f(x)=C a(x) + D a(x) ...
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2answers
49 views

gamma function question relating to normal distribution

I'm trying to show that $\Gamma(1/2)=\sqrt\pi$. A hint I've been given is to use a change of variable and then relate it to normal distribution density. However, I'm really confused as to how I would ...
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1answer
279 views

Using calculus to determine maximum height

Use calculus to determine how long it takes the sphere to reach its maximum height, also determine what the maximum height is. The original question is A life form standing on the surface of an ...
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2answers
199 views

Getting rid of square root in this integration

How do I get rid of this square root so that I can integrate it? $$S = 2 \pi a ^2 \int_0^{\sqrt{2}} t \sqrt{4t^2 + 1}\ dt$$
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1answer
115 views

Volumes of solids of revolution - guide to solve similar problems

So, I'm preparing for an exam and I'm stuck with problems with volumes of solids of revolution. I have two examples: a, find the volume of a solid, described as: $T=\{(x,y,z) : x^2 + y^2 - z^2 ...
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1answer
74 views

Indefinite Integration of $(\arctan(x))^2$

$$ \mbox{Hello. I was wondering how to integrate this:}\quad \int\arctan^{2}\left(x\right)\,{\rm d}x $$ Do I first do a u-substitution first ?.
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2answers
38 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
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2answers
47 views

Derivative $\operatorname{arccos}x +\operatorname{arcsec}\frac{1}{x}$

My calculus text has this problem, but it seems to me that the inverse cosine and inverse secant functions do not share domains. How can this function be defined? Have a derivative? What is ...
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1answer
102 views

non-removable singularities of a function are essential singularities of the composition function

Let $f$ be a non-constant enrire function on $\mathbb{C}$ such that $f(z+i)=f(z)$ for all $z$. Let $U$ be an open subset of $C$ and $z_0\in U$. Let $g:U\setminus\{z_0\}\longrightarrow\mathbb{C}$ be ...
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0answers
128 views

A farmer with $800$ ft of fencing wants to enclose a rectangular area and then divide it into four pens

A farmer with $800$ ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of ...
3
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1answer
75 views

Line Integral with Lemniscate

I am being told to integrate the function $$f(x,y) = x + y$$ over the right loop of the lemniscate $$r^2 = a^2cos(2\theta)$$ Now, we take $x = rcos(\theta)$ and $y = rsin(\theta)$, and as a result ...
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1answer
96 views

an amazing complex integral along unit circle

Let $f$ be an entire function on $\mathbb{C}$. Let $z_0\in\mathbb{C}$. Let $C=\{z\in\mathbb{C}\mid |z-z_0|=1\}$. Suppose $f(z)\neq f(z_0)$ for any $|z-z_0|\leq 1$, $z\neq z_0$. Given $f'(z_0)=2$, ...
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1answer
77 views

Proving the mean value inequality in higher dimensions for a differentiable function (rather than $C^1$)

For a function $f: [a,b] \to \mathbb R$, the mean value theorem is an equality. Further, the differentiability imposed on $f$ is fairly weak: $f$ need only be continuous on $[a,b]$ and differentiable ...
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2answers
87 views

Solving the Integral using $\ln|u|$ [closed]

Can Someone help me solve this $$ \int\frac{19\tan^{-1}x}{x^{2}}\,dx $$ We have been told to use $\ln|u|$ and $C$. Thanks!
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1answer
74 views

The only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x)$ is $f(x)=ce^{x}$

Prove that the only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x) \mspace{1ex} \forall x\in \mathbb R$ is $f(x)=ce^{x}, \forall x\in \mathbb R$, and for some ...
2
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1answer
137 views

does this sequence of functions converges uniformly to Dirichlet function?

Let $r_{1},r_{2},...$ a sequence that includes all rational numbers in $[0,1]$. Define $$f_n(x)=\begin{cases}1&\text{if }x=r_{1},r_{2},...r_{n}\\0&\text{otherwise}\end{cases}$$ this sequence ...
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2answers
55 views

Cylinderical Shell Calculus Question

How should i solve this? Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y = $19x^2$, $y = 0$ , $x=1$ I Know ...
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3answers
91 views

How to calculate $\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}} dx$?

How to calculate? $$\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\, \mathrm dx$$ I try to let $x=\cos^2 t$, then $$\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}=\tan\frac t2,\; dx=-2\sin t\cos t\,\mathrm dt $$ so ...
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2answers
79 views

Sum Of Multiple Series in Closed Form

Consider Following Expression: $S(k)=\frac{a_1 (-1)^k}{k!}+a_2 \sum _{l_1=0}^k \frac{(-1)^k}{l_1! \left(k-l_1\right)!}+a_3 \sum _{l_1=0}^k \sum _{l_2=0}^{l_1} \frac{(-1)^k}{l_2! \left(l_1-l_2\right)! ...
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1answer
113 views

Compute $\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$

Compute $$\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$$ Let $t=\tan\frac x2$, then $$\sin x=\frac{2t}{1+t^2},\; \cos x=\frac{1-t^2}{1+t^2},\; \mathrm dx =\frac{2\mathrm ...
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1answer
37 views

How do I prove this statement?

I have to prove that if $$u=t^{\lambda}y(z)$$ and $$z=\frac{x}{\sqrt{t}} \,\,,$$ then $$\frac{\partial{u}}{\partial{t}}=\frac{\partial ^{2}{u}}{\partial{x}^{2}} \Rightarrow ...
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1answer
122 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
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1answer
61 views

Finding the integral of a trig function using a matrix

It can be shown that Ɓ = {1, $\cos(t)$,…$\cos(6t)$ and Ƈ = (1,$\cos(t)$,…$\cos^6$(t)} span the same subspace of Ƈ(ℝ) a. Use an appropriate change of coordinate matrix to find $cos^6$(t) in terms of ...
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1answer
34 views

Difference between derivative and its approximation

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
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4answers
74 views

Differentiating fraction $\phi(\theta) = \frac{\theta}{(1+\theta^2)}.$

I graduate high school many years ago, and I have no idea where to find how to differentiate this formulae. $$\phi(\theta) = \frac{\theta}{(1+\theta^2)}.$$ I have a solution of it, but I know not ...
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2answers
96 views

Definite Integral of $e^{\large x^2}$

I know there's no elementary antiderivative of $e^{\large x^2}$. But what if there's a definite integral like $$\int_0^1e^{\large x^2}\ dx\ ?$$ I tried using basic definite integral property like ...
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3answers
124 views

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found?

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy $ be found, if $D$ is $x$ O $y$ axis? So far I have done it this far: ...
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3answers
74 views

How to find $\frac{dy}{dx}$ at the point $(-2,0)$?

Could someone please demonstrate this problem? I can't seem to get it right. Find $\dfrac{dy}{dx}$ at the point $(-2,0)$, if $y^3 = x^3 + e^x \sin y +8$.
2
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2answers
75 views

Finding the radius of convergence for $\sum_{n=0}^{\infty} \frac{(n!)^kx^n}{(kn)!}$

I'm reviewing some calculus concepts right now, and this series stumped me. I'm trying to figure out what its radius of convergence is, but it's not falling for any of my old tricks. Using the ...
2
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2answers
76 views

Limit of a function with a defined integral

I have the next limit: \begin{equation} \lim_{x\to\ 0}\displaystyle\frac{\displaystyle\int_0^{x^2}{\frac{1-\cos{t^2}+at^4}{t}}dt}{(1-\cos{(\frac{x}{3})})^4} \end{equation} I've tried to solve it by ...
3
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1answer
153 views

Does the sum $\sum _{n=1}^{\infty }\left(\frac{\sin\left(n\right)}{n}\sum _{k=1}^n\left(\frac{1}{k}\right)\:\right)$ absolutely converge?

I have the following sum: $$\sum _{n=1}^{\infty }\left(\frac{\sin\left(n\right)}{n}\sum _{k=1}^n\left(\frac{1}{k}\right)\:\right)$$ And I need to determine if it absolutely converges, conditionally ...
3
votes
4answers
88 views

Showing that $3x^2+2x\sin(x) + x^2\cos(x) > 0$ for all $x\neq 0$

I got this question: Show that for all $x\neq 0$, $3x^2+2x\sin(x) + x^2\cos(x) > 0$ I tried to show it but got stuck.