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

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1
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2answers
24 views

what is the value of $\theta$ used in calculate volume bounded by $z=x^2+y^2$ and $x^2+y^2=2x$

This is an example from my textbook, it explain everything well except the reason why $\frac{\pi}{2} < \theta < \frac{-\pi}{2}$ but not $2\pi < \theta < 0$. It's not explained and I can't ...
3
votes
10answers
107 views

Point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and the $x$-axis

How can I algebraically (without looking at the graph) find the point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and $x$-axis, in the interval $[0, \pi]$?
1
vote
1answer
36 views

Fourier series of constant on $2\pi$ intervals

I want to find a fourier expansion of only sines representing $g(x) = 1$ on the interval $[0, \pi]$. So I extend the function on $[-\pi, \pi]$ such that it is odd, and calculate $$b_k = \frac 1\pi ...
1
vote
2answers
42 views

Maxima and Minima of Functions of Two Variables $ f(x,y) = e^{x+y^2}\cdot y $ and $ f(x,y) = e^{x^2-y^2}\cdot y $

I'm having trouble finding the local minimum and maximum of the next functions: $$1. f(x,y) = e^{x+y^2} \cdot y $$ $ f_x'= (e^{x+y^2}\cdot y) ; $ $ f_y'= (e^{x+y^2}(1+2y^2)) $ $$ 2. f(x,y) = ...
2
votes
5answers
66 views

$U_n=\int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ .

$U_n= \int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ where Find $\lim_{n\to \infty} U_n$ without finding the integration I don't know how to start
-1
votes
1answer
66 views

Find the x-coordinates of two other points of inflection of $f(x)= \int \frac{x+1}{x^2+1}$, given there is an inflection point at $(1,1) $

$$f(x)= \int {\frac{x+1}{x^2+1}}$$ I have to find the x-coordinates of two other points of inflection, given there is an inflection point at (1,1). My approach is to differentiate the equation, and ...
1
vote
1answer
82 views

Calculating in closed form an integral in Airy function

Can we hope for a nice closed form for the integral below? $$\int_0^1 \frac{\displaystyle \text{Ai}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 t}}\right)^2+\text{Bi}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 ...
1
vote
1answer
35 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
1
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2answers
67 views

$\Delta u$ is bounded. Can we say $u\in C^1$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary. Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial ...
2
votes
0answers
27 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
0
votes
1answer
34 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
4
votes
3answers
76 views

Simple Logarithms Equation

$$3^x = 3 - x$$ I have to prove that only one solution exists, and then find that one solution. My approach has been the following: $$\log 3^x = \log (3 - x)$$ $$x\log 3 = \log (3 - x)$$ $$\log 3 ...
-2
votes
0answers
58 views

Does $f(x)=x^3+x^4\sin\frac{\pi}{x}$ have the inflection point? [closed]

Does the function $f(x)= x^3+x^4\sin\frac{\pi}{x} \,\,\,$ have an inflection point? Is it $(0,0)$?
1
vote
1answer
14 views

Question about r(t), movement along line

So I'm studying for an exam in calculus when i came across the concept of objects moving along a curve. I have a general idea of how to calculate speed, velocity and such when r(t)(position vector I ...
0
votes
2answers
31 views

Piecewise $\mathscr C^1$ and piecewise continuous

I'm a little bit confused in piecewise continuity of a function. Say, if we have an odd function like $f(x) = x$ defined on the open interval $(0, \pi)$. We then extend it to a period $2\pi$ function ...
1
vote
2answers
57 views

Proving a theorem about Fourier coefficients

I need to prove this: Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$. Can someone please let me ...
1
vote
3answers
31 views

First principle derivative of a square root and conjugates

I'm trying to find the derivative of the equation: $$g(x)=\sqrt {x+2}-3x^2$$. I can find the solution just fine using the power rule but am finding trouble with First Principles. Essentially, I ...
2
votes
4answers
382 views

Sum function of a series

Does anyone know what is the sum function $f(x)$ of the series $\displaystyle\sum_{n=1}^\infty \frac{\cos(nx)}{n^2}$? I have no idea how to find a sum function... Any help would be appreciated.
0
votes
1answer
16 views

Coefficients of general Fourier Series

I know how to compute coefficients of Fourier Series on an interval of $2\pi$. But in this case I need to find the sine series of $f(x)=b$ on the interval $x \in [-L,L]$. Can someone please let me ...
1
vote
1answer
23 views

Question regarding path independence

I've been wracking my brain to try and figure out why the following works: The question is asking whether $$\int F \, dr $$ is independent of path. We have a hint, that is (compute $$\int_a F \, dr ...
1
vote
5answers
138 views

Tangent line parallel to another line

At what point of the parabola $y=x^2-3x-5$ is the tangent line parallel to $3x-y=2$? Find its equation. I don't know what the slope of the tangent line will be. Is it the negative reciprocal?
1
vote
1answer
28 views

General Chain Rule

Product Rule: For a two $\mathcal{C}^{\infty}(\mathbb{R})$ functions, $u(x)$, $v(x)$ we have $$\frac{d^k}{dx^{k}}[u(x)v(x)]=\sum_{j=0}^k ...
-3
votes
1answer
16 views

Sketching the graph of a sequence of functions [closed]

Does anyone know how to sketch graphs of a sequence of functions? Say, I need to sketch $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ for $x \in [0, 2]$. But I have no idea where to start. Can someone please help? ...
4
votes
2answers
43 views

Example of a non-polynomial function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x)$ is negative for $x<0$ and positive for $x \ge 0$.

I have a bunch of polynomial functions example easily (e.g. $x^2$), but have trouble coming up with a non-polynomial function. I was thinking of defining $f(x) = e^{-x}$ for $x<0$ and $f(x) = ...
0
votes
3answers
43 views

How do I solve this deceleration problem?

Question: A car is traveling at 100km/hr, when the driver sees an accident 80 meters ahead. What constant deceleration is required to stop the car in time to avoid a pileup? So far I have approached ...
0
votes
0answers
43 views

Math Extended Essay Topics [closed]

I am doing an IB Extended Essay in Math and need some help finding a viable topic. It needs to be 4000 words and high school level (calculus). I'm interested in game theory and triangulation, but ...
0
votes
1answer
41 views

How to show $G$ is a perfect set that contains no rational points?

For $E:=[0,1]$, since $\Bbb Q\cap E$ is enumerable, let it be $\{q_1,q_2,\cdots\}$. If I remove the elements of $V_1:=(q_1-\frac1{10},q_1+\frac1{10})$ from $E$, I obtain a closed (and compact) set ...
4
votes
1answer
83 views

Prove that for every positive integer $n, \exists c_n$ such that $f(c_n) = f(c_n+1/n)$

Let $f:[0,1] \to [0,1]$ be a continuous function with $f(0) = f(1)$. Prove that for every positive integer, $n, \exists c_n \in [0,1]$ such that $f(c_n) = f(c_n+1/n), c_n \in [0,1-\frac{1}{n}].$ With ...
1
vote
2answers
64 views

A curve has equation $\arctan(x^2)+\arctan(y^2)=\pi/4$ [closed]

A curve has equation $$\arctan(x^2)+\arctan(y^2)=\frac{\pi}4$$ a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$. b) Find the gradient of the curve at the point where $x=\frac{1}{\sqrt{2}}$ and ...
2
votes
1answer
35 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 ...
0
votes
0answers
24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
-2
votes
0answers
30 views

hello, as I simplify to get a (1 - cos ( x ) ) , can not remember the exact passage or identity. help [closed]

as I simplify to get a (1 - cos ( x ) ) , can not remember the exact passage or identity. help -cos(x)+1 = 1-cos(x)
1
vote
1answer
93 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
1
vote
1answer
31 views

Need help with this question concerning compact spaces

Let the set be given like in the following manner: $$\{x_n: n\in\mathbb N\}\subset \mathbb{R^n}$$ $$l^2=\left\{\{x_{n}\}_{n=1}^{\infty}\,\Big|\, \sum_{n=1}^{\infty}|x_n|^2<\infty\right\}.$$ Prove ...
0
votes
2answers
24 views

Moment of inertia about the origin of an ellipsoid?

Find the moment of inertia about the origin of an ellipsoid. Heres what I did but I believe it is incorrect: $$I_o= \iiint_{V_e}{(x^2 +y^2 +z^2)\rho dx dy dz} $$ Making Substitution of $aX=x \ bY=y \ ...
0
votes
0answers
16 views

End-point as point of tangency

Can an end-point be a point of tangency? For example in the function $f(x)=8x^{3/2}$ can the point $(0,0)$ be a tangent point?
1
vote
2answers
36 views

Upper and lower Riemann sums problem

Let $c > 0$ and $f(x) = x, x\in [0,c].$ Let $P = \{x_0, x_1, x_2,...,x_n\}$ be a partition of $[0,c]$ with $x_i = \frac{i}{n}c, i = 0,1,2,...,n.$ Find $U(P,f).$ Find $\lim_{n \to \infty} U(P,f).$ ...
1
vote
2answers
96 views

Find a closed form of the series $\sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2}$

The question I've been given is this: Using both sides of this equation: $$ \frac{x}{1-x} = \sum_{n=1}^{\infty}x^n $$ Find an expression for: $$ \sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2} $$ Any help ...
1
vote
2answers
43 views

How do I solve a double integral with an absolute value?

Given the following integral $$\int_{y=0}^1 \int_{x=0}^1|x-y|(6x^2y) \, dx \, dy$$ how do I change the limits of integration? According to my textbook, it is $$\int_{y=0}^1 ...
2
votes
2answers
98 views

Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$

The integral is $$\int \dfrac{(1 + x)\sin x}{(x^2 + 2x)\cos^2 x-(1 + x)\sin2x}dx.$$I've tried the problem by first multiplying both the numerator and denominator by $\sec^2 x$ but couldn't do justice. ...
0
votes
0answers
41 views

Help in partial derivative during maximization for estimation problem

The joint pdf is: $$P((\mathbf{X,y}) |y_n, \theta) = \frac{1}{\sqrt{2 \pi \sigma^2_c}} \exp \big(\frac{-(c_0)^2}{2 \sigma^2_c} \big) \prod_{n=1}^{N-1} \frac{1}{\sqrt{2 \pi \sigma^2_w}} \exp ...
4
votes
6answers
154 views

Finding $\frac {a}{b} + \frac {b}{c} + \frac {c}{a}$ where $a, b, c$ are the roots of a cubic equation, without solving the cubic equation itself

Suppose that we have a equation of third degree as follows: $$ x^3-3x+1=0 $$ Let $a, b, c$ be the roots of the above equation, such that $a < b < c$ holds. How can we find the answer of the ...
1
vote
4answers
131 views

Explain the integral of $1/x = \ln |x| + \mathrm{C}$ graphically as sum of area?

I am unable to interpret the integral $$\int {1\over x}{\rm d}x=\ln|x|+\mathrm{C}$$ Graphically as area under the curve of $1/x$ (as the definition of the integral). Can somebody please ...
0
votes
0answers
25 views

Use the Minkowski inequality to show that, for each given location $(a_i , b_i )$, $d_i(x, y) = \sqrt{(x-a_i )^2+(y-b_i)^2}$ is convex.

Use the Minkowski inequality to show that, for each given location $(a_i , b_i )$, $d_i(x, y) = \sqrt{(x-a_i )^2+(y-b_i)^2}$ is convex. I know the Minkowski inequality: $||f+g||_{p} \leq ...
-1
votes
3answers
37 views

Line touching a curve at a single point

A straight line $y=2x-b$ touches a curve $y=3x^2+2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$? I don't know where to begin. Please help. Thanks!
0
votes
2answers
28 views

Use the graph of f to find the limit

How can I use the graph of f to find the limit? $\displaystyle \lim_{x \to 1^−}f(x)$ $\displaystyle \lim_{x\to 1^+}f (x)$ I am not sure where to start
0
votes
2answers
46 views

Prove that the sum of convex functions is again convex.

I must to prove that the sum of convex functions is again convex. I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq f(x_1)+(1-t)f(x_2)$ - this the first convex function, then I have the ...
0
votes
0answers
21 views

Laplace transform of $\left(1 - \exp(-t^y)\right)^n$

Does anybody know how to determine the Laplace Transform of $$f(t) = \left(1 - \exp(-t^y)\right)^n$$ where $y$ is a positive real number, $n$ is an integer. If we use the logarithm, can lead to a ...
1
vote
2answers
47 views

Maximizing area under curve

I came across this problem in TMH mathematics for jee.I tried finding the derivative to the curve but I got stuck while evaluating the area of triangle in terms of tangent to the curve.How should I ...
4
votes
7answers
82 views

Differentiate the Function: $y=2x \log_{10}\sqrt{x}$

$y=2x\log_{10}\sqrt{x}$ Solve using: Product Rule $\left(f(x)\cdot g(x)\right)'= f(x)\cdot\frac{d}{dx}g(x)+g(x)\cdot \frac{d}{dx}f(x)$ and $\frac{d}{dx}(\log_ax)= \frac{1}{x\ \ln\ a}$ $(2x)\cdot ...