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

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1
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0answers
8 views

What is the general Taylor Expansion for the following function of a function.

guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: ...
0
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0answers
5 views

How do I give an upper bound of interpolating error?

So i am stuck trying to find upper bound on the absolute value of interpolating error.I calculated the second taylor polynomial and I am completely lost as to finding an upper bound for interpolating ...
-3
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0answers
13 views

infinite series convergence test 2

Test the convergence of the series $\sum_{0}^{\infty}e^{-n^2}$ Does the integral $\int_{-1}^{1}\sqrt{\frac{1+x}{1-x}}dx$ exist?
0
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2answers
22 views

Finding the points on a curve, closest to a specific point

Find the point(s) on the curve $y^3=x^2$ closest to the point $P=(0,4).$ I understand that there is a way to solve this, using the distance formula, however this turns out to seem rather complicated. ...
0
votes
1answer
19 views

Determine the Fourier series considering the derivative of a function

Let $f\left(x\right)=x^2+1$ on the interval $\left[-\pi,\pi\right]$, which is extended periodically to $\mathbb{R}$. I have calculated the Fourier series of $f$ to be ...
0
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2answers
37 views

Question on Rolle's theorem involving roots

Use Rolle's theorem to show that $f(x)=x^3-\frac{3}{2}x^2+\lambda$, $\lambda \in \mathbf{R}$ never has 2 zeroes in $[0,1]$. I started by assuming that $\exists$ $2$ zeroes in$[0,1]$ Then ...
0
votes
1answer
20 views

Uniform convergence of a function composition

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
1
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2answers
27 views

Simplification idea for finding antiderivative

Is there a simple way of finding the anti-derivative $F$ (i.e. $F(x)=\int f(x)dx$) of $$f(x)=\frac{1}{(\sqrt{x}-1)\sqrt{x}}$$ I've managed to do it by 2 by parts integrations in row, but that took ...
0
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1answer
24 views

Finding an antiderivative

$\newcommand{\dx}{\,\mathrm dx}$ I need to find the following: $$\int \sqrt{\frac{1-x}{1+x}} \cdot \frac 1x \dx$$ Firstly, it has to be that $x\in (-1, 0)\cup (0,1]$. From this, it is implied ...
0
votes
1answer
10 views

How do I examine the function on continuity? How do I discuss and sketch the level lines of f?

How do I examine the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $f (x, y) = (2x- y)\ \rm{sign}(4x-y)^2$ for continuity? How do I discuss and sketch the level lines of $f$?
0
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0answers
18 views

Harmonic function using Green's identity

Let$\ u:\mathbb{R}^{3}\rightarrow \mathbb{R}\ $ be a harmonic function and $\ 0< \varepsilon < \frac{1}{2}\ $ such that the following holds: $$\left | z \right |^{1-\varepsilon }\left \| ...
0
votes
2answers
30 views

Find the point of function closest to a point not on the function

I am asked to find the point on the graph of $y = x^2 $ that is closest to the point Q $(0,3)$. Let P be the point on the graph with coordinates $(x,x^2)$ I searched on Google and found this answer ...
1
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4answers
28 views

An equality of sets involving two directions

LEt $A,B$ be sets. Prove that $A \subseteq B \iff P(A) \subseteq P(B) $. Attempt: First, take any element of $P(A)$, say $Y$, we know by definition that $Y \subseteq A$ and so $Y \subseteq B$ ...
1
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0answers
22 views

Stirling's approximation from Euler-Maclaurin formula

I try to derive Stirling's approximation from Euler-Maclaurin formula with form: ...
3
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0answers
27 views

Why did the number of types of integrals got lower from the beginning of the $20^{th}$ to this day?

There is an old $(\text{circa } 1930)$ and interesting book in calculus: Edwards' Treatise on Integral Calculus. This book has a very complete list of cases of integrals, for example, these ...
0
votes
4answers
36 views

derivative of complex function of higher order

i am not able to find the value of the 2014th order derivative of the function $f(z)=z^2e^{kz}$ at zero. Any help will be appreciated. thank you
2
votes
0answers
30 views

How do you show that $\int_{0}^1\frac{dx}{x^x}=\sum_{k=1}^\infty\frac{1}{k^k}$?

My task is this: i) Find the sum to$$1-x\ln x +\frac{1}{2}(x\ln x)^2-\ldots+\frac{(-1)^k}{k!}(x\ln x)^k+\ldots$$ (ii) The great norwegian mathematician Atle Selberg showed that ...
0
votes
1answer
13 views

An identity regarding symmetric difference of sets

Let $A,B$ be sets and define $A \triangle B = (A \setminus B) \cup (B \setminus A )$. Show that $A \triangle B = (A \cup B) \setminus (A \cap B )$. Attempt: Suppose $x \in A \triangle B$. By ...
0
votes
1answer
11 views

Is there a analytic algorithm to solve for a partially specified constant present in function & its derivative with the rate of change at an x value?

I have a Calculus problem that I am not entirely happy with how I solved it. Given the following information: $$y = x^{k} + x^{k-2}$$ $$(k \in \mathbb{N}) \wedge (k \mod{2} \neq 0) \wedge (k > ...
4
votes
3answers
136 views

Why is $\log(1+e^x) - \frac{x}{2}$ even?

I'm dealing with Fourier series and I'm trying to figure out $\log(1+e^x) - \frac{x}{2}$ is even??? I've tried the $f(-x) = f(x)$ method but it doesn't give me the equality. But I've plotted it, and ...
0
votes
2answers
45 views

Is substitution an axiom (Apostol Calculus Vol. 1)?

I am just starting Apostol's Calculus Vol. 1, and I have no experience with rigorous mathematics. In his introduction, the first proof he gives is that if $a+b=a+c$, then $b=c$. It says, "Given ...
0
votes
1answer
26 views

Prove that if $A \subseteq [0,\infty)$ and $B \subseteq (-\infty,0]$, then $\sup(AB) = \sup{B} \inf{A}$

Prove that if $A \subseteq [0,\infty)$ and $B \subseteq (-\infty,0]$, then $\sup(AB) = \sup{B} \inf{A}$. Note: Assume $A,B$ are nonempty subsets of $\mathbb{R}$ and $\alpha \in \mathbb{R}$ and ...
0
votes
1answer
12 views

How to derive this equality by change of variables for the Riemann integral?

Define Diff$_hf = \frac{f(x+h)-f(x)}{h}$. Why is the following true? $$ \int_u^vDiff_hf=\frac{1}{h}\int_0^hf(v+t)-f(u+t)dt $$ Note: this is from royden's real analysis page 122.
1
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3answers
33 views

Non-compact subsets of a metric space $(X,d)$.

I'm trying to come up with an example of a metric space $(X,d)$ such that a subset $A \subset X$ is not compact, but is closed and bounded. Essentially I want to find an example that shows that a ...
2
votes
2answers
37 views

How to find the set of values $S$ where $f$ is not differentiable?

Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no ...
0
votes
2answers
33 views

Prove that $\sup(AB) = \inf{A} \cdot \inf{B}$

If $A,B \subseteq (-\infty,0]$, prove that $\sup(AB) = \inf{A} \cdot \inf{B}$. For this question you can assume that if $A,B \subseteq [0,\infty)$ then $\sup(AB) = \sup{A} \sup{B}$ as that was an ...
-1
votes
2answers
20 views

Calculate the plane equation of 2 vectors. [on hold]

Which type should I use in order to calculate the plane equation that is defined by 2 vectors, let's say V1 $\langle{1,2,3}\rangle$ V2 $\langle{4,5,6}\rangle$.
0
votes
2answers
43 views

Decreasing integral sequence

How does one show that $I_n = \int\limits_0^1 x^n e^x dx$ is decreasing? The best I came up with is this: $I_{n-1} - I_n= \int\limits_0^1 e^xx^{n-1}(1-x)dx$, but how do we go from here? I'd ...
3
votes
2answers
24 views

Relative Extrema - First-derivative test of : $f(x)=x^5-5x^3-20x-2$

Find the relative extrema of the function by applying the first-derivative test: $$f(x)=x^5-5x^3-20x-2$$ So I found the $f'(x)$ $$f'(x) = 5x^4-15x^2-20$$ Now, I'm trying to find the critical ...
0
votes
0answers
14 views

Factoring $R(ry')'-y(rR')'=[r(Ry'-R'y)]'$

In a problem this formula was used and I'm not seeing how this factor using the chain rule was derived. Other than calculating the derivative of the two that someone else already solved and showing ...
0
votes
1answer
37 views

If $f\colon\mathbb{R}\to\mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2$ for every $x\in\mathbb{R}$, then $f$ is differentiable at 0.

If $ f\colon \mathbb{R} \to \mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2 $ for every $x \in \mathbb{R} $, then $f$ is differentiable at $0$. The solution provided uses delta-epsilon to prove ...
3
votes
1answer
67 views

Linear ODE with non-constant coefficients

I have encountered some problem in computing the explicit solution for the following ODE: $$x^\prime(t) = (2t-1) x(t)-1, \quad x(0) =: x_{0}$$ The formula that I have used to solve it is: ...
1
vote
1answer
55 views

What sum to $\sum_{n=0}^\infty\frac{x^n}{(n+1)!}$ in its convergence radus?

My task is this: Find the sum to $$\sum_{n=0}^\infty\frac{x^n}{(n+1)!}.$$ in its convergence radus. My work so far: By ration test we get ...
0
votes
0answers
28 views

Mean value theorem, finding value of log [on hold]

With the help of mean value theorem, if $0<\theta<1$, then the value of $\log_{10}(x+1)$ is
1
vote
2answers
23 views

Density of real numbers and density function

In Quantum mechanics, given a certain material, it is possible to write the density of energy states $\rho (E)$ as a function of $E$. That is: let's consider all the real values contained in the ...
0
votes
0answers
58 views

Trivial equation but with integral

Let's consider the following equation: $$\int_{a}^{x} f(t)dt = K$$ where $a, K \in \mathbb{R}$. Suppose that $a, K$ and the function $f$ are known and that the equation should be used to determine ...
-3
votes
0answers
40 views

Please help me with these 4 questions, thanks. [on hold]

This is the image with the questions: (Large version)
1
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1answer
25 views

Why the radius of curvature of a curve is independent of the choice of the coordinate axes.

The radius of curvature of a curve $y=f(x)$ is given by $\rho=\frac{(1+(\frac{dy}{dx})^2)^{\frac{3}{2}}}{\frac{d^2y}{dx^2}}$. I know this formula.Its derivation is also given in the book and i ...
1
vote
0answers
18 views

Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...
1
vote
1answer
19 views

Find the surface area of the shape formed by the boundary of $\frac{z^2}{4}=\frac{x^2}{2}+\frac{y^2}{4},z=2x+4y, z\geq 0$

$$\frac{z^2}{4}=\frac{x^2}{2}+\frac{y^2}{4},z=2x+4y, z\geq 0$$ I know that this is a cone that is cut by a plane, but I do not know how to find the projection onto $xOy$. I need this because then I ...
0
votes
0answers
19 views

If $y=[\log{x+√(1+x)²)}]²$ and n is positive integer then (yn+2)(0) is

If $y=[\log{x+√(1+x)²)}]²$ and n is positive integer then $(y_n+2)(0)$ is
4
votes
5answers
102 views

Roots of $x^{101}-100x^{100}+100=0$

I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots. Some can give me hint for solving this please. Thanks for your time.
0
votes
3answers
33 views

Prove that if $A,B \subseteq [0,\infty)$, then $\sup(AB) = \sup(A)\sup(B)$

Prove that if $A,B \subseteq [0,\infty)$, then $\sup(AB) = \sup(A)\sup(B)$ and $\inf(AB) = \inf(A)\inf(B)$. We show the property for the supremum first. we see that since for any $a \in A$ and $b ...
0
votes
1answer
40 views

Continuous function, finding its value [duplicate]

If a function $f: \mathbb{R}\to\mathbb{R}$ is continuous and $f(x+y) = f(x) + f(y)$ for all $x,y\in\mathbb{R}$, then what is this function $f(x)$?
0
votes
1answer
22 views

Parametric equation of tangent

Find parametric equation for the tangent line at $(1,3,3)$ to the curve of intersection of the surface $z=x^2y$ and a) the plane $x=1$ b) the plane $y=3$ I found out $\frac{dz}{dx}$ and ...
0
votes
1answer
57 views

Finding value of functions $f(x) g(x)$ [on hold]

If $f(x)=2x^3+4x^2+3x+2$ and $g(x)=2x^3+x^2+4$, where $f(x), g(x) \in \mathbb{Z}_5[x]$ then $f(x) g(x)$ is equal to ?
5
votes
2answers
60 views

A tricky limit (indeterminate form)

While tutoring I came upon this limit I know that this limit is obviously 1, but how would I show this formally $$\lim_{\eta\rightarrow\infty}(2\eta + 5)^x-(2\eta)^x + 1$$ where $x\in (0,1)$ I've ...
1
vote
1answer
17 views

signature of the quadratic form: $f(x,y,z) = xy+yz+xz$

I am asked to find the signature of the following quadratic form: $f(x, y, z) = xy+yz+xz$ I have found that matrix wise, $f(x,y,z)= \begin{bmatrix}x&y&z\end{bmatrix}. ...
1
vote
1answer
31 views

When do I use the 'plus-minus' sign when square rooting both sides of an equation? (example in main body).

Above is the image of an integration by substitution question that I was doing; the answer can only have a plus sign in front (if you were to differentiate the answer to check if it's correct). ...
0
votes
1answer
52 views

Solve equation $\lim_{n \to \infty}\cos (nx)=1$

Solve equation $\lim_{n \to \infty}\cos(nx)=1$ Ok, $ \cos0 = 1$, but $0\times \infty \neq 0$ So, i think $x = \frac{0}{n}$ will work: $$\lim_{n \to \infty}\cos(n \times \frac{0}{n})=\lim_{n \to ...