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

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-3
votes
2answers
46 views

Find the Taylor's Series for $f(x)=x^3-10x^2+6$ about $x_0=3$ [on hold]

Please help me. I want a solution for this question Find the Taylor's Series for $$f(x)=x^3-10x^2+6$$ about $x_0=3$.
1
vote
2answers
32 views

Prove that $\sup (-A) = -\inf A$

Prove that $\sup (-A) = -\inf A$. Note: Assume $A$ is a nonempty subset of $\mathbb{R}$ and $\alpha \in \mathbb{R}$ and define $\alpha A = \{\alpha A \mid a \in A\}$. I said let $x \in A$. Then ...
5
votes
2answers
82 views

Is the series convergent

Is series $\sum_1^\infty \frac{\ln(1+1/2) \ln(1+1/4) \cdots \ln(1+1/(2n))}{\ln(1+1/1) \ln(1+1/3) \cdots \ln(1+1/(2n-1))} = \sum_{n=1}^\infty \prod_{m=1}^n \ln(1+1/(2m))/(\ln(1+1/(2m-1))$ convergent ?
1
vote
2answers
24 views

How to simplify from this thing to this (double derivative, stuck in the alegba part)

In my maths class I am doing double derivative to find concavity of the equation if I graph it, and getting these big functions. Plugging in even on online calculators skips from this big thing to ...
16
votes
4answers
1k views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
-4
votes
0answers
36 views

Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
2
votes
1answer
600 views

Question on using chain rule or product rule to find Jacobian of function with matrices times a vector…

Suppose we have a function consisting of a series of matrices multiplied by a vector: $$f(x) = ABb,$$ where $x$ is a vector containing elements that are contained within $A, b$, and/or $b$, $A$ is a ...
63
votes
10answers
12k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
0
votes
1answer
346 views

Tetrahedron volume in the first octant

The surface is given: $xyz = 2$ It is in the first octant so $x > 0, y > 0, z > 0$. The tangent plane taken at any point of this surface binds with the coordinate axes to form a ...
0
votes
0answers
17 views

Are there any $k$-valued functions with first $k$ integrals all $0$? [on hold]

Let $f(x)$ be a step function on the domain $[0, 1]$, which changes value at most $k-1$ times (and thus takes at most $k$ different values). Suppose that the first $k$ integrals are all $0$ (i.e. the ...
1
vote
1answer
31 views

Logarithmic Taylor series question [on hold]

Consider the transformation of variables, x = $\frac{y+1}{y-1}$ How would you develop log(x) as a Taylor Series in y about zero?
0
votes
0answers
27 views

Show that $\{f_n(x) \}_{n \in \mathbb{N}}$ doesnt converge in M.

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 ...
6
votes
3answers
91 views

How to see $\cos x \leq \exp(-x^2/2)$ on $x \in [0,\pi/2]$?

Can anyone help me with the above inequality? I tried looking at the series expansion and I guess the answer indeed lies there, but I fail to see it. Thanks
2
votes
2answers
91 views

Prove that $\int_0^\infty \frac{\sin nx}{x}dx=\frac{\pi}{2}$

There was a question on multiple integrals which our professor gave us on our assignment. QUESTION: Changing order of integration, show that $$\int_0^\infty \int_0^\infty e^{-xy}\sin nx \,dx ...
1
vote
2answers
30 views

$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$

I need to prove the following: $$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$$ It looks pretty intuitive since I can make $\frac{1}{n}$ as small as I want, thusk making $a$ as close as to ...
0
votes
1answer
29 views

Showing $\mathbb{B}_{\mathbb{Q}}$ is a bases for $\mathbb{R}_{\text{usual}}$

Show that the collection $\mathbb{B}_{\mathbb{Q}} := \{(p, q) \subseteq \mathbb{R} : p, q \in \mathbb{Q}, p < q \}$ is a basis for the usual topology on $\mathbb{R}$. Solution: We know that ...
1
vote
1answer
31 views

Integrating using polar co-ordinates

Hey I've just finished an exam paper and just am stuck with one question. It's something that usually makes sense to me but for some reason I can't get this one: Let $R = \{(x,y) : x,y ≥ 0, x^2 + ...
2
votes
1answer
27 views

Mean-value Theorem $f(x)=\sqrt{x+2}; [4,6]$

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for $c$ that satisfies the conclusion of the ...
1
vote
1answer
36 views

prove a finite limit exists

Let $f$ be differentiable for any $x$. Given that $$\lim_{x\to \infty} f'(x) = 0,$$ prove there exists a finite $L$ such that $$\lim_{x\to \infty} f(x)=L.$$ By definition: ...
1
vote
1answer
49 views

What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ ...
0
votes
3answers
24 views

Rearrangement of alternating harmonic series that does not converge

From Riemann's series theorem, we know that, given a conditionally convergent series, we can permute the elements of the series in order to basically do whatever we want. I have seen a rearrangement ...
2
votes
1answer
53 views

$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{x^2}{b^2}}e^{-\frac{x^2}{a^2}+\frac{y^2}{b^2}}dxdy$

$$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}\,e^{-\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)}\,dx\,dy$$ Basically I have done problems similar to this, using the theorem that if ...
0
votes
0answers
13 views

Kullback-Leibler Divergence (KL) and Approximation Symmetry Property

The Kullback-Leibler Divergence doesn't satisfy the symmetric property. But, it can be approximated (bounded) to such a value. in this paper: Compressing Interactive Communication under product ...
1
vote
2answers
77 views

Why does $\lim_{h \to 0^-} \frac{f(x+h) - f(x)}{h} \neq \lim_{h \to 0} \frac{f(x-h) - f(x)}{h} $

I realize that the only reason one-sided limits arise is as a result of the $\epsilon-\delta$ definition of a limit, applied to the real field $\mathbb{R}$, and that one-sided limits aren't even well ...
0
votes
2answers
33 views

Difficult projectiles question

A particle $P$ of mass $m$ lies on a plane inclined at an angle $\alpha$ to the direction vector $\mathbf{i}$. At $t=0$ the particle is projected from the origin of the coordinate system with speed ...
1
vote
2answers
83 views

Am I using sandwich theorem incorrectly?

I saw this question and wondered how OP of that question was able to do : $$0<\sin x+1<2$$ this $$\frac 0{|x|}<\frac{\sin x+1}{|x|}<\frac 2{|x|}$$ and when $x\to \infty$ he got the limit ...
2
votes
2answers
395 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
0
votes
0answers
28 views

Proving absolute or conditionally convergence

I have the following partial series - $$ \sum_{n = 0}^\infty {a_n} $$ when $$a_n=\begin{cases} \frac{1}{n} &; \quad n \ \text{is even}\\ \frac{-1}{n^2} &; \quad n \ \text{is odd}\ . ...
0
votes
1answer
10 views

Finding the Components of a Hessian Matrix of a Quadratic Form

I'm trying to find the Hessian form of the following quadratic form: $f(x,y) = x^2y+y^2+xy$. I know that it's in the form of a matrix and that the elements of $H_f(a)_{i,j}=\dfrac{\delta^2f}{\delta ...
0
votes
3answers
2k views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
9
votes
3answers
98 views

Mean Value theorem functional equation

I need help solving the following functional equation. Suppose $\,f : \mathbb{R} \to \mathbb{R}$ is differentiable and $$f'\Big(\frac{x + y}{2}\Big) = \frac{f(x) - f(y)}{x -y}$$ holds for all $x ...
0
votes
2answers
37 views

Differentiate a Function (Help me Solve?!)

Find $\dfrac{d}{dx}$ for: $C(1+Ae^{-bt})^{-1}$ I have tried and arrived at: $-C(1+Ae^{-bt})^{-2}$ however that is not the correct answer.
0
votes
0answers
44 views

Topology bases for $\mathbb{R}_{\text{usual}}$

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
1
vote
1answer
28 views

show that $ \left | y^{2} f(y)-x^{2} f(x) \right | \leq \left | y^{3} -x^{3} \right | (\forall x,y\in \mathbb{R})$

Q: If $f(x)$ differentiable function and $ \left | f '(x) \right | \leq 1 (\forall x\in \mathbb{R})$, $f(0)=0$ then show that $$ \left | y^{2} f(y)-x^{2} f(x) \right | \leq \left | y^{3} -x^{3} ...
1
vote
2answers
40 views

Constructing topology on $\Bbb{Z}$

Fix an infinite subset $A$ of $\mathbb Z$ whose complement $\mathbb{Z}\setminus A$ is also infinite. Construct a topology on $\mathbb{Z}$ in which: (a) $A$ is open (b) Singletons are never open (i.e ...
0
votes
1answer
48 views

How do I evaluate $\displaystyle\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?

I am not being able to find the specific product $\prod_{r=1}^{k} \left(1-\frac{1}{\sqrt {r+1}}\right)$ so to evaluate the given problem when $k \to \infty $.
1
vote
2answers
69 views

Show that $f^{-1}$ is differentiable $f^{-1}(x)+f^{-1}(y)=f^{-1}(xy)$

Q: $f^{-1}(x)$ satisfing 1,2,3 ${f^{-1}}(x)$ is the inverse function to $f(x)$ $f(x)$ is differentiable $f^{-1}(x)+f^{-1}(y)=f^{-1}(xy)$, $x,y\in \mathbb{R}^{+}$ Show that $f^{-1}$ is ...
0
votes
1answer
36 views

Verification on finding the radius of convergence of a Laurent series, “the largest R”.

Question: Determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2}{z^2-1} + \dfrac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$? Attempt: The radius ...
1
vote
1answer
33 views

Sequence existing for a set of conditions

Let function $f$ is continous and limited on the interval $(x_0, +\infty)$. Prove $\forall \ number \ T \ \exists \ sequence \ \{x_n\},\ \lim_{n \to\infty}{\{x_n\}} = +\infty $: ...
-5
votes
0answers
34 views

Find $\oint_C x^2y\,dy+x\,dy.$ [on hold]

MultiVariable Calculus Question ( about green's Theorem and etc.) 5) \begin{align*} C_1 &: \vec r(t) = \langle t, 0\rangle&0\leq t\leq 1\\ C_2 &: \vec r(t) = \langle1, ...
2
votes
2answers
242 views

Changing the bounds of integration

I have a question that asks me to find the derivative of this integral, with out evaluation the intergral. $$\int_{\sin x}^{\cos x}\frac {1}{1-t^2}dt$$ I think I need to use U-substitution and the ...
5
votes
3answers
116 views

Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$ \int_a^bf\,dx\leq\int_a^bg\,dx $$ now, imagine that we have $f<g$, is it true that $$ \int_a^bf\,dx<\int_a^bg\,dx $$
2
votes
2answers
28 views

How can I prove the concavity of $f(p_1,p_2,\ldots,p_n) = \sum_{i = 1}^n p_i(1-p_i)$?

Assume $p_n$ is the probability of being in class $n$ which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$ I need to come up with a concave function that show the relation between $p_1$ and ...
0
votes
1answer
62 views

Differential Equation: Solve $y''-4y'+3y=4e^x$

For this question it has an initial condition of $y(0)=5$ and $y'(0)=3$. I managed to get $y = Ae^x + Be^{3x} - 2xe^x$. Solving for $A$ and $B$ I would get $A = 7$ and $B = -2$. However, the real ...
2
votes
0answers
27 views

Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
3
votes
1answer
51 views

How to prove that $E\subset [0,1]$ with some property is countable

Let $E$ be a subset of $[0,1]$. For every sequence $(a_n)$ who's elements are in $E$ and different from each other, the series $\sum\limits_{n=1}^{\infty} a_n$ converges. prove that $E$ is countable. ...
0
votes
0answers
44 views

Series $\sum \lambda^{n-k} c_k $ converges to zero

Let $(c_n)$ a sequence of real number, such that $\lim_{n \to \infty} c_n=0$, Let $0<\lambda<1 $ and $\lambda^nc_0+\lambda^{n-1}c_1+\cdots+\lambda c_{n-1}+c_n=y_n$ a sequence. I have to prove ...
0
votes
2answers
38 views

Extrema Where the Derivative is Undefined

Say we are given the derivative of a function say, $$f'(x)=\begin{cases} 5 & x<3 \\ -5 & x>3 \end{cases}$$ Notice that the derivative has opposite signs on either side of $x=3$, so you ...
0
votes
2answers
63 views

How to Find the Global Minimum and Maximum of this Multivariable Function?

We have the set $$M=\{(x,y,z)\in\mathbb R^3: x^2 + y^2 = z \wedge x+y+z=12\}$$ and the function $$F(x,y,z) = xy+ z^2.$$ How can we find the global maximum and global minimum of F on M and prove ...
0
votes
2answers
33 views

How to show $f$ is Riemann Integrable, finding the $m_i$ and $M_i$ values?

I keep getting stuck on the same sort of question on Riemann Integrals, I am trying to show that a function f is Riemann Integrable on an interval. e.g. Let $f : [−4, 4] \to \mathbb{R} $ be the ...