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18.020 derivatives questions.

Prove that if $f: (a,b) \rightarrow \Re$ is differentiable on the open interval $(a,b)$, and $f'(x)$ is bounded on the interval $(a,b)$, then $f$ is uniformly continuous on $(a,b)$. Also, prove the ...

Suppose the sequence of vector valued functions $\{ {\bf f}_n \}$ are equidifferentiable at ${\bf x}_0$. In other words: $$\lim_{{\bf h} \to {\bf 0}} \max_n \frac{\left\Vert {\bf f}_n({\bf x}_0+{...

Consider the function $$f(x,y) =(x^2 + y^2)\sin\left(\frac{1}{\sqrt{x^2 + y^2}}\right)$$ The partial derivative with respect to $x$ are equal to $$\frac{\partial}{\partial x}f(x,y) =\left\{\begin{...

Suppose the sequence of vector valued functions $\{ {\bf f}_n \}$ are equidifferentiable at ${\bf x}_0$. In other words: $$\lim_{{\bf h} \to {\bf 0}} \max_n \frac{\left\Vert {\bf f}_n({\bf x}_0+{...

For every twice differentiable function $𝑓: \mathbb R → [−2, 2]$ with $(𝑓(0))^ 2 + (𝑓 ′ (0))^ 2 = 85$, which of the following statement(s) is (are) TRUE? (A) There exist 𝑟, 𝑠 ∈ ℝ, where 𝑟...

Let $U \subset \mathbb R^{d}$ open, and both $f: U \to \mathbb R^{\nu}$ and $g: U \to \mathbb R$ are differentiable. Show: the function $F: U \to \mathbb R^{\nu}$, $F(x):=g(x)f(x)$ is differentiable ...

I want to compute the derivative determinant of Schur Complement $\frac{\partial}{\partial C}\det(C^*-P^*C^{-1}P)$ Where $C$ and $P$ are $N\times N$ square symmetric complex matrices. I am not ...

Is the function $$ \log \log |\sin x|$$ differentiable? How can I justify the answer?

Let the $x$'s be vectors and $A$ be a matrix \begin{align} \nabla xx'Ax &= \partial(xx')Ax + xx'\partial(Ax)\\ &= \partial(xx')Ax + xx'A\textbf{1} \tag{i} \\ &= \partial xx'Ax + x\partial ...

guys this is the question i come across- to find the value of $$\int_{-∞}^{∞}e^{-2t} \delta'(t).dt$$ where $ \delta(t) $ is the usual delta function or impulse function. The problem is solved as the ...

Let $f:\mathbb R^n\to\mathbb R^n$ be given by the equation $f(\mathbf x)=\|\mathbf x\|^2 \mathbf x$. Show that $f$ is of class $C^\infty$ and that $f$ carries the unit ball $B(\mathbf 0;1)$ onto ...

So here we have $\textbf b =\frac{\textbf B}{|B|}$ so that $\nabla \textbf B = B\nabla \textbf b + \textbf b \nabla B$. Then $$\nabla p = \frac{B^2}{\mu_o}(\textbf b.\nabla)\textbf b -\frac{1}{2\mu_o}...

Given: $f$ has a derivative for every $x \in (0,\infty)$ $f'(x) > x$ for every $x>0$ Can I prove that there exists an $M \in \mathbb R$, such that $f(x+1) - f(x)$ is a monotonic increasing ...

For some function, $x = f(t)$ $y = g(t)$ If we wish to calculate second order derivative of y with respect to x would it be right to approach in the following manner: $\frac{d^2y}{dx^2} = \frac{d^...

Let $f: U \subset \mathbb{R}^n\to \mathbb{R}^m$ be a differentiable map in $U$, where $U$ is convex and open, such that $Df(x)$ is injective for all $x \in U$ and $$\langle Df(x)(x-y), Df(y)(x-y) \...

Let $G\subset \mathbb{C}$ be a domain and $f:G\to \mathbb{C}$ be continuously differentiable. Let $H\subset G$ dense s.t. $f$ is holomorphic in every $z\in H$. Then $f$ is already holomorphic on $G$. ...

I have a very simple algorithm, and I need to calculate the derivative of an error function, and it gets a bit messy with chain rule. I have a question whether I'm doing this correctly. To be more ...

Theorem: Suppose that $f : A \to \mathbb{R}$ where $A \subseteq \mathbb{R}$. If $f$ is differentiable at $x \in A$, then $f$ is continuous at $x$. This theorem is equivalent (by the contrapositive) ...

Let $\varphi: U \to \mathbb{R}^{n}$ of class $C^{k}$ ($k\geq 1$) in the open $U \subset \mathbb{R}^{m}$. If $a \in U$ is such that $\varphi'(a): \mathbb{R}^{m} \to \mathbb{R}^{n}$ is injective, then ...

The thermal sensation can be modeled by $$W =35.74 + 0.6215T +(0.4275T - 35.75)v^{0.16}$$ where $W$ (thermal sensation) and the temperature $T$ are given in Fahrenheit and v (wind speed) in miles per ...

$$y=(\sin x)^{\sqrt{x}}.$$ I know that I suppose to apply the chain rule here, but I can't get clearly what is the composition here.

Given: $f$ has a derivative for every $x \in (0,\infty)$ $f'(x) > x$ for every $x>0$ Can I prove that $f(x+1) - f(x)$ is a monotonic increasing function? From Lagrange I know that in every $...

In Deep Learning (adapted from page 108), explaining linear regression as a machine learning algorithm, there is a passage for the solution of this expression: To minimize $MSE$, we can simply ...

I have come across this as a fundamental theorem of calculus: $\frac{d}{dx}\int_{a}^{x} f(t) dt = f(x)\tag{1}$ for any constant $a$. For example here: http://mathmistakes.info/facts/CalculusFacts/...

Let $x \geq 0, y \geq 0$ and $p > 0, q > 0$ with $\dfrac{1}{p} + \dfrac{1}{q} = 1$. Show that $$xy \leq \frac{1}{p}x^p + \frac{1}{q}y^q.$$ Given $v = (v_1,\cdots,v_m) \in \mathbb{R}^{m}$, ...

Solve for $x$ in the equation algebraically $$ 2^x=2x. $$ The solutions are $x =\{1,2\}.$ I have solved it but no one has validated my method. So I thought this website can help. I converted ...

I've just come across this function when playing with the Desmos graphing calculator and it seems that it has turning points for many values of $a$. So I pose the following problem: Given $a \in \...

Let $f: V \to \mathbb{R}$ of class $C^{2}$ and $b \in V$ a critical point of $f$. If $\varphi: U \to V$ is a diffeomorphism $C^{2}$ with $\varphi(a) = b$, then the hessian of $f$ in the point $b$ and ...

Let $g: V \to \mathbb{R}^{m}$ be a $C^{2}$ function in the open $V \subset \mathbb{R}^{m}$. Given $b \in V$, suppose that $g'(b): \mathbb{R}^{m} \to \mathbb{R}^{m}$ be an isomorphism. Prove that there ...

Let $u: \mathbb{R}^N \to \mathbb{R}$ be a smooth function. Does $\nabla u$ has null tangential component to $\mathrm{supp}\, u $ ? That is, do we have $$\frac{\partial u}{\partial \nu} = 0 \...

given a function $$f(x) = \frac{1}{1+e^{-x}}$$ we can express its derivative in terms of the function's output: $$\frac{df}{dx} = f(x) - f(x)\cdot f(x)$$ But is it possible to express the ...

(I know this is my second question today, but I'm explaining what I'm doing so I hope it's okay) Consider the graph of $f(x) = x^4-6x^2$. a) Find the relative maxima and minima (both x and y ...

I have the function $$f(\mathbf{x}) = \sqrt{\frac{1}{n}\sum_{i=1}^n\left(\log_e(x_i+1)-c_i\right)^2}$$ where $c_i$ is a constant, and I want to find $f'(\mathbf{x})$, more explicitly, $$\frac{\...

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...

I want to understand the following: Let $\pi:X \to Y$ a fiber bundle and $\omega$ a closed smooth differential form. Define $I: Y\to \mathbb C$, $y\mapsto I(y)=\int_{\pi^{-1}(y)} \omega$. Then Stoke'...

I have a formula for an exponentially weighted moving average function defined recursively as: $S_t = a*Y_t+(1-a)*S_{t-1}$ Where: $a\in (0,1)\cap \mathbb{Q}$ $t$ represents time $Y_t$ is the value ...

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = \sum_{n=...

$\omega$ is a 3×1 vector and $I$ is a constant 3×3 matrix. What is the derivative respect to vector $\omega$? It would be appreciated if you could show me any references I can refer to. $$\frac{d(\...

Boyle's Law for enclosed gases states that if the volume is kept constant, the pressure P and temperature T are related by the equation P/T=k, where k is a constant. Suppose that the rate of change of ...

The Lie derivative $L_X$ of a vector field w.r.t. to another is defined as $$ L_X(Y)= \lim_{ε\to 0} [(σ_{-ε})_*(Y|_{σ_ε(x)}-Y|_x] $$ Where $σ$ denotes flow of the vector field X and $_*$ denotes ...

I'm in a single-variable calculus course, in which we recently covered logarithmic differentiation. The professor proved it that works when $f(x)>0$, and when $f(x)<0$. I've been trying to ...

I was working along with http://jimherold.com/2012/04/20/least-squares-bezier-fit/ to see if I could understand each step of the way for fitting a Bezier curve to a set of coordinates, and I ...

I want to implicitly differentiate this twice with respect to x (y is the dependent variable), (finding y''): $$(xy)^3 = 4x$$ using the derivative rules we would get $$y'' = -\frac{3x^3y^3+8x - 3x^2y^...

Given the open connected $U \subset \mathbb{R}^{m}$, define the geodesic distance $d_{U}(x,y)$ between $x,y \in U$ by the infimum of lengths of paths (continuous) and rectifiable contained in $U$, of $...

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