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1d
accepted Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.
1d
comment Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.
Is there an error in this solution? Not sure why it has a downvote as it looks correct to me.
1d
asked Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.
2d
asked Where is $f_x$ continuous for $f(x,y) = (x^2-y^2)/\sqrt[3]{x^2+y^2}$?
Apr
12
awarded  Popular Question
Apr
12
comment Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Thanks for clearing up my understanding.
Apr
12
accepted Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Apr
12
comment Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Don't we also have that $y$ is a function of $x$ and vice versa when implicitly differentiating? (i.e. $y(x)$ so requires the product rule when differentiating $yf'(a)$?)
Apr
12
asked Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Apr
12
accepted What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$?
Apr
12
accepted Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Apr
12
comment Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Thanks for this. What is the best way to solve the system of equations by hand? At first glance, it seems quite messy.
Apr
11
asked Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Apr
11
comment What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$?
This is a good answer also. Although, I see we want to set $d/dx (d^2) =0$ and solve for $t$. What is the best way to do this by hand? It again seems cumbersome.
Apr
11
asked What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$?
Apr
11
accepted How to solve $3y\cos(\theta) - 2x\sin(\theta) = 5 \sin(\theta)\cos(\theta)$?
Apr
11
asked How to solve $3y\cos(\theta) - 2x\sin(\theta) = 5 \sin(\theta)\cos(\theta)$?
Apr
8
accepted Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.
Apr
8
accepted Prove $f(x,y) = \frac{x^2+y^2}{x+y}$ is not continuous at $(0,0)$.
Apr
8
accepted Prove that if $E$ is measurable then $\forall \epsilon > 0$ $\exists F \subset E$ closed such that $m(E \setminus F) < \epsilon$.