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Assuming that you are referring to part a) of the exercise 2.12, did you try this? \begin{align*} \frac{D Z}{dt}(t)&=\frac{D}{dt}\big(\cos(\theta_0-\beta t)\,X(t)+\sin(\cos(\theta_0-\beta t)\,Y(t)\big)\\ &=\frac{D}{dt}\big(\cos(\theta_0-\beta t)\,X(t)\big)+\frac{D}{dt}\big(\sin(\cos(\theta_0...

If you write the subadditive property for this function, it would help.

Is this \[C\big((qx_1,qy_1)+(qx_2,qy_2)\big)\leqslant C(qx_1,qy_1)+C(qx_2,qy_2)\] the right one?

...Is it simply $100 + qx^{0.5} + qx$ and $100 + qy^{0.5} + qy$ ? In your fist post you wrote \[C(x,y)=\begin{cases} 100+qx\,0.5+qy\,0.5+qx+qy\,, & qx>0,qy>0\\ 0\,, & qx=qy=0 \end{cases}\] which is different from \[C(x,y)=\begin{cases} 100+qx^{0.5}+qy^{0.5}+qx+qy\,, & qx>0,qy>0\\ 0\,, &...

To be the function \[C(x,y)=\begin{cases} 100+1.5\,qx+1.5\,qy\,, & qx>0,qy>0\\ 0\,, & qx=qy=0 \end{cases}\] subadditive it should be: \begin{align*} C\big((qx_1,qy_1)+(qx_2,qy_2)\big)\leqslant C(qx_1,qy_1)+C(qx_2,qy_2) \end{align*} for all $(qx_1,qy_1),(qx_2,qy_2)\in X\times Y$. But to proce...

The surface $F_2=\big\{(x,y,z)\in\mathbb{R}^3\; |\; (x-2)^2 + y^2 =4,\; z\in\mathbb{R}\big\}$ is a cylinder with base the circle $p=\big\{(x,y,0)\in\mathbb{R}^3\; |\; (x-2)^2 + y^2 =4\big\}$ and axis perpedicular to $xy$-plane. The curve obtained by the intersection of the cylinder $F_2$ by any surf...

\begin{align*} \int_0^1 \frac{\sqrt{1-x^2}}{(1+x)^2}\, dx&=\int_0^1 \sqrt{\frac{1-x}{1+x}}\,\frac{1}{1+x}\, dx\\ &\mathop{=\!=\!=\!=\!=\!=\!=\!=\!=\!=}\limits^{\begin{subarray}{c} {t\,=\,\sqrt{\frac{1-x}{1+x}}}\\ {dx\,=-\frac{4t}{(1+t^2)^2}\,dt} \\ \end{subarray}} \,-\int_1^0 t\,\frac{1}{1+\...

Because $S$ is infinite we can construct a sequence $\{a_n\}_{n\in\mathbb{N}}$ with terms distinct elements of $S$; i.e. a sequence such that \[(\forall\,i,j\in\mathbb{N}) \quad i\neq j \quad \Rightarrow\quad a_i\neq a_j.\] Because $S$ is bounded, $\{a_n\}_{n\in\mathbb{N}}$ is bounded and, by the Bo...

Let $E_1\,,\; E_2$ be metric spaces and $f:E_1\longrightarrow E_2$ a map such that for every compact subspace $S$ of $E_1$ the restriction $f|_S:S\subseteq E_1\longrightarrow E_2$ is continuous. Prove that $f$ is continuous on $E_1$.

Prove that the set \[A=\big\{(x,y)\in\mathbb{R}^2\;|\; y\cos{x}+x\sin{y}=1\big\}\] is not path-connected set with respect to the relative topology of $A$ in $\mathbb{R}^2$.

Evaluate \[\displaystyle\int_{0}^{\frac{\pi}{2}} \frac{x^{k}}{\tan{x}}\, {\rm d}x\,,\] for $k=1,3,5,7$.