Obtain an approximate value for \(\sqrt{105}\) to within \(.01\) by using the Mean Value Theorem.

Rewrite the polynomial $^{\mathrm{n}} \sum_{\mathrm{i}=0} \alpha_{\mathrm{i}} \mathrm{t}^{\mathrm{t}}$ as a polynomial in \(\mathrm{x}=\mathrm{t}-1\) Verify this for the polynomial $1+\mathrm{t}+3 \mathrm{t}^{4}$.

Let the vector-valued function $\mathrm{f}: \mathrm{R}^{\mathrm{n}} \rightarrow \mathrm{R}^{\mathrm{m}}$ be defined by $\mathrm{f}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)=\left\\{\mathrm{f}_{1}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right), \ldots, \mathrm{f}_{\mathrm{m}}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)\right\\}$ (a) Show that \(\mathrm{f}\) is differentiable at $\mathrm{a} \in \mathrm{R}^{\mathrm{n}}\( if and only if each \)\mathrm{f}_{\mathrm{i}}(1 \leq \mathrm{i} \leq \mathrm{m})$ is differentiable at a and \(J_{f}(a)=\left\\{J_{(f) 1}(a), \ldots, J_{(f) m}(a)\right\\}\) (b) Show that this derivative \(\mathrm{J}_{\mathrm{f}}\) (a) \((\mathrm{x}-\mathrm{a})\) is unique. \\{Note: (b) does not depend on (a).\\}

(a) State and prove Euler's Theorem on positively homogeneous functions of two variables. (b) Let \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) be positively homogeneous of degree 2 and $\mathrm{u}=\mathrm{r}^{\mathrm{m}} \mathrm{F}(\mathrm{x}, \mathrm{y})\( where \)\mathrm{r}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{1 / 2}$. Show that $\left(\partial^{2} \mathrm{u} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{u} / \partial \mathrm{y}^{2}\right)$ $=\mathrm{r}^{\mathrm{m}}\left\\{\left(\partial^{2} \mathrm{~F} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{~F} / \partial \mathrm{y}^{2}\right)\right\\}+\mathrm{m}(\mathrm{m}+4) \mathrm{r}^{\mathrm{m}-2} \mathrm{~F}$

Show that the functions $\mathrm{f}, \mathrm{g} \in \mathrm{C}^{1}(\mathrm{E}), \mathrm{E}\( open in \)\mathrm{R}^{2}$, are functionally dependent (i.e., there exists a function \(\mathrm{F}\) such that \(g=F^{\circ} \mathrm{f}\) ) if det \(J \phi(x, y)=0\) for \(\Phi=(f, g)\) and $(x, y)\( in some neighborhood of \)(a, b)\(, where \)(\partial f / \partial x)(a, b) \neq 0$

State and prove L'Hospital's Rule for the indeterminant forms $(O / O) ;(\infty / \infty)$