Chapter 5

State and prove Rolle's Theoorem.

(a) State and prove the Mean Value Theorem for the derivative of a real valued function of a single real variable. (b) Give a geometrical interpretation to this result.

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

State and prove the Cauchy Mean Value Theorem.

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

Use L'Hospital's Rule to evaluate (a) \(\lim _{x \rightarrow 2}\left\\{\left(2 x^{2}-4 x\right) /(x-2)\right\\}\) (b) \(\lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \ln \mathrm{x}\) (c) \(\lim _{\mathrm{x} \rightarrow 0}(1 / \sin \mathrm{x})-(1 / \mathrm{x})\) (d) \(\lim _{\mathrm{x} \rightarrow \infty}(1+\mathrm{x})^{1 / \mathrm{x}}\)

Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{x}^{3}\). Find a suitable \((\mathrm{u}, \mathrm{v})\) on the line segment connecting \((\mathrm{a}, b)\) with \((c, d)\) such that \(\mathrm{f}(\mathrm{c}, \mathrm{d})-\mathrm{f}(\mathrm{a}, b)=(\partial \mathrm{f} / \partial \mathrm{x})(\mathrm{u}, \mathrm{v})(\mathrm{c}-\mathrm{a})+(\partial \mathrm{f} / \partial \mathrm{y})(\mathrm{u}, \mathrm{v})(\mathrm{d}-\mathrm{b})\) if \((a, b)=(1,2)\) and \((c, d)=(1+h, 2+k)\).

Show that if a function \(\mathrm{f}: \mathrm{V} \rightarrow \mathrm{R}, \mathrm{V} \subseteq \mathrm{R}^{\mathrm{n}}\), is \(\mathrm{C}^{2}\) locally at \(\mathrm{a}\), then \(\left[\left(\partial^{2} \mathrm{f}\right) /\left(\partial \mathrm{x}_{\mathrm{i}} \partial \mathrm{x}_{\mathrm{j}}\right)\right](\mathrm{a})=\left[\left(\partial^{2} \mathrm{f}\right) /\left(\partial \mathrm{x}_{j} \partial \mathrm{x}_{\mathrm{i}}\right)\right]\) (a) for all \(i, j\) between 1 and \(n\) inclusive.

(a) Let \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}\) be defined by \(f(x, y)=2 x y\left\\{\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right)\right\\}, x^{2}+y^{2} \neq 0\) and \(=0, \quad \mathrm{x}=\mathrm{y}=0\). Show that \(\left(\partial^{2} \mathbf{f} / \partial \mathrm{x} \partial \mathrm{y}\right) \neq\left(\partial^{2} \mathrm{f} / \partial \mathrm{x} \partial \mathrm{y}\right)\) and explain why. (b) Does there exist a function \(\mathrm{f}\) with continuous second partial derivatives (i.e., an element of \(\mathrm{C}^{2}\) ) such that of \(/ \partial \mathrm{x}=\mathrm{x}^{2}\) and \partialf \(/ \partial \mathrm{y}=\mathrm{xy}\) ?

Prove Taylor's Theorem for \(\mathrm{f} \in \mathrm{C}^{\mathrm{T}}(\mathrm{E})\) where \(\mathrm{E} \subseteq \mathrm{R}^{\mathrm{n}}\) is an open convex set.

Give a Taylor expansion of \(f(x, y)=e^{x} \cos y\) on some compact convex domain \(\mathrm{E}\) containing \((0,0)\).

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 \(1+\mathrm{t}+3 \mathrm{t}^{4}\).

Using Taylor's Theorem, approximate \(\sqrt{40}\) to three decimal places.

Calculate \(\mathrm{e}^{4}\) within an error of \(10^{-3}\)

State and prove the Chain Rule.

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) Define "homogeneous of degree \(\mathrm{n}^{\prime \prime}\) and "positively homogeneous of degree \(\mathrm{n}^{\prime \prime}\) for a function of two variables \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) and give a geometric interpretation of homogeneity (b) Determine whether the following functions are homogeneous: (i) \(\mathrm{x}^{2} \mathrm{y} \log (\mathrm{y} / \mathrm{x})\); (ii) \(x^{1 / 3}+x y^{-2 / 3}\); (iii) \(\left[\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right)\right]\) (iv) \(\mathrm{Ar}^{n}\) where \(\mathrm{A}\) is any constant and \(\mathrm{r}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{1 / 2}\)

(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 \(\left(\partial^{2} \mathrm{u} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{u} / \partial \mathrm{y}^{2}\right)\) \(=\mathrm{r}^{\mathrm{m}}\left\\{\left(\partial^{2} \mathrm{~F} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{~F} / \partial \mathrm{y}^{2}\right)\right\\}+\mathrm{m}(\mathrm{m}+4) \mathrm{r}^{\mathrm{m}-2} \mathrm{~F}\)

State and prove the Implicit Function Theorem.

Represent the contour line \(y-x e^{y}=1\) near \((-1 / 0)\) as a function \(\mathrm{x}=\psi(\mathrm{y})\). Compute \(\Phi^{\prime}(\mathrm{x})\) and \(\Phi^{\prime}(-1)\) for \(\mathrm{x}\) near \(-1\), where \(\mathrm{y}=\Phi(\mathrm{x})\).

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 \((\partial f / \partial x)(a, b) \neq 0\)