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Problem 121

(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.

Expert verified

The Mean Value Theorem states that if a function \(f(x)\) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \(c \in (a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\). In the proof, we define a new function \(g(x) = f(x) - \frac{f(b) - f(a)}{b - a}(x - a)\), show that it satisfies the conditions of Rolle's theorem, and use it to demonstrate the theorem's claim. The geometrical interpretation is that there exists a point within the given interval where the tangent line to the curve is parallel to the secant line connecting the points \((a, f(a))\) and \((b, f(b))\), meaning it has the same slope as the average rate of change between the interval's endpoints.

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Chapter 5

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 \((c, d)=(1+h, 2+k)\).

Chapter 5

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.

Chapter 5

(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 \(/ \partial \mathrm{y}=\mathrm{xy}\) ?

Chapter 5

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.

Chapter 5

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$

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