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Chapter 12: Problem 9
The domain of \(Q=f(u, v, w, x, y, z)\) lies in \(\mathbb{R}^{n}\) for what value of \(n ?\) Explain.
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The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\). a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\). b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\). c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(\bar{Q}\). d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3. e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?
Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=\cos (2(x+c t))$$
Recall that Cartesian and polar coordinates are related through the transformation equations $$\left\\{\begin{array}{l} x=r \cos \theta \\ y=r \sin \theta \end{array} \quad \text { or } \quad\left\\{\begin{array}{l} r^{2}=x^{2}+y^{2} \\ \tan \theta=y / x \end{array}\right.\right.$$ a. Evaluate the partial derivatives \(x_{r}, y_{r}, x_{\theta},\) and \(y_{\theta}\) b. Evaluate the partial derivatives \(r_{x}, r_{y}, \theta_{x},\) and \(\theta_{y}\) c. For a function \(z=f(x, y),\) find \(z_{r}\) and \(z_{\theta},\) where \(x\) and \(y\) are expressed in terms of \(r\) and \(\theta\) d. For a function \(z=g(r, \theta),\) find \(z_{x}\) and \(z_{y},\) where \(r\) and \(\theta\) are expressed in terms of \(x\) and \(y\) e. Show that \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\)
The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1\). $$u(x, t)=e^{-t}(2 \sin x+3 \cos x)$$
Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14 .) In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi .\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\).
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