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Chapter 11: Problem 61
Define a function of two variables.
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The temperature at the point \((x, y)\) on a metal plate is \(T=\frac{x}{x^{2}+y^{2}}\). Find the direction of greatest increase in heat from the point (3,4) .
Moment of Inertia An annular cylinder has an inside radius of \(r_{1}\) and an outside radius of \(r_{2}\) (see figure). Its moment of inertia is \(I=\frac{1}{2} m\left(r_{1}^{2}+r_{2}^{2}\right)\) where \(m\) is the mass. The two radii are increasing at a rate of 2 centimeters per second. Find the rate at which \(I\) is changing at the instant the radii are 6 centimeters and 8 centimeters. (Assume mass is a constant.)
If \(f(x, y)=0,\) give the rule for finding \(d y / d x\) implicitly. If \(f(x, y, z)=0,\) give the rule for finding \(\partial z / \partial x\) and \(\partial z / \partial y\) implicitly.
Find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(r\) and \(\boldsymbol{\theta}\) before differentiating. \(w=\frac{y z}{x}, \quad x=\theta^{2}, \quad y=r+\theta, \quad z=r-\theta\)
Ideal Gas Law The Ideal Gas Law is \(p V=m R T,\) where \(R\) is a constant, \(m\) is a constant mass, and \(p\) and \(V\) are functions of time. Find \(d T / d t,\) the rate at which the temperature changes with respect to time.
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