Chapter 11: Q22E (page 632)
Determine the set of points at which the function is continuous.
\(f(x,y) = \cos \sqrt {1 + x - y} \)
The function
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Show the equation \({\left( {\frac{{\partial z}}{{\partial x}}} \right)^2} - {\left( {\frac{{\partial z}}{{\partial y}}} \right)^2} = \frac{{\partial z}}{{\partial s}}\frac{{\partial z}}{{\partial t}}\) holds true.
Calculate the values of \({g_u}(0,0)\) and \({g_v}(0,0)\) using the given table of values if \(g(u,v) = f\left( {{e^u} + \sin v,{e^u} + \cos v} \right)\) where \(f\)is a differentiable function of \(x\) and \(y.\)
Determine the speed of temperature rising on the bug's path after 3 seconds. The temperature function is, \(T(x,y)\) it is measured in degrees Celsius.
The value of \(x = \sqrt {1 + t} \) and \(y = 2 + \frac{1}{3}t\) which is measured in centimeters.
The function satisfies \({T_x}(2,3) = 4\) and\({T_y}(2,3) = 3\).
(a) Determine the rate of change of the volume of the box whose length \(l\) increase from \(1\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\), width \(w\) increase from \(2\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\) and height \(h\) decrease from \(2\;{\rm{m}}/{\rm{s}}\) to \(3\;{\rm{m}}/{\rm{s}}\).
(b) Determine the rate of change of the surface of the box whose length \(l\) increase from \(1\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\), width \(w\) increase from \(2\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\) and height \(h\) decrease from \(2\;{\rm{m}}/{\rm{s}}\) to \(3\;{\rm{m}}/{\rm{s}}\).
(c) Determine the rate of change of the length of a diagonal of the box whose length \(l\) increase from \(1\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\), width \(w\) increase from \(2\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\) and height \(h\) decrease from \(2\;{\rm{m}}/{\rm{s}}\) to \(3\;{\rm{m}}/{\rm{s}}\).
Find the value of \(\frac{{dy}}{{dx}}\) using equation 6.
\(\cos (xy) = 1 + \sin y\)
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