Chapter 11: Q6E (page 648)
Find an equation of the tangent plane to the given surface at the specified point..
\(z = \ln (x - 2y),(3,1,0)\)..
The tangent plane Equation is \(x - 2y - z = 1\)..
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Graph and discuss the continuity of the function
\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{sinxy}}{{xy}}, if xy \ne 0\\1, if xy = 0\end{aligned} \right.\)
Determine the derivative\(\frac{{dz}}{{dt}}\) with the help of the chain rule. The functions are\(z = \cos (x + 4y),x = 5{t^4}\) and\(y = \frac{1}{t}.\)
Let \(g(x,y,z) = {x^3}{y^2}z\sqrt {10 - x - y - z} \)
a) Evaluate g(1,2,3)
b) Find and describe the domain of g.
Suppose \(z = f(x,y)\), where \(x = g(s,t)\) and \(y = h(s,t)\).
(a) Show that
\(\frac{{{\partial ^2}z}}{{\partial {t^2}}} = \frac{{{\partial ^2}z}}{{\partial {x^2}}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + 2\frac{{{\partial ^2}z}}{{\partial x\partial y}}\frac{{\partial x}}{{\partial t}}\frac{{\partial y}}{{\partial t}} + \frac{{{\partial ^2}z}}{{\partial {y^2}}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + \frac{{\partial z}}{{\partial x}}\frac{{{\partial ^2}x}}{{\partial {t^2}}} + \frac{{\partial z}}{{\partial y}}\frac{{{\partial ^2}y}}{{\partial {t^2}}}\)
(b) Find a similar formula for \(\frac{{{\partial ^2}z}}{{\partial s\partial t}}\).
Find the value of \(\frac{{\partial T}}{{\partial p}},\frac{{\partial T}}{{\partial q}}\) and\(\frac{{\partial T}}{{\partial r}}\),using the chain rule if \(T = \frac{v}{{2u + v}},u = pq\sqrt r \) and \(v = p\sqrt q r\)when \(p = 2,q = 1\) and \(r = 4.\)
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