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Chapter 21: Problem 3
If \(z=f(x, y)=3 \mathrm{e}^{x}-2 \mathrm{e}^{y}+x^{2} y^{3}\) find \(z(1,1)\)
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If \(z=f(x, y)=-11 x+y\) find (a) \(f(2,3)\) (b) \(f(11,1)\).
Find all the second partial derivatives in each of the following cases: (a) \(z=\frac{1}{x}\) (b) \(z=\frac{y}{x}\) (c) \(z=\frac{x}{y}\) (d) \(z=\frac{1}{x}+\frac{1}{y}\)
In each case, given \(z=f(x, y)\) find \(z_{x}\) and \(z_{y}\). (a) \(z=x y\) (b) \(z=3 x y\) (c) \(z=-9 y x\) (d) \(z=x^{2} y\) (e) \(z=9 x^{2} y\) (f) \(z=8 x y^{2}\)
Consider the function \(f(x, y)=5 x^{2} y\). (a) Evaluate this function and its first partial derivatives at the point \(A(2,3)\). (b) Suppose we consider point \(A\). Suppose small changes, \(\delta x, \delta y\), are made in the values of \(x\) and \(y\) so that we move to a nearby point \(B\). It is possible to show that the corresponding change in \(f\) is given approximately by \(\delta f \approx \frac{\partial f}{\partial x} \delta x+\frac{\partial f}{\partial y} \delta y\), where the partial derivatives are evaluated at the original point \(A\). Use this result to find the approximate change in the value of \(f\) if \(x\) is increased to \(2.1\) and \(y\) is increased to \(3.2\). (c) Compare your answer in (b) to the value of \(f\) at \((2.1,3.2)\)
Given \(z=f(x, y)=7 x+2 y\) find the output when \(x=8\) and \(y=2\).
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