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Chapter 21: Problem 2

$$ \text { If } w=5 y-2 x \text { state } \frac{\partial^{2} w}{\partial x^{2}} \text { and } \frac{\partial^{2} w}{\partial y^{2}} \text {. } $$

Short Answer

Expert verified
Answer: The second-order partial derivatives are $$\frac{\partial^{2}w}{\partial x^{2}} = 0$$ and $$\frac{\partial^{2}w}{\partial y^{2}} = 0$$.

Step by step solution

01

Differentiate $$w$$ with respect to $$x$$ and $$y$$

First, we will find the first-order partial derivatives of $$w$$ with respect to $$x$$ and $$y$$. To do this, we need to differentiate the function $$w = 5y - 2x$$ partially with respect to each variable, treating the other variable as constant. Differentiating with respect to $$x$$: $$\frac{\partial w}{\partial x} = -2$$ Differentiating with respect to $$y$$: $$\frac{\partial w}{\partial y} = 5$$
02

Differentiate the first-order partial derivatives with respect to $$x$$ and $$y$$

Now that we have the first-order partial derivatives, we can find the second-order partial derivatives by differentiating these derivatives with respect to $$x$$ and $$y$$ again. Differentiating $$\frac{\partial w}{\partial x} = -2$$ with respect to $$x$$: $$\frac{\partial^{2}w}{\partial x^{2}} = 0$$ Differentiating $$\frac{\partial w}{\partial y} = 5$$ with respect to $$y$$: $$\frac{\partial^{2}w}{\partial y^{2}} = 0$$
03

Conclusion

The second-order partial derivatives of the function $$w = 5y - 2x$$ are: $$\frac{\partial^{2}w}{\partial x^{2}} = 0$$ and $$\frac{\partial^{2}w}{\partial y^{2}} = 0$$.

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Most popular questions from this chapter

Find \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y^{2}}\) and \(\frac{\partial^{2} f}{\partial x \partial y}\) if \(f=(x-y)^{2}\).

If \(z=14-4 x y\) evaluate \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((1,2)\).

Calculate \(\frac{\partial z}{\partial x}\) when (a) \(z=\frac{y}{x^{2}}-\frac{x}{y^{2}}\), (b) \(z=\mathrm{e}^{x^{2}-4 x y}\)

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 \((2.1,3.2)\)

Calculate \(\frac{\partial z}{\partial y}\) when \(z=\frac{x^{2}-3 y^{2}}{x^{2}+y^{2}}\).
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