Chapter 11: Problem 24
In Exercises 23 and \(24,\) find the highest point on the curve of intersection of the surfaces. Cone: \(x^{2}+y^{2}-z^{2}=0, \quad\) Plane: \(x+2 z=4\)
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Consider the function \(w=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta .\) Prove each of the following. (a) \(\frac{\partial w}{\partial x}=\frac{\partial w}{\partial r} \cos \theta-\frac{\partial w}{\partial \theta} \frac{\sin \theta}{r}\) \(\frac{\partial w}{\partial y}=\frac{\partial w}{\partial r} \sin \theta+\frac{\partial w}{\partial \theta} \frac{\cos \theta}{r}\) (b) \(\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\left(\frac{1}{r^{2}}\right)\left(\frac{\partial w}{\partial \theta}\right)^{2}\)
Differentiate implicitly to find the first partial derivatives of \(w\). \(x^{2}+y^{2}+z^{2}-5 y w+10 w^{2}=2\)
In Exercises 87 and \(88,\) use the function to prove that (a) \(f_{x}(0,0)\) and \(f_{y}(\mathbf{0}, \mathbf{0})\) exist, and (b) \(f\) is not differentiable at \((\mathbf{0}, \mathbf{0})\). \(f(x, y)=\left\\{\begin{array}{ll}\frac{3 x^{2} y}{x^{4}+y^{2}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{array}\right.\)
The function \(f\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) Determine the degree of the homogeneous function, and show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\) \(f(x, y)=\frac{x^{2}}{\sqrt{x^{2}+y^{2}}}\)
In Exercises 21-26, find the gradient of the function and the maximum value of the directional derivative at the given point. $$ \frac{\text { Function }}{h(x, y)=x \tan y} \frac{\text { Point }}{\left(2, \frac{\pi}{4}\right)} $$
What do you think about this solution?
We value your feedback to improve our textbook solutions.
