Problem 42
If \(f(x, y, z)=(x y / z)^{1 / 2}\), find \(f_{x}(-2,-1,8)\).
Problem 42
Let \(f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \quad\) if \(\quad(x, y) \neq(0,0)\) and \(f(0,0)=0\) Show that \(f_{x y}(0,0) \neq f_{y x}(0,0)\) by completing the following steps: (a) Show that \(f_{x}(0, y)=\lim _{h \rightarrow 0} \frac{f(0+h, y)-f(0, y)}{h}=-y\) for all \(y\). (b) Similarly, show that \(f_{y}(x, 0)=x\) for all \(x\). (c) Show that \(f_{y x}(0,0)=\lim _{h \rightarrow 0} \frac{f_{y}(0+h, 0)-f_{y}(0,0)}{h}=1\). (d) Similarly, show that \(f_{x y}(0,0)=-1\).
Problem 43
Draw the graph and the corresponding contour plot. \(f(x, y)=\sin \sqrt{2 x^{2}+y^{2}} ;-2 \leq x \leq 2,-2 \leq y \leq 2\)
Problem 44
Plot the graphs of each of the following functions on \(-2 \leq x \leq 2,-2 \leq y \leq 2\), and determine where on this set they are discontinuous. (a) \(f(x, y)=x^{2} /\left(x^{2}+y^{2}\right), f(0,0)=0\) (b) \(f(x, y)=\tan \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=0\)
Problem 44
Let \(A(x, y)\) be the area of a nondegenerate rectangle of dimensions \(x\) and \(y\), the rectangle being inside a circle of radius 10. Determine the domain and range for this function.
Problem 44
Draw the graph and the corresponding contour plot. \(f(x, y)=\sin \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=1 ;\) \(-2 \leq x \leq 2,-2 \leq y \leq 2\)
Problem 45
Draw the graph and the corresponding contour plot. \(f(x, y)=\left(2 x-y^{2}\right) \exp \left(-x^{2}-y^{2}\right) ;-2 \leq x \leq 2\) \(-2 \leq y \leq 2\)
Problem 46
Give definitions of continuity at a point and continuity on a set for a function of three variables.
Problem 46
Draw the graph and the corresponding contour plot. \(f(x, y)=(\sin x \sin y) /\left(1+x^{2}+y^{2}\right) ;-2 \leq x \leq 2\) \(-2 \leq y \leq 2\)
Problem 46
The wave equation \(c^{2} \partial^{2} u / \partial x^{2}=\partial^{2} u / \partial t^{2}\) and the heat equation \(c \partial^{2} u / \partial x^{2}=\partial u / \partial t\) are two of the most important equations in physics ( \(c\) is a constant). These are called partial differential equations. Show each of the following: (a) \(u=\cos x \cos c t\) and \(u=e^{x}\) cosh \(c t\) satisfy the wave equation. (b) \(u=e^{-c t} \sin x\) and \(u=t^{-1 / 2} e^{-x^{2} /(4 c t)}\) satisfy the heat equation.
