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Chapter 12: Problem 65
Use the method of your choice to evaluate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x-y|}{|x+y|}$$
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Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$
The angle between two planes is the angle \(\theta\) between the normal vectors of the planes, where the directions of the normal vectors are chosen so that \(0 \leq \theta<\pi\) Find the angle between the planes \(5 x+2 y-z=0\) and \(-3 x+y+2 z=0\)
a. Consider the function \(w=f(x, y, z)\). List all possible second partial derivatives that could be computed. b. Let \(f(x, y, z)=x^{2} y+2 x z^{2}-3 y^{2} z\) and determine which second partial derivatives are equal. c. How many second partial derivatives does \(p=g(w, x, y, z)\) have?
Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=\cos (2(x+c t))$$
Use the definition of differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x+y$$
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