91Ó°ÊÓ

Gives a function \(f(x, y)\) and a positive number \(\varepsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y),\) $$\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\varepsilon.$$ \(f(x, y)=(x+y) /(2+\cos x), \quad \varepsilon=0.02\)

Find the value of \(\partial x / \partial z\) at the point \((1,-1,-3)\) if the equation $$x z+y \ln x-x^{2}+4=0$$ defines \(x\) as a function of the two independent variables \(y\) and \(z\) and the partial derivative exists.

Parametrized Surfaces Just as you describe curves in the plane parametrically with a pair of equations \(x=f(t), y=g(t)\) defined on some parameter interval \(I,\) you can sometimes describe surfaces in space with a triple of equations \(x=f(u, v), y=g(u, v), z=h(u, v)\) defined on some parameter rectangle \(a \leq u \leq b, c \leq v \leq d .\) Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section \(16.5 .\) ) Use a CAS to plot the surfaces in Exercises \(77-80 .\) Also plot several level curves in the \(x y\) -plane. $$\begin{array}{l}{x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2} \\ {0 \leq v \leq 2 \pi}\end{array}$$

Gives a function \(f(x, y)\) and a positive number \(\varepsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y),\) $$\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\varepsilon.$$ \(f(x, y)=\frac{x y^{2}}{x^{2}+y^{2}} \quad\) and \(\quad f(0,0)=0, \quad \varepsilon=0.04\)

Parametrized Surfaces Just as you describe curves in the plane parametrically with a pair of equations \(x=f(t), y=g(t)\) defined on some parameter interval \(I,\) you can sometimes describe surfaces in space with a triple of equations \(x=f(u, v), y=g(u, v), z=h(u, v)\) defined on some parameter rectangle \(a \leq u \leq b, c \leq v \leq d .\) Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section \(16.5 .\) ) Use a CAS to plot the surfaces in Exercises \(77-80 .\) Also plot several level curves in the \(x y\) -plane. $$\begin{array}{l}{x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2} \\ {0 \leq v \leq 2 \pi}\end{array}$$

Gives a function \(f(x, y)\) and a positive number \(\varepsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y),\) $$\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\varepsilon.$$ \(f(x, y)=\frac{x^{3}+y^{4}}{x^{2}+y^{2}} \quad\) and \(\quad f(0,0)=0, \quad \varepsilon=0.02\)

Express \(v_{x}\) in terms of \(u\) and \(y\) if the equations \(x=v \ln u\) and \(y=u \ln v\) define \(u\) and \(v\) as functions of the independent variables \(x\) and \(y,\) and if \(v_{x}\) exists. (Hint: Differentiate both equations with respect to \(x\) and solve for \(v_{x}\) by eliminating \(u_{x}\).)

Gives a function \(f(x, y, z)\) and a positive number \(\varepsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y, z),\) $$\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\varepsilon.$$ \(f(x, y, z)=x^{2}+y^{2}+z^{2}, \quad \varepsilon=0.015\)

Parametrized Surfaces Just as you describe curves in the plane parametrically with a pair of equations \(x=f(t), y=g(t)\) defined on some parameter interval \(I,\) you can sometimes describe surfaces in space with a triple of equations \(x=f(u, v), y=g(u, v), z=h(u, v)\) defined on some parameter rectangle \(a \leq u \leq b, c \leq v \leq d .\) Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section \(16.5 .\) ) Use a CAS to plot the surfaces in Exercises \(77-80 .\) Also plot several level curves in the \(x y\) -plane. $$\begin{array}{l}{x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u} \\ {0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi}\end{array}$$

Gives a function \(f(x, y, z)\) and a positive number \(\varepsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y, z),\) $$\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\varepsilon.$$ \(f(x, y, z)=x y z, \quad \varepsilon=0.008\)