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Chapter 11: Q40E (page 624)

Use a computer to graph the function using various domains and viewpoints.

\(f\left( {x,y} \right) = \cos x\cos y\)

Short Answer

Expert verified

Answer is not given in the drive.

Step by step solution

01

Sketching the curve from a number of viewpoints.

02

The curve is sketched below from a number of viewpoints.

03

The curve is drawn here from different viewpoints.

04

Plotting continuous lines.

A continuous map for the function look like this:

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Most popular questions from this chapter

Find an equation of the tangent plane to the given surface at the specified point..

\(z = 3{y^2} - 2{x^2} + x,(2, - 1, - 3)\)..

Determine the derivative \(\frac{{dz}}{{dt}}\)at the given value of \(t.\)The functions are \(z = f(x,y),x = g(t)\) and \({\rm{ }}y = h(t){\rm{.}}\)

Suppose that the equation \(F(x,y,z) = 0\) implicitly defines each of the three variables \(x,y\), and \(z\) as functions of the other two: \(z = f(x,y),y = g(x,z),x = h(y,z)\). If \(F\) is differentiable and \({F_s},{F_y}\), and \({F_z}\) are all nonzero, show that

\(\frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial y}}\frac{{\partial y}}{{\partial z}} = - 1\).

Suppose \(z = f(x,y)\), where \(x = g(s,t)\) and \(y = h(s,t)\).

(a) Show that

\(\frac{{{\partial ^2}z}}{{\partial {t^2}}} = \frac{{{\partial ^2}z}}{{\partial {x^2}}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + 2\frac{{{\partial ^2}z}}{{\partial x\partial y}}\frac{{\partial x}}{{\partial t}}\frac{{\partial y}}{{\partial t}} + \frac{{{\partial ^2}z}}{{\partial {y^2}}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + \frac{{\partial z}}{{\partial x}}\frac{{{\partial ^2}x}}{{\partial {t^2}}} + \frac{{\partial z}}{{\partial y}}\frac{{{\partial ^2}y}}{{\partial {t^2}}}\)

(b) Find a similar formula for \(\frac{{{\partial ^2}z}}{{\partial s\partial t}}\).

Find the limit, if it exists, or show that the limit does not exist.

\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{x{y^4}}}{{{x^2} + {y^8}}}\)
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