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Chapter 11: Q17E (page 623)

Sketch the graph of the function \(f\left( {x,y} \right) = 9 - {x^2} - 9{y^2}\)

Short Answer

Expert verified

The graph of the function \(f\left( {x,y} \right) = 9 - {x^2} - 9{y^2}\)

Step by step solution

01

Step-1: Graph

02

Step-2: Consideration

The function \(f\left( {x,y} \right) = 9 - {x^2} - 9{y^2}\)

03

Step-3: Observation

The domain of the function can be found through graph of the function also

Hence, the graph of the function \(f\left( {x,y} \right) = 9 - {x^2} - 9{y^2}\)

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Most popular questions from this chapter

(a) Find the values of \(\frac{{\partial z}}{{\partial r}}\) and \(\frac{{\partial z}}{{\partial \theta }}\) if \(z = f(x,y)\), where \(x = r\cos \theta \) and \(y = r\sin \theta \).

(b) Show the equation\({\left( {\frac{{\partial z}}{{\partial x}}} \right)^2} + {\left( {\frac{{\partial z}}{{\partial y}}} \right)^2} = {\left( {\frac{{\partial z}}{{\partial r}}} \right)^2} + \frac{1}{{{r^2}}}{\left( {\frac{{\partial z}}{{\partial \theta }}} \right)^2}\).

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

\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{yz}}{{{x^2} + 4{y^2} + 9{z^2}}}\)

Determine the set of points at which the function is continuous.
\(f\left( {x,y,z} \right) = \sqrt {y{\rm{ - }}{x^2}} {l_n}z\)

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.

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