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Chapter 11: Q20E (page 632)

Find h(x, y) = g(f(x, y)) and the set on which h is continuous.

\(g(t) = t + \ln t{\rm{ , }}f(x,y) = \frac{{1 - xy}}{{1 + {x^2}{y^2}}}\)

Short Answer

Expert verified

The set of all points at which h(x, y) is continuous is

Step by step solution

01

Finding Composition of Function:

Consider the following functions \(g(t) = t + \ln t\) and \(f(x,y) = \frac{{1 - xy}}{{1 + {x^2}{y^2}}}\) to find composition of functions h(x, y) = g(f(x, y))

\(\begin{aligned}{l}h(x,y) = g(f(x,y)) = g\left( {\frac{{1 - xy}}{{1 + {x^2}{y^2}}}} \right)\\{\rm{ }} = \frac{{1 - xy}}{{1 + {x^2}{y^2}}} + \ln \left( {\frac{{1 - xy}}{{1 + {x^2}{y^2}}}} \right)\end{aligned}\)

The domain of function \(f(x,y) = \frac{{1 - xy}}{{1 + {x^2}{y^2}}}\) is the set of all points in R2 except denominator is zero. But \(1 + {x^2}{y^2} > 0{\rm{ , }}\forall {\rm{ }}(x,y) \in {R^2}\)

The domain of function f(x, y) is R2

02

The Set of Points where h(x, y) is continuous:

The domain of function g is equal to domain of logarithmic function is set of all positive real numbers.

The domain of h(x, y) = g(f(x, y)) is the set of all points where it is f(x, y) > 0

\(\begin{aligned}{l}\frac{{1 - xy}}{{1 + {x^2}{y^2}}} > 0\\1 - xy > 0{\rm{ since }}1 + {x^2}{y^2} > 0{\rm{ , }}\forall {\rm{ }}(x,y) \in {R^2}\end{aligned}\)

The domain of function h(x, y) is

The set of all points at which h(x, y) is continuous is

03

The Graph of Function:

The region of domain D as shown below:

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Most popular questions from this chapter

Find the value of \(\frac{{\partial T}}{{\partial p}},\frac{{\partial T}}{{\partial q}}\) and\(\frac{{\partial T}}{{\partial r}}\),using the chain rule if \(T = \frac{v}{{2u + v}},u = pq\sqrt r \) and \(v = p\sqrt q r\)when \(p = 2,q = 1\) and \(r = 4.\)

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 \(\frac{{{\partial ^2}z}}{{\partial s\partial t}}\).

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

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

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

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

\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{5{y^4}co{s^2}x}}{{{x^4} + {y^4}}}\)
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