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Chapter 11: Q15E (page 623)

Sketch the graph of the function \(f\left( {x,y} \right) = {y^2} + 1\)

Short Answer

Expert verified

The graph of the function

Step by step solution

01

Step-1: Graph

02

Step-2: Consideration

Given the function \(f\left( {x,y} \right) = {y^2} + 1\)

03

Step-3: Observation

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

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Most popular questions from this chapter

Determine the equation \(\left( {\frac{{{\partial ^2}z}}{{\partial {x^2}}}} \right) + \left( {\frac{{{\partial ^2}z}}{{\partial {y^2}}}} \right) = \left( {\frac{{{\partial ^2}z}}{{\partial {r^2}}}} \right) + \frac{1}{{{r^2}}}\left( {\frac{{{\partial ^2}z}}{{\partial {\theta ^2}}}} \right) + \frac{1}{r} \cdot \frac{{\partial z}}{{\partial r}},\) where \(x = r\cos \theta \) and \(y = r\sin \theta .\)

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 \(r = 4.\)

Calculate the values of \({g_u}(0,0)\) and \({g_v}(0,0)\) using the given table of values if \(g(u,v) = f\left( {{e^u} + \sin v,{e^u} + \cos v} \right)\) where \(f\)is a differentiable function of \(x\) and \(y.\)
  1. Determine the partial derivative \(\frac{{\partial z}}{{\partial r}}.\)
  2. Determine the partial derivative \(\frac{{\partial z}}{{\partial \theta }}.\)
  3. Determine the partial derivative \(\frac{{\partial _z^2}}{{\partial r\partial \theta }}.\)

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

\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {1,0} \right)} \frac{{xy - y}}{{{{(x - 1)}^2} + {y^2}}}\)
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