Find the differential of f(x, y)= Vr2 + y3 at the point (2, 3).df =Then use the differential to estimate f(1.9, 3.06).f(1.9, 3.06) =

given function is f(x,y) =x2+y3 we know, for the function z = f(x,y), the total differential of the…

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Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Find the differential of f(x, y)
= Vr2 + y3 at the point (2, 3).
df =
Then use the differential to estimate f(1.9, 3.06).
f(1.9, 3.06) =

Expert Solution

Step 1

given function is

$f (x, y) = \sqrt{x^{2} + y^{3}}$

we know, for the function z = f(x,y), the total differential of the function is given by

$d z = \frac{\partial z}{\partial x} d x + \frac{\partial z}{\partial y} d y$

calculating the partial derivatives, we have

$\frac{\partial z}{\partial x} = \frac{2 x}{2 \sqrt{x^{2} + y^{3}}} \frac{\partial z}{\partial y} = \frac{3 y^{2}}{2 \sqrt{x^{2} + y^{3}}}$

Therefore, the total differential function is given by

$d z = \frac{2 x}{2 \sqrt{x^{2} + y^{3}}} d x + \frac{3 y^{2}}{2 \sqrt{x^{2} + y^{3}}} d y$

At point (2,3)

$d z = \frac{2 \times 2}{2 \sqrt{2^{2} + 3^{3}}} d x + \frac{3 \times 3^{2}}{2 \sqrt{2^{2} + 3^{3}}} d y d z = \frac{2}{\sqrt{31}} d x + \frac{27}{2 \sqrt{31}} d y$

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