7. Approximate V(2.03)2 + (0.8)² + (2.1)² using a linear approximation.?

To approximate: 2.032+0.82+2.12 Consider a function, fx,y,z=x2+y2+z2 The linear approximation of a…

Answered: Approximate V(2.03)² + (0.8)² + (2.1)²…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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7. Approximate V(2.03)2 + (0.8)² + (2.1)² using a linear approximation.?

Expert Solution

Step 1

To approximate:

$\sqrt{{(2.03)}^{2} + {(0.8)}^{2} + {(2.1)}^{2}}$

Consider a function,

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

The linear approximation of a function f(x, y, z) at point (x₀, y₀, z₀) is given by

$L (x, y, z) = f (x_{0}, y_{0}, z_{0}) + f_{x} (x_{0}, y_{0}, z_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}, z_{0}) (y - y_{0}) + f_{z} (x_{0}, y_{0}, z_{0}) (z - z_{0})$

Formulas used:

$\frac{d}{d x} x^{n} = n x^{n - 1} \frac{d}{d x} constant = 0 Chain rule of differentiation, \frac{d}{d x} (f (g (x))) = f' (g (x)) \cdot g' (x)$

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Approximate V(2.03)² + (0.8)² + (2.1)² using a linear approximation.