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…

Math

Calculus

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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 41E

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Question

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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