Find the linearization L (x, y, z) of the function f (x, y, z) at the given points. f (x, y, z) =...


Find the linearization {eq}L (x,\ y,\ z) {/eq} of the function {eq}\displaystyle f (x,\ y,\ z) {/eq} at the given points.

{eq}\displaystyle f (x,\ y,\ z) = e^x + \cos (y + z) {/eq}.

(a) {eq}(0,\ 0,\ 0) {/eq} .

(b) {eq}\displaystyle \bigg(0,\ \pi,\ \frac \pi 2\bigg) {/eq}.

(c) {eq}\displaystyle \bigg(0,\ \frac \pi 4,\ \frac {3 \pi} 4\bigg) {/eq}.

Linearization of Three-variable Function

The linearization of three variable functions is done by first getting the three partial derivatives of the function and evaluating it at the given point. Consider a function {eq}w = f(x, y,z) {/eq}, the linear approximation at {eq}P(a,b,c) {/eq} is

$$L(x,y,z) = f(a,b,c) + f_x(a,b,c)(x-a) + f_y(a,b,c)(y-b) + f_z(a,b,c)(z-c) $$

Answer and Explanation:

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Let's begin by getting the partial derivatives of the function.

{eq}\displaystyle f (x,\ y,\ z) = e^x + \cos (y + z) {/eq}

With respect to x:


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Learn more about this topic:

Linearization of Functions


Chapter 10 / Lesson 1

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