Find the linearization of the function f(x,y) = 1 - 8x ln(xy-19) at the point (4,5). Use the...


Find the linearization of the function {eq}f(x,y) = 1 - 8x \ln(xy-19) {/eq} at the point {eq}(4,5) {/eq}.

Use the linearization to approximate {eq}f(3.97, 4.8) {/eq}.

Linearization in Two Variables:

Given a function {eq}f(x, y) {/eq} of two variables that is continuous and differentiable at the point {eq}P(x_0, y_0), {/eq} the linearization of {eq}f {/eq} at {eq}P {/eq} is given by the formula {eq}L(x, y) = f(x_0, y_0) + f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0). {/eq} This can be used to approximate the value of {eq}f {/eq} at points near P.

Answer and Explanation:

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We start by calculating the two first-order partial derivatives:

{eq}f_x(x, y) = -8 \ln (xy - 19) - \displaystyle\frac{8xy}{xy - 19} \\ f_y(x, y) =...

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Linearization of Functions


Chapter 10 / Lesson 1

Over the river and through the woods to Grandmother's house we go ... Are we there yet? In this lesson, apply linearization to estimate when we will finally get to Grandma's house!

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