(a) Let f(x,y) = sqrt{x^2 + y^2.} Find an equation of the tangent plane of the graph z =...

Question:

(a) Let {eq}f(x,y) = \sqrt{x^2 + y^2.} {/eq} Find an equation of the tangent plane of the graph {eq}z = f(x,y) {/eq} at (3, 4, 5).

(b) Use a linear approximation to estimate {eq}f(3.1, 4.2). {/eq}

Linearization:

The equation of the tangent plane to a function of two variables {eq}f(x,y) {/eq} at point {eq}(x_0,y_0) {/eq}

is found by arresting the series of Taylor at the first order terms, i.e.

{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}

where {eq}f_x, \; f_y {/eq} are the first partial derivatives of the function.

Answer and Explanation:

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(a) Consider the function

{eq}f(x,y) = \sqrt{x^2 + y^2.} {/eq}

The equation of the tangent plane at point at (3, 4, 5) is calculated as

{eq}f(3,4)...

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How to Estimate Function Values Using Linearization

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Chapter 10 / Lesson 2
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