# For { g(x,y) = \sqrt{(x^2+y+14)}, \\ (a)\Delta g(3,2)=...................... } (b) Find the...

## Question:

For {eq}g(x,y) = \sqrt{(x^2+y+14)}, \\ (a)\Delta g(3,2)=...................... {/eq}

(b) Find the best linear approximation of g(x,y) for (x,y) near (3,2). Linear approximation =....................

(c) Use the approximation in part (b) to estimate g(2.97,1.97). g(2.97,1.97)=...........................

## Linearization of a multivariable function

One of the applications of differentiation is Linearization.

Any function can be approximated as a linear function using derivative. Usually linearization is calculated at a given value of x and y.

If {eq}f(x,y) {/eq} is the given function, then it's linearization at {eq}x = a, y = b, {/eq} is given by,

{eq}L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) \\ \therefore \Delta f(a,b)=f_x(a,b)(x-a) + f_y(a,b)(y-b) {/eq}

## Answer and Explanation:

Given,

{eq}\begin{align} \displaystyle g(x,y) &= \sqrt{x^2+y+14} \\ \displaystyle g_x &= \frac{x}{\sqrt{x^2+y+14}} \\ \displaystyle g_y &= \frac{1}{2\sqrt{x^2+y+14}} \\ \end{align} {/eq}

(a)

{eq}\begin{align} \displaystyle \Delta g(3,2) &= g_x(3,2)(x-3) + g_y(3,2)(y-2) \\ \displaystyle &= \frac{3}{\sqrt{3^2+2+14}}(x-3) + \frac{1}{2\sqrt{3^2+2+14}}(y-2) \\ \displaystyle &= \color{blue}{\frac{3}{5}(x-3) + \frac{1}{10}(y-2)} \end{align} {/eq}

(b)

Linear approximation of g(x,y) near (3,2) is given by,

{eq}\begin{align} \displaystyle L(x,y) &= g(3,2) + g_x(3,2)(x-3) + g_y(3,2)(y-2) \\ \displaystyle &= g(3,2) + \Delta g_x(3,2) \\ \displaystyle &= \sqrt{3^2+2+14} + \frac{3}{5}(x-3) + \frac{1}{10}(y-2) &&....\text{From the result in (a)}\\ \displaystyle &= \color{blue}{5 + \frac{3}{5}(x-3) + \frac{1}{10}(y-2)} \end{align} {/eq}

(c)

Using the result in (b),

{eq}\begin{align} \displaystyle L(x,y) &= 5 + \frac{3}{5}(x-3) + \frac{1}{10}(y-2) \\ \displaystyle g(2.97,1.97) & \approx 5 + \frac{3}{5}(2.97-3) + \frac{1}{10}(1.97-2) \\ \displaystyle & \approx 5 + \frac{3}{5}(-0.03) + \frac{1}{10}(-0.03) \\ \displaystyle & \approx 5-0.021 \\ \displaystyle & \approx \color{blue}{4.979} \end{align} {/eq}