# For \ a \ function \ f(x,y), \ we \ are \ given \ f(100,20)=2750, \ and \ f \ x(100,20)=4 \ and \...

## Question:

{eq}For \ a \ function \ f(x,y), \ we \ are \ given \ f(100,20)=2750, \ and \ f \ x(100,20)=4 \ and \ f \ y(100,20)=7. \ Estimate \ f(105,21) {/eq}

## Estimating the Value of the Function:

In this problem, we have an actual value of the function and the values of the partial derivatives of the function. By using these, we have to write the linear approximation function. Then, it should be estimated using the given value of the function.

Let us use the given functions {eq}\displaystyle f(100,20)=2750 {/eq}, {eq}\displaystyle f_{x}(100,20)=4 {/eq} and {eq}\displaystyle f_{y}(100,20)=7 {/eq} to estimate {eq}\displaystyle f(105,21) {/eq}.

Detecting Linear approximation function:

{eq}\begin{align*} \\ \displaystyle L(x, y) &=f(A, B)+f_{x}(A, B)(x-A)+f_{y}(A, B)(y-B) \\ \displaystyle &=2750+(4)(x-100)+(7)(y-20) \\ \displaystyle &=2750+4x-400+7y-140 \\ \displaystyle L(x, y) &=4x+7y+2210 \end{align*} {/eq}

The linear approximation function is {eq}\ \displaystyle \mathbf{\color{blue}{ L(x, y) =4x+7y+2210 }} {/eq}.

Estimating {eq}\displaystyle f(105,21) {/eq}:

{eq}\begin{align*} \displaystyle f(105,21) &\approx L(x, y) =4x+7y+2210 \\ \displaystyle f(105,21) &\approx 4(105)+7(21)+2210 \\ \displaystyle f(105,21) &\approx 2777 \end{align*} {/eq}

Therefore, the estimated value is {eq}\ \displaystyle \mathbf{\color{blue}{ f(105,21) \approx 2777 }} {/eq}.