Question

Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, fx(−6,7)=−1 and fy(−6,7)=4. Given that f(−6,7)=4, use this information to estimate the value of f(−5,8). Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation.

Answer 1

7

Explanation:

We are given that

`f_x(-6,7)=-1`

`f_y(-6,7)=4`

`f(-6,7)=4`

We have to find the value of f(-5,8),

We know that

`f(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)`

We have

`x=-5,y=8`

`x_0=-6,y_0=7`

`f(x,y=f(-6,7)+f_x(-6,7)(x+6)+f_y(-6,7)(y-7)`

`f(x,y)=4+(-1)(x+6)+4(y-7)`

`f(x,y)=4-x-6+4y-28=-x+4y-30`

`f(-5,8)=-(-5)+4(8)-30=5+32-30=7`