INSTRUCTOR: Find the differentials dz of the function f of x comma y equal to 4y squared plus x squared times y minus 2xy and use it to approximate delta z at the 0.1 comma negative 1. Use delta x equal to 0.03 and delta y equal to negative 0.02. What is the exact value of delta z?

Now, the equation for dz is that dz equals fx evaluated at x0 comma y0 multiplied by dx plus fy evaluated at x0 y0 multiply by dy. So in order to find dz, we need to find the partial with respect to x of our function and the partial with respect to y of our function and evaluate it at our given 0.1 negative 1.

Well, fx is equal to 2xy minus 2y. And evaluating that at the 0.1 negative 1, we get 0. Also, fy for our function is equal to 8y plus x squared minus 2x. So evaluating that at the 0.1 negative 1, fy of 1, negative 1 is negative 9.

Using delta x as our dx-- that's how we're going to do that-- delta x will be our dx of 0.03, and delta y for our dy of negative 0.02. Substituting that into our equation for dz, needless to see that dz is equal to 0.18.

Now, in order to find the exact value of delta z, we use the equation delta z equals f of x plus x0 comma y plus y0 minus f of x0 comma y0. So we can evaluate our function at the point 1.03 comma negative 1.02. Then subtract f of 1, negative 1.

Now, f of 1.03 comma negative 1.02 is 5.180682. And f of 1 negative 1 is 5, which means that delta z is equal to 0.180682. And as we can see, delta z is approximating dz. Those two values are-- or rather, dz approximates delta z, so they are very close.