INSTRUCTOR: Given the function f of xy equal to e to the exponent of 5 minus 2x plus 3y end exponent, approximate f of 4.1 comma 0.9 using the point 4 comma 1 for x0 comma y0. What is the approximate value of f of 4.1 comma 0.9 to four decimal places? We're going to use the idea of a tangent plane approximation here.

So the equation for a tangent plane is L of xy equal to f of x0 comma y0 plus fx evaluated at x0 comma y0 multiplied by parentheses x minus x0 plus fy parentheses x0 comma y0 close parentheses open parentheses y minus y0 close parentheses. That would be the equation of the tangent plane.

Now we have some values to input here. First off, we have f of 4 comma 1. That will be equal to e to the exponent of 5 minus 2 times 4 plus 3 times 1, which is equal to e to the 0. So f of 4 1 is equal to 1. Also to find this we need to find fx and fx is equal to negative 2 e to the exponent of 5 minus 2x plus 3y end exponent.

Knowing that, we can say fx at the point 4 1 is equal to negative 2. Now we can find fy. That would be equal to 3 e to the exponent of 5 minus 2x plus 3y end exponent. So that fy evaluated at 4 1 is equal to 3. This means that our linear approximation, L of xy, is equal to 1 minus 2 open parenthesis x minus 4 close parentheses plus 3 open parentheses y minus 1 close parentheses.

And we can evaluate this to find L of 4.1 comma 0.9 is approximately 0.5. So on our tangent plane, f of 4.1 comma 0.9, is about 0.5. However, if we take the function and say f of 4.1 comma 0.9 and evaluate, that is approximately 0.6065, which means that our tangent plane doesn't do a terrible job at predicting a value. But, of course, the tangent plane approximation is off. There's some error.