Question

How do you find the linear approximation of #f(x,y) = sqrt(53-9x^2-y^2)# at (2,-1)?

Answer 1

The linear approximation to a function `f` of two variables (at a point) is the equation of the tangent plane to the surface (at that point).

The equation of that tangent plane depends on the slope in each direction; the partial derivatives `f_x` and `f_y`. If the surface is

`z = f(x, y)`, then the tangent plane at `(x_0, y_0)` is

`z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)`.

You need to take the partials of your `f(x, y)`, evaluate them at `(2,-1)`, and simplify the equation.

• Hint: If you get stuck, look for previous examples of derivatives of a function to a power. Use the Chain Rule to do the job.*

dansmath to the (partial) rescue!