Question

A differentiable function f(x,y) has the property that f(4,2)=4 and fx​(4,2)=2 and fy​(4,2)=5. Find the equation of the tangent plane at the point on the surface z=f(x,y) where x=4,y=2.

Answer 1

To find the equation of the tangent plane at the point (x=4,y=2) on the surface z=f(x,y), substitute the given values into the general equation of a plane.

Step-by-step explanation:

To find the equation of the tangent plane at the point (x=4,y=2) on the surface z=f(x,y), we need to use the given information about the function. The equation of a plane is of the form z=ax+by+c, where a, b, and c are coefficients. Since we know that f(4,2)=4, the point (4,2,4) lies on the tangent plane, so we can substitute these values into the equation. Also, fx(4,2)=2 and fy(4,2)=5 give us the partial derivatives with respect to x and y at the point (4,2). We can substitute these values into the equation to find the coefficients a and b. Therefore, the equation of the tangent plane is z=2x+5y-6.