Final answer:
To find the linear approximation for the function f(x,y)=3x+2yxy at the point (2,1), you would compute the partial derivatives, find the gradient, and use the tangent plane equation.
Step-by-step explanation:
To find the linear approximation for the function f(x,y)=3x+2yxy at the point (2,1), we can use partial derivatives, the gradient, and the tangent plane equation.
First, find the partial derivatives with respect to x and y: fx = 3 + 2yx and fy = 2xy.
Then, calculate the gradient by combining the partial derivatives: ∇f = (fx, fy) = (3 + 2yx, 2xy). Next, substitute the point (2,1) into the gradient to find its value: ∇f(2,1) = (3 + 2(2)(1), 2(2)(1)) = (7,4).
Finally, use the tangent plane equation z - f(a,b) = ∇f(a,b) · (x-a, y-b), where (a, b) is the point of approximation and z is the linear approximation.
Substitute the given values and the calculated gradient from the previous steps to find the linear approximation at (2,1).