Final answer:
To find the equation of the tangent plane to the surface z=x⁴ + y⁴ at the point (1,0,0), we need to find the partial derivatives of the surface equation with respect to x and y. Evaluating these partial derivatives at the given point, we can write the equation of the tangent plane as z = 4x-4.
Step-by-step explanation:
To find the equation of the tangent plane to the surface z=x⁴ + y⁴ at the point (1,0,0), we need to find the partial derivatives of the surface equation with respect to x and y. Taking the partial derivative with respect to x, we get dz/dx = 4x³. Taking the partial derivative with respect to y, we get dz/dy = 4y³. Evaluating these partial derivatives at the point (1,0,0), we have dz/dx = 4(1)³ = 4, and dz/dy = 4(0)³ = 0. Therefore, the equation of the tangent plane is z = f(a,b) + (x-a)(df/da) + (y-b)(df/db), where (a,b) is the point of tangency and f(a,b) is the value of the surface equation at that point. Plugging in the values, we get z = 0 + (x-1)(4) + (y-0)(0), which simplifies to z = 4x-4.