Equation of the tangent plane to a surface

Find the equation of the tangent plane to the surface $z = 4x^2 - y^2 + 2y$ at $(-1, 2, 4)$.

Finding the equation of the tangent plane to a surface at a given point is an important concept in multivariable calculus. This problem explores this concept in the context of functions of two variables, where the surface is defined by an equation of the form $z = f(x, y)$. The tangent plane can be thought of as the plane that just touches the surface at the given point, providing the best linear approximation of the surface at that point.

To find the equation of the tangent plane, it is essential to understand the role of partial derivatives. These derivatives provide the slope of the tangent lines to the curves obtained by intersecting the surface with planes parallel to the coordinate planes. The partial derivative with respect to x gives the slope of the tangent line in the x-direction, while the partial derivative with respect to y gives the slope in the y-direction. Together, they determine the orientation of the tangent plane.

In this context, examining how the point of tangency and the values of these partial derivatives inform the final equation emphasizes the significance of linear approximation methods in calculus. Such techniques are foundational for more advanced applications, such as optimization and differential equations, showcasing how a local linear model can approximate and solve more complicated models.

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