Tangent planes

Section: Understanding Tangent Planes

• A tangent plane is a plane that just ‘touches’ a surface at a particular point, without cutting into the surface. It provides an approximation of the surface near that point.

• The tangent plane to a surface at a point is identical to the tangent plane to the level curve of the function at that point.

• The concept of a tangent plane extends the idea of a tangent line to a function of two variables, instead of one.

• Crucially, the direction of quickest change of the function at a point is represented by the vector normal (perpendicular) to the tangent plane.

Section: Constructing Tangent Planes

• The equation of the tangent plane to a surface at a point can be found using partial derivatives.

• If a function f(x, y) represents a surface, and P(a, b, f(a, b)) is a point on this surface, the equation of the tangent plane at P can be given as: f(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b).

• The expressions f_x(a, b) and f_y(a, b) are the partial derivatives of the function f with respect to x and y respectively, evaluated at the point (a, b).

Section: Interpreting and Using Tangent Planes

• Tangent planes allow us to make linear approximations to functions of several variables. This is useful when dealing with complex functions where exact values are difficult to calculate.

• The accuracy of the approximation depends on how well the function can be approximated by a plane in the vicinity of the particular point.

• Tangent planes can also be used to understand and illustrate the behaviour of functions, and solve problems in diverse scientific and mathematical fields.

• As tangent planes are used to calculate gradients, you’ll often encounter tangent planes when dealing with optimisation problems.

Section: Application of Tangent Planes

• The concept of tangent planes is critical in a variety of real-world applications, including physics, engineering, computer graphics, and more.

• For example, plane tangents are used in the physics to study rates of change and in engineering for approximating the measurements.

• In calculus, tangent planes play a vital role in problem-solving tools such as Newton’s method, used to find the zeros of a function (places where the function equals zero).