Sec 15.4 Tangent Planes and Linear Approximations

1. Sec 15.4 Tangent Planes and Linear Approximations Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z = f (x, y) at the point The linear function: whose graph is the tangent plane at the point (a, b) is called the linearization of f at (a, b). The approximation: is called the linear approximation or the tangent plane approximation of f at (a, b).

2. Theorem: If the partial derivatives exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b). Definitions: Differentials If z = f (x, y), we define the differentialsdx and dy to be independent variables (that can be given any values) The differential (also called the total differential) dzis defined by For w = f (x, y, z),

3. Sec 15.5 The Chain Rule The Chain Rule (Case 1): Suppose z = f (x, y) is a differentiable function of x and y, where x = g(t) and y = h(t) are both differentiable functions of t. Then z is a differentiable function of t and The Chain Rule (Case 2): Suppose z = f (x, y) is a differentiable function of x and y, where x = g(s, t) and y = h(s, t) are both differentiable functions of s and t. Then

4. Implicit Differentiation Suppose an equation of the form F(x, y) = 0 defines y implicitly as a differentiable function of x, then Suppose an equation of the form F(x, y, z) = 0 defines z implicitly as a differentiable function of x and y, then