Chapter 14 – Partial Derivatives

1. Chapter 14 – Partial Derivatives 14.4 Tangent Planes & Linear Approximations • Objectives: • Determine how to approximate functions using tangent planes • Determine how to approximate functions using linear functions 14.4 Tangent Planes & Linear Approximations

2. Definition – Tangent Plane • Suppose a surface S has equation z = f(x, y), where f has continuous first partial derivatives. • Let P(x0, y0, z0) be a point on S. • let C1 and C2 be the curves obtained by intersecting the vertical planes y = y0 and x = x0 with the surface S. • Then, the point Plies on both C1 and C2. • Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P. 14.4 Tangent Planes & Linear Approximations

3. Tangent Plane • Then, the tangent planeto the surface S at the point P is defined to be the plane that contains both tangent lines T1 and T2. 14.4 Tangent Planes & Linear Approximations

4. Equation of a tangent plane 14.4 Tangent Planes & Linear Approximations

5. Example 1 • Find an equation of the tangent plane to the given surface at the specified point. 14.4 Tangent Planes & Linear Approximations

6. Visualization • Tangent Plane of a Surface 14.4 Tangent Planes & Linear Approximations

7. Linearization • The linear function whose graph is this tangent plane, namely is called the linearization of f at (a, b). 14.4 Tangent Planes & Linear Approximations

8. Linear Approximation • The approximation is called the linear approximation or the tangent plane approximation of f at (a, b). 14.4 Tangent Planes & Linear Approximations

9. Differentiable • This means that the tangent plane approximates the graph of f well near the point of tangency. 14.4 Tangent Planes & Linear Approximations

10. Theorem 14.4 Tangent Planes & Linear Approximations

11. Example 2 • Find the linear approximation of the function and use it to approximate f(6.9,2.06). 14.4 Tangent Planes & Linear Approximations

12. Total differential • For a differentiable function of two variables, z = f(x, y), we define the differentials dx and dy to be independent variables. • Then the differential dz, also called the total differential, is defined by: 14.4 Tangent Planes & Linear Approximations

13. Example 3 • Find the differential of the function below: 14.4 Tangent Planes & Linear Approximations

14. Example 4 – pg. 923 # 34 • Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm think and the metal in the sides is 0.05 cm thick. 14.4 Tangent Planes & Linear Approximations

15. More Examples The video examples below are from section 14.4 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 1 • Example 2 • Example 4 14.4 Tangent Planes & Linear Approximations

16. Demonstrations Feel free to explore these demonstrations below. • Tangent Planes on a 3D Graph • Total Differential of the First Order • Limits of a Rational Function of Two Variables 14.4 Tangent Planes & Linear Approximations