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Chapter 14 – Partial Derivatives

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dinesh
16 diapositives · 591 Views · Jul 27, 2014

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. Definition – Tangent Plane.

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Chapter 14 – Partial Derivatives
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  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