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

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Presentation on theme: "Chapter 14 – Partial Derivatives 14.4 Tangent Planes & Linear Approximations 1 Objectives:  Determine how to approximate functions using tangent planes."— Presentation transcript:

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

2 Definition – Tangent Plane Suppose a surface S has equation z = f(x, y), where f has continuous first partial derivatives. Let P(x 0, y 0, z 0 ) be a point on S. let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S. ◦ Then, the point P lies on both C 1 and C 2. Let T 1 and T 2 be the tangent lines to the curves C 1 and C 2 at the point P. 14.4 Tangent Planes & Linear Approximations2Dr. Erickson

3 Tangent Plane Then, the tangent plane to the surface S at the point P is defined to be the plane that contains both tangent lines T 1 and T 2. 14.4 Tangent Planes & Linear Approximations3Dr. Erickson

4 Equation of a tangent plane 14.4 Tangent Planes & Linear Approximations4Dr. Erickson

5 Example 1 Find an equation of the tangent plane to the given surface at the specified point. 14.4 Tangent Planes & Linear Approximations5Dr. Erickson

6 Visualization Tangent Plane of a Surface 14.4 Tangent Planes & Linear Approximations6Dr. Erickson

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 Approximations7Dr. Erickson

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 Approximations8Dr. Erickson

9 Differentiable This means that the tangent plane approximates the graph of f well near the point of tangency. 14.4 Tangent Planes & Linear Approximations9Dr. Erickson

10 Theorem 14.4 Tangent Planes & Linear Approximations10Dr. Erickson

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 Approximations11Dr. Erickson

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 Approximations12Dr. Erickson

13 Example 3 Find the differential of the function below: 14.4 Tangent Planes & Linear Approximations13Dr. Erickson

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 Approximations14Dr. Erickson

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 1 ◦ Example 2 Example 2 ◦ Example 4 Example 4 14.4 Tangent Planes & Linear Approximations15Dr. Erickson

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 Limits of a Rational Function of Two Variables 14.4 Tangent Planes & Linear Approximations16Dr. Erickson


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