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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

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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

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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

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

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

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

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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

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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

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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

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Theorem 14.4 Tangent Planes & Linear Approximations10Dr. Erickson

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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

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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

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

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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

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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

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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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