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**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 Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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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(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 P lies on both C1 and C2. Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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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 T1 and T2. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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**Equation of a tangent plane**

Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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

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**Visualization Tangent Plane of a Surface**

Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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

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**Linear Approximation The approximation**

is called the linear approximation or the tangent plane approximation of f at (a, b). Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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

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Theorem Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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

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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: Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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**Example 3 Find the differential of the function below:**

Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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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. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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More Examples The video examples below are from section 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 Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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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 Dr. Erickson 14.4 Tangent Planes & Linear Approximations

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Chapter 14 – Partial Derivatives 14.8 Lagrange Multipliers 1 Objectives: Use directional derivatives to locate maxima and minima of multivariable functions.

Chapter 14 – Partial Derivatives 14.8 Lagrange Multipliers 1 Objectives: Use directional derivatives to locate maxima and minima of multivariable functions.

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