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
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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 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

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

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

5. 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

6. Visualization Tangent Plane of a Surface
Dr. Erickson
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).
Dr. Erickson
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).
Dr. Erickson
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.
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14.4 Tangent Planes & Linear Approximations

10. Theorem
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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).
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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:
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14.4 Tangent Planes & Linear Approximations

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

15. 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
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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
Dr. Erickson
14.4 Tangent Planes & Linear Approximations