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Chapter 14 – Partial Derivatives14.4 Tangent Planes & Linear Approximations
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
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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.
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14.4 Tangent Planes & Linear Approximations
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Tangent PlaneThen, 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.
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14.4 Tangent Planes & Linear Approximations
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Equation of a tangent plane
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14.4 Tangent Planes & Linear Approximations
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Example 1Find an equation of the tangent plane to
the given surface at the specified point.
)2,2,2()cos( yxyz
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14.4 Tangent Planes & Linear Approximations
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VisualizationTangent Plane of a Surface
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14.4 Tangent Planes & Linear Approximations
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LinearizationThe linear function whose graph is
this tangent plane, namely
is called the linearization of f at (a, b).
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14.4 Tangent Planes & Linear Approximations
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Linear ApproximationThe approximation
is called the linear approximation or the tangent plane approximation of f at (a, b).
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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.
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14.4 Tangent Planes & Linear Approximations
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Theorem
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14.4 Tangent Planes & Linear Approximations
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Example 2Find the linear approximation of the function
and use it to approximate f (6.9,2.06). 2,7at 3ln),( yxyxf
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14.4 Tangent Planes & Linear Approximations
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Total differentialFor 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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Example 3Find the differential of the function
below:xyyv cos
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Example 4 – pg. 923 # 34Use 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.
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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 2◦Example 4
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Demonstrations
Feel free to explore these demonstrations below.
Tangent Planes on a 3D GraphTotal Differential of the First OrderLimits of a Rational Function of Two Vari
ables
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