Section 10.4: Tangent Planes and Linearization
Let S be the surface defined by z = f (x, y), where f has continuous first partial derivatives
fx and fy . For each point P = (x0 , y0 , z0 ) on the surface, the vertical planes x = x0 and
y = y0 intersect S in curves C1 and C2 . Let T1 and T2 denote the tangent lines to C1 and
C2 at the point P . Then the plane defined by T1 and T2 is called the tangent plane to the
surface S at the point P .
Theorem: (Equation of the Tangent Plane)
An equation of the tangent plane to the surface z = f (x, y) at (x0 , y0 , z0 ) is
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ).
Example: Find an equation of the tangent plane to the surface z = x2 − 3y 2 at (−1, 1, −2).
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Example: Find an equation of the tangent plane to the surface z = ex−y at (1, 1, 1).
Example: Find an equation of the tangent plane to the surface z = ex cos y at (0, 0, 1).
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Definition: The linear approximation or linearization of f (x, y) at (x0 , y0 ) is the equation
of the tangent plane to the surface z = f (x, y) at (x0 , y0 ). That is,
L(x, y) = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ).
The linearization is a good approximation of f (x, y) for values of (x, y) near (x0 , y0 ).
Example: Find the linearization of f (x, y) = x2 + y 2 + sin(xy) at (0, 2).
Example: Find the linear approximation of f (x, y) = sin(x + 2y) at (0, 0) and use it to
approximate f (−0.1, 0.2).
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Definition: A two-dimensional vector function is a function F~ (x, y) = hf (x, y), g(x, y)i
that assigns a unique vector in R2 to every value of (x, y) in its domain. The functions f
and g are called the component functions of F~ .
Consider the vector function defined by
F~ (x, y) = hf (x, y), g(x, y)i.
The component functions can be linearized at (x0 , y0 ) as
f (x, y) = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
g(x, y) = g(x0 , y0 ) + gx (x0 , y0 )(x − x0 ) + gy (x0 , y0 )(y − y0 ).
Thus, the linearization of the vector function F~ (x, y) at (x0 , y0 ) can be written in matrix
form as
f (x0 , y0 )
fx (x0 , y0 ) fy (x0 , y0 ) x − x0
~
L(x, y) =
+
.
g(x0 , y0 )
gx (x0 , y0 ) gy (x0 , y0 ) y − y0
Definition: The Jacobian matrix or derivative matrix of F~ (x, y) is the matrix of partial
derivatives
fx (x, y) fy (x, y)
J(x, y) =
.
gx (x, y) gy (x, y)
Example: Find the Jacobian matrix for the vector function F~ (x, y) = h2x3 − 3y 2 , 4x2 − 5yi.
Example: Find the Jacobian matrix for the vector function F~ (x, y) = hln(x + y), ex+y i.
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p
2
2
Example: Find the Jacobian matrix for the vector function F~ (x, y) = h x2 + y 2 , e−(x +y ) i.
Example: Find the linearization of F~ (x, y) = h(x + y)2 , xyi at (−1, 1).
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√
Example: Find the linear approximation of F~ (x, y) = h 2x + y, x − y 2 i at (1, 2) and use it
to approximate F~ (1.05, 2.05).
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