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Tangent Planes and Linear Approximations
Just as in 2-space we can visualize the line tangent to a curve at a
point, in 3-space we can picture the plane tangent to a
surface at a point.
Consider the surface given by z = f(x,y). Let (x0,y0,z0) be any point on this surface.
If f(x,y) is differentiable at
(x0,y0), then the surface has a tangent plane at
(x0,y0,z0). The equation of the tangent plane at (x0,y0,z0)
is given by
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fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0) = 0 |
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Notes
- Recall that the equation of the plane containing a point (x0,y0,z0)
and normal to the vector n = (a,b,c) is
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a(x-x0)+b(y-y0)+c(z-z0) = 0. |
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The derivation of the equation for the tangent plane just involves
showing that the tangent plane is normal to the vector n = (fx(x0,yy), fy(x0,y0),-1).
- For surfaces F(x,y,z) = 0 that are not easily solved for z,
the equation of the tangent plane at (x0,y0,z0) is
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Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0 |
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provided that ÑF(x0,y0,z0) ¹ 0. Note that if we let
F(x,y,z) = f(x,y)-z, we obtain the equation given for the tangent
plane to z = f(x,y) at (x0,y0,z0).
Example
Let's find the equation of the plane tangent to the surface
z = 4x3y2+2y at the point (1,-2,12).
Since f(x,y) = 4x3y2+2y ,
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fx(x,y) = 12x2y2 and fy(x,y) = 8x3y +2. |
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With x = 1 and y = -2,
Thus, the tangent plane has normal vector n = (48,-14,-1) at
(1,-2,12) and the equation of the tangent plane is given by
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48(x-1)-14( y-(-2)) -(z-12) = 0. |
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Simplifying,
The following Exploration allows you to plot the surface from the
previous example (as well as a few other surfaces) and the tangent
plane at various points. Zoom in on a particular point. What do you
notice about the surface and the tangent plane near the point?
Exploration
The tangent plane to a surface at a point stays close to the surface
near the point. In fact, if f(x,y) is differentiable at the point
(x0,y0), the tangent plane to the surface z = f(x,y) at
(x0,y0) provides a good approximation to f(x,y) near
(x0,y0):
- Solving fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)-(z-z0) = 0
for z,
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z = z0 +fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0). |
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- Since z0 = f(x0,y0), we have that
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z = f(x0,y0)+fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0). |
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- Near (x0,y0), the surface is close to the tangent plane.
Thus,
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f(x,y) » f(x0,y0)+fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0) |
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We call this the linear approximation or local
linearization of f(x,y) near (x0,y0).
Notes
- The linear approximation is really just the multivariable Taylor
polynomial of degree 1 for f(x,y) about (x0,y0). It is only
accurate near (x0,y0). Better approximations can be obtained by
using higher-order Taylor polynomials.
- These concepts can be extended to functions of more than two
variables:
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f(x,y,z) » f(x0,y0,z0)+fx(x0,y0,z0)(x-x0) +fy(x0,y0,z0)(y-y0) + fz((x0,y0,z0)(z-z0) |
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where f(x,y,z) is differentiable at (x0,y0,z0).
Example
From our work in the previous example, the linear approximation to
f(x,y) = 4x3y2+2y near x = 1, y = -2 is
This is, of course, exact at x = 1, y = -2:
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f(1,-2) = 12 = 48(1)-14(-2)-64. |
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At x = 1.1 and y = -1.9, according to the linear approximation,
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f(1.1,-1.9) » 48(1.1) -14(-1.9)-64 = 15.4, |
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which is very close to the exact value f(1.1,-1.9) = 15.41964!
Key Concepts
- Tangent Plane to a Surface
Let (x0,y0,z0) be any point on the surface z = f(x,y). If
f(x,y) is differentiable at (x0,y0), then the surface has a
tangent plane at (x0,y0,z0) given by
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fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0) = 0 |
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- Linear Approximation to a Surface
If f(x,y) is differentiable at (x0,y0), then near (x0,y0)
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f(x,y) » f(x0,y0)+fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0). |
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