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Show All NotesHide All NotesI have just been informed that Lamar University needs to perform emergency work on the internet here. Because of that there is a high probability that the site will not be reachable on Saturday, October 24 from 8:00 - 10:00 PM Central Standard Time. I apologize for the inconvienence.
Paul
October 23, 2020
October 23, 2020
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Section 3-1 : Tangent Planes and Linear Approximations
Earlier we saw how the two partial derivatives ({f_x}) and ({f_y}) can be thought of as the slopes of traces. We want to extend this idea out a little in this section. The graph of a function (z = fleft( {x,y} right)) is a surface in ({mathbb{R}^3})(three dimensional space) and so we can now start thinking of the plane that is “tangent” to the surface as a point.
Let’s start out with a point (left( {{x_0},{y_0}} right)) and let’s let ({C_1}) represent the trace to (fleft( {x,y} right)) for the plane (y = {y_0}) (i.e. allowing (x) to vary with (y) held fixed) and we’ll let ({C_2}) represent the trace to (fleft( {x,y} right)) for the plane (x = {x_0}) (i.e. allowing (y) to vary with (x) held fixed). Now, we know that ({f_x}left( {{x_0},{y_0}} right)) is the slope of the tangent line to the trace ({C_1}) and ({f_y}left( {{x_0},{y_0}} right)) is the slope of the tangent line to the trace ({C_2}). So, let ({L_1}) be the tangent line to the trace ({C_1}) and let ({L_2}) be the tangent line to the trace ({C_2}).
The tangent plane will then be the plane that contains the two lines ({L_1}) and ({L_2}). Geometrically this plane will serve the same purpose that a tangent line did in Calculus I. A tangent line to a curve was a line that just touched the curve at that point and was “parallel” to the curve at the point in question. Well tangent planes to a surface are planes that just touch the surface at the point and are “parallel” to the surface at the point. Note that this gives us a point that is on the plane. Since the tangent plane and the surface touch at (left( {{x_0},{y_0}} right)) the following point will be on both the surface and the plane.
[left( {{x_0},{y_0},{z_0}} right) = left( {{x_0},{y_0},fleft( {{x_0},{y_0}} right)} right)] What we need to do now is determine the equation of the tangent plane. We know that the general equation of a plane is given by, Records 1 5 6 – innovative personal database login.
[aleft( {x - {x_0}} right) + bleft( {y - {y_0}} right) + cleft( {z - {z_0}} right) = 0] where (left( {{x_0},{y_0},{z_0}} right)) is a point that is on the plane, which we have. Let’s rewrite this a little. We’ll move the (x) terms and (y) terms to the other side and divide both sides by (c). Doing this gives,
[z - {z_0} = - frac{a}{c}left( {x - {x_0}} right) - frac{b}{c}left( {y - {y_0}} right)] Now, let’s rename the constants to simplify up the notation a little. Let’s rename them as follows,
[A = - frac{a}{c}hspace{0.25in}B = - frac{b}{c}] With this renaming the equation of the tangent plane becomes,
[z - {z_0} = Aleft( {x - {x_0}} right) + Bleft( {y - {y_0}} right)] and we need to determine values for (A) and (B).
Let’s first think about what happens if we hold (y) fixed, i.e. if we assume that (y = {y_0}). In this case the equation of the tangent plane becomes,
[z - {z_0} = Aleft( {x - {x_0}} right)] This is the equation of a line and this line must be tangent to the surface at (left( {{x_0},{y_0}} right)) (since it’s part of the tangent plane). In addition, this line assumes that (y = {y_0}) (i.e. fixed) and (A) is the slope of this line. But if we think about it this is exactly what the tangent to ({C_1}) is, a line tangent to the surface at (left( {{x_0},{y_0}} right)) assuming that (y = {y_0}). In other words,
[z - {z_0} = Aleft( {x - {x_0}} right)] is the equation for ({L_1}) and we know that the slope of ({L_1}) is given by ({f_x}left( {{x_0},{y_0}} right)). Therefore, we have the following,
[A = {f_x}left( {{x_0},{y_0}} right)] If we hold (x) fixed at (x = {x_0}) the equation of the tangent plane becomes,
[z - {z_0} = Bleft( {y - {y_0}} right)] However, by a similar argument to the one above we can see that this is nothing more than the equation for ({L_2}) and that it’s slope is (B) or ({f_y}left( {{x_0},{y_0}} right)). So,
[B = {f_y}left( {{x_0},{y_0}} right)] The equation of the tangent plane to the surface given by (z = fleft( {x,y} right)) at (left( {{x_0},{y_0}} right)) is then,
[z - {z_0} = {f_x}left( {{x_0},{y_0}} right)left( {x - {x_0}} right) + {f_y}left( {{x_0},{y_0}} right)left( {y - {y_0}} right)] Also, if we use the fact that ({z_0} = fleft( {{x_0},{y_0}} right)) we can rewrite the equation of the tangent plane as,
[begin{align*}z - fleft( {{x_0},{y_0}} right) & = {f_x}left( {{x_0},{y_0}} right)left( {x - {x_0}} right) + {f_y}left( {{x_0},{y_0}} right)left( {y - {y_0}} right) z & = fleft( {{x_0},{y_0}} right) + {f_x}left( {{x_0},{y_0}} right)left( {x - {x_0}} right) + {f_y}left( {{x_0},{y_0}} right)left( {y - {y_0}} right)end{align*}] We will see an easier derivation of this formula (actually a more general formula) in the next section so if you didn’t quite follow this argument hold off until then to see a better derivation.
Example 1 Find the equation of the tangent plane to (z = ln left( {2x + y} right)) at (left( { - 1,3} right)). Show SolutionThere really isn’t too much to do here other than taking a couple of derivatives and doing some quick evaluations.
[begin{align*}fleft( {x,y} right) & = ln left( {2x + y} right)hspace{0.25in}& {z_0}& = fleft( { - 1,3} right) = ln left( 1 right) = 0 {f_x}left( {x,y} right) & = frac{2}{{2x + y}}hspace{0.25in} &{f_x}left( { - 1,3} right) & = 2 {f_y}left( {x,y} right) & = frac{1}{{2x + y}}hspace{0.25in} & {f_y}left( { - 1,3} right) & = 1end{align*}] The equation of the plane is then,
[begin{align*}z - 0 & = 2left( {x + 1} right) + left( 1 right)left( {y - 3} right) z & = 2x + y - 1end{align*}] One nice use of tangent planes is they give us a way to approximate a surface near a point. As long as we are near to the point (left( {{x_0},{y_0}} right)) then the tangent plane should nearly approximate the function at that point. Because of this we define the linear approximation to be,
[Lleft( {x,y} right) = fleft( {{x_0},{y_0}} right) + {f_x}left( {{x_0},{y_0}} right)left( {x - {x_0}} right) + {f_y}left( {{x_0},{y_0}} right)left( {y - {y_0}} right)]
and as long as we are “near” (left( {{x_0},{y_0}} right)) then we should have that, Aiseesoft dvd ripper lite 6 5 13.
[fleft( {x,y} right) approx Lleft( {x,y} right) = fleft( {{x_0},{y_0}} right) + {f_x}left( {{x_0},{y_0}} right)left( {x - {x_0}} right) + {f_y}left( {{x_0},{y_0}} right)left( {y - {y_0}} right)] Example 2 Find the linear approximation to (z = 3 + frac{{{x^2}}}{{16}} + frac{{{y^2}}}{9}) at (left( { - 4,3} right)). Show SolutionSo, we’re really asking for the tangent plane so let’s find that.
[begin{align*}fleft( {x,y} right) & = 3 + frac{{{x^2}}}{{16}} + frac{{{y^2}}}{9}hspace{0.25in} & fleft( { - 4,3} right) & = 3 + 1 + 1 = 5 {f_x}left( {x,y} right) & = frac{x}{8}hspace{0.25in} & {f_x}left( { - 4,3} right) & = - frac{1}{2} {f_y}left( {x,y} right) & = frac{{2y}}{9}hspace{0.25in} & {f_y}left( { - 4,3} right) & = frac{2}{3}end{align*}] The tangent plane, or linear approximation, is then,
[Lleft( {x,y} right) = 5 - frac{1}{2}left( {x + 4} right) + frac{2}{3}left( {y - 3} right)] Fotofuze 2 0 1 Sezonas
For reference purposes here is a sketch of the surface and the tangent plane/linear approximation.
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Chronos Fotofuse v2.0.1 (Mac OSX) | 475.72 MB
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