```DAY 20
Plan: We want to find
p2 (x, y) = a0 + a1 (x
x0 ) + b1 (y
y0 ) + a2 (x
x0 )2 + b2 (x
x0 )(y
y0 ) + c2 (y
y0 ) 2
which matches f to second order at (x0 , y0 ). This can be done by choosing the coefficients
appropriately.
The formula: Near (x0 , y0 ) we have
f (x, y) ⇡ f (x0 , y0 ) + @x f (x0 , y0 ) (x
x0 ) + @y f (x0 , y0 ) (y
y0 )
1
1
+ @x2 f (x0 , y0 ) (x x0 )2 + @x @y f (x0 , y0 ) (x x0 )(y y0 ) + @x2 f (x0 , y0 ) (y y0 )2
2
2
Interpretation: View above formula in three parts: Constant term, linear contributions to
form tangent plane, quadratic correction to include some bending.
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Example: Write out second-order approximation of cos (x2 + y 2 ) near ( 2⇡ , 0).
Mathematica: We can easily view the graph of a function together with various approximations. It is helpful to give each piece a name. Consider this code for the function
cos (x2 + y 2 ). . .
graph = Plot3D[Cos[x^2 + y^2],
{x, -3.5, 3.5}, {y, -3.5, 3.5},
RegionFunction -> Function[{x, y}, x^2 + y^2 < 5], PlotRange -> {-2, 5},
PlotStyle -> Opacity[.5], Mesh -> None, ColorFunction -> Function[{x, y}, Red],
AxesLabel -> {"x", "y", "z"}];
plane = Plot3D[
1/Sqrt[2] - Sqrt[Pi/2] (x - Sqrt[Pi]/2),
{x, -3.5, 3.5}, {y, -3.5, 3.5},
PlotStyle -> Opacity[.5],ColorFunction -> Function[{x, y}, Yellow]];
1/Sqrt[2]-Sqrt[Pi/2](x - Sqrt[Pi]/2)-((2+Pi)/Sqrt[2])(x-Sqrt[Pi]/2)^2-Sqrt[2] y^2,
{x, -3.5, 3.5}, {y, -3.5, 3.5},
ColorFunction -> Function[{x, y}, Green]];
point = ListPointPlot3D[{{Sqrt[Pi]/2, 0, 1/Sqrt[2]}},
PlotStyle -> Directive[Blue, PointSize[Large]]];
Show[graph, plane, point]
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HOMEWORK PROBLEMS
49
Homework problems
(1) Find the following approximations:
(a) Linear approximation of the function f (x, y) = x3 3xy + y 3 near the point (2, 1).
(b) Linear approximation of the function f (x, y) = arctan xy near the point (1, 1).
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(c) Linear approximation of the function f (x, y, z) =
x2 + y 2 + z 2 near the point
(3, 4, 12).
(d) Quadratic approximation of the function f (x, y) = ln(x2 + y 2 ) near the point (1, 0);
(e) Quadratic approximation of the function f (x, y, z) = x2 y 2 z 2 near the point (1, 1, 1).
(2) This problem concerns the function
f1 (x, y) = e
x2 +y 2
2
.
(a) Use Mathematica to graph this function in the vicinity of the domain point (0, 0).
(b) Based on the appearance of this graph predict the equation of the tangent plane to
the graph of the function f1 at the domain point (0, 0).
(c) Confirm your prediction with computation. That is, find the equation of the tangent
plane to the graph of (or find the linearization of) the function f1 at the domain point
(0, 0).
(d) Use Mathematica to graph the function and its tangent plane (linearization) in the
vicinity of the domain point (0, 0).
(e) Find the equation of the quadratic (second-order) approximation of the function f1
at the domain point (0, 0).
(f) Use Mathematica to graph the function, its tangent plane (linearization), and its
quadratic (Taylor) approximation in the vicinity of the domain point (0, 0).
(g) Based on the quadratic approximation you found estimate the value of f1 (0.1, 0.2)
without using your calculator. Compare with the answer one gets for f1 (0.1, 0.2)
using the actual rule for it, and your calculator.
4
4
(3) This problem concerns the function f2 (x, y) = xy x16 y16 .
(a) Use Mathematica to graph this function in the vicinity of the domain point (2, 2).
(b) Based on the appearance of this graph predict the equation of the tangent plane to
the graph of the function f2 at the domain point (2, 2).
(c) Confirm your prediction with computation. That is, find the equation of the tangent
plane to the graph of (or find the linearization of) the function f2 at the domain point
(2, 2).
(d) Use Mathematica to graph the function and its tangent plane (linearization) in the
vicinity of the domain point (2, 2).
(e) Find the equation of the quadratic (Taylor) approximation of the function f2 at the
domain point (2, 2).
(f) Use “technology” to graph the function, its tangent plane (linearization), and its
quadratic (second order) approximation in the vicinity of the domain point (2, 2).
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(4) This problem concerns the function f (x, y) = x2 + y 2 nearby the point (3, 4). (Think
HOMEWORK PROBLEMS
50
@f
(a) Compute @f
@x (3, 4) and @y (3, 4). What do these two quantities represent?
(b) Complete the following sentence. “The distance between a particle and the origin as
the particle leaves the point (3, 4) increases at the rate of
per each unit
of displacement in the x-direction and
in the y-direction.
(c) A particle is leaving the point (3, 4) with velocity vector h1, 2i. At what rate is the
particle’s distance away from the origin changing?
(d) Complete the following sentences: “A particle is leaving the point (3, 4) with velocity
vector hv1 , v2 i. The distance between a particle and the origin as the particle leaves the
. This example is meant to illustrate
point (3, 4) changes at the rate of
the concept we call
.”
(e) Find the linear approximation of the function f at the point (3, 4).
(f) Judging by the linear approximation you just found, how far way from the origin is
the point (3.1, 3.9)? Please do not use a calculator to answer this question.
(g) Express the total di↵erential of the function f at the point (3, 4) in terms of dx and
dy, and interpret it in words.
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(5) Repeat the previous problem for the function f (x, y, z) = x2 + y 2 + z 2 and the point
( 5, 0, 12).
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