Chapter 4: Partial Differentiation
Section 4.10: Optimization on Closed Domains
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Example 4.10.4
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Find the extreme values of the function on the domain consisting of the interior and boundary of the disk whose radius is 4, and whose center is at the origin.
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Solution
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Mathematical Solution
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The unconstrained optimization problem requires solving the equations , that is, the equations
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for , a point interior to the disk. At this point, . See Figure 4.10.4(a) where the points and are shown as black dots.
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For the constrained problem, describe the boundary in polar coordinates , , , and find the extrema of
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The distinct critical numbers are , as corroborated by the graph of in Figure 4.10.4(a).
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use plots, plottools in
module()
local p1,p2,p3,p4,f,d;
f:=(x^2-x*y+y^2+4*y+10)*(1/10);
d:=sqrt(16-x^2);
p1:=plot3d(f,x=-4..4,y=-d..d,shading=xy);
p2:=plot3d(0,x=-4..4,y=-d..d,color=green);
p3:=pointplot3d([[-4/3,-8/3,0],[-4/3,-8/3,7/15]],symbol=solidsphere,symbolsize=20,color=black);
p4:=display(p1,p2,p3,labels=[x,y,z],tickmarks=[4,4,5],style=surfacecontour,lightmodel=none,orientation=[-55,75,0],axes=frame);
print(p4);
end module:
end use:
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Figure 4.10.4(a) Surface over disk
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plot((13-8*cos(t)*sin(t)+8*sin(t))/5,t=0..2*Pi,0..5);
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Figure 4.10.4(b)
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Table 4.10.4(a) lists the boundary extrema in Cartesian coordinates.
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Table 4.10.4(a) Boundary extrema
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Therefore, the absolute minimum is , and the absolute maximum is = ≐ .
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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Context Panel: Assign Function
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Numerically find any unconstrained extrema
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Context Panel: Optimization≻Minimize (local)
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Analytically find any unconstrained extrema
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Calculus palette: Partial-derivative operator
Press the Enter key.
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Context Panel: Solve≻Solve
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Context Panel: Evaluate and Display Inline
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Optimization Assistant: Optimize numerically subject to the boundary constraint
The Optimization Assistant is no longer listed in Tools≻Assistants. It is now found in Tools≻Tutors≻Optimization
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Form the sequence and press the Enter key.
Context Panel: Optimization≻Optimization Assistant
Use the
to obtain extrema on the boundary.
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Figure 4.10.4(c) shows the assistant finding a maximum on the circumference of the circle.
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Figure 4.10.4(d) shows a graph of the surface determined by , with the circular boundary of lifted to this surface.
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Select "Minimize" and/or add additional constraints on and to find other extrema on the boundary.
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Figure 4.10.4(c) Constrained maximum
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Figure 4.10.4(d) Surface and constraint
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Apply the boundary constraint analytically
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Write , where describe the boundary of ; press the Enter key.
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Context Panel: Simplify≻Simplify
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Context Panel: Differentiate≻With Respect To≻
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Context Panel: Student Calculus1≻Solve≻Find Roots
See Figure 4.10.4(e)
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Context Panel: Evaluate and Display Inline
(Evaluate at each distinct root.)
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Figure 4.10.4(e) Roots dialog
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Maple Solution - Coded
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Initialize
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Find any unconstrained extrema
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Obtain the equations via the diff command, and their solution via the solve command.
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Find extrema on the circumference of the disk
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Representing the circumference of the disk in polar coordinates, use the diff command to obtain the derivative with respect to .
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Set that derivative equal to zero and, for the resulting equation, use the Roots command in the Student Calculus1 package to find all solutions in the interval .
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Evaluate at each distinct solution.
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