Sometimes rather than knowing the highest or lowest point it is important to know the highest or lowest point under certain constraints. For example knowing the highest point of the mountain is unimportant if your hike does not take you over the mountain’s highest point. Instead it is interesting to know the highest point on the mountain (surface) that is also on your path (constraint).
Subsection2.7.1Derivation
If you are unfamiliar with contour maps such as topographical maps, you may wish to learn about them before trying Activity 44
Figure2.7.1.Guido’s Path on a Surface
Activity44.
We begin to derive a method for determining constrained extrema by considering what should be happening on the curve in the surface at its highest and lowest points.
Guido is walking along the path marked in gray in Figure 2.7.1. The map is a contour map with approximate heights listed. We want to know when Guido is at the highest and lowest altitudes on his path.
In these questions an extreme altitude refers to a maximum or minimum altitude that is on Guido’s path.
(a)
Could he have been at an extreme altitude anywhere his path crosses from red to orange?
(b)
Could he have been at an extreme altitude anywhere his path crosses from orange to yellow?
(c)
Could he have been at an extreme altitude anywhere his path changes from gray to black?
(d)
Could a high or low occur where some contour line is crossed?
(e)
Where could a high or low on a path occur?
The problems above illustrate where extrema on a surface can occur when constrained by a curve. The next problems illustrate two means for calculating these constrained surface extrema.
The max and min temperature question in Section 2.5 demonstrated one means for calculating extrema along a curve on a surface by parameterizing the curve and using single variate extrema techniques.
Activity45.
The following illustrates a second technique, which is extensible to higher dimensions.
(a)
If a path osculates (barely touches) a contour line what is true of the tangents of both curves there?
(b)
If a path osculates (barely touches) a contour line what is true of the normals of both curves there?
(c)
Remember the following.
(i)
From Calculus 1 what does \(f^\prime(x) \gt 0\) and \(f^\prime(x) \lt 0\) imply about the curve at \(x\text{?}\)
(ii)
Why then does \(f^\prime(x)=0\) at a maximum or minimum?
Therefore what is the relation between \(\nabla f\) and the contour lines?
(d)
If \(f(x,y)\) is a surface and \(g(x,y)=k\) is some path, what does \(\nabla f = \lambda \nabla g\) have to do with the above statements?
Note this technique is called Lagrange Multipliers.
(e)
Use the idea above to determine the following. Suppose Guido is walking a path given by \(g(x,y)=x^2+y^2=1.\) If the surface on which he is walking is given by \(f(x,y)=x^2-y^2,\) where is Guido at the highest and lowest altitude?
(f)
Suppose Guido is flying such that his position is always on \(g(x,y,z)=x^2+y^2+z^2=1.\) If the temperature at each point is given by \(f(x,y,z)=2x+4y+6z,\) what is the hottest temperature experienced by Guido?
(g)
Suppose Guido is flying such that his position is always on \(g(x,y,z)=x^2+y^2+z^2=1,\) and Swen is flying such that his position is always on \(h(x,y,z)=x^2-y^2+z^2=1.\) If the temperature at each point is given by \(f(x,y,z)=2x+4y+6z,\) what is the hottest temperature experienced by both pilots?
(h)
Suppose Guido is flying such that his position is always on \(g(x,y,z)=x^2+y^2+z^2=1,\) and Swen is flying such that his position is always on \(h(x,y,z)=x^2+y^2=1.\) If the temperature at each point is given by \(f(x,y,z)=2x+4y+6z,\) what is the hottest temperature experienced by both pilots?
