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Chapter 3: Applications of Differentiation
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Section 3.8: Optimization
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Essentials


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The unconstrained optimization problem in differential calculus of a single variable consists in the search for the absolute extreme of an objective function of one variable. If the objective function contains no parameters, its graph can be inspected for the extreme value, and the tools of Section 3.7 used to determine the exact location and value of the absolute extremum. Hence, the unconstrained optimization problem has essentially been considered in Section 3.7.
The constrained optimization problem in differential calculus of a single variable can come in one of two forms. If the domain of the objective function whose extremum is sought is restricted, then in addition to any extrema in the interior of the domain, the endpoints of the interval restricting the domain must also be examined. Again, this problem was examined in Section 3.7.
The more prevalent form of the constrained optimization problem consists in an objective function of two variables, $f\left(x\,y\right)$, with a constraint of the form $g\left(x\,y\right)\=0$ that is used to eliminate one of the variables in $f$. If, for example, the equation $g\=0$ is solved for $y\=y\left(x\right)$, then the objective function can be written as $f\left(x\,y\left(x\right)\right)\equiv F\left(x\right)$, and the extreme value of $F$ sought by the techniques discussed in Section 3.7. The challenge for the student is in translating a description of a task into a formulation of the objective function $f$ and the constraint $g\=0$.
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Examples


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Example 3.8.1

Of all rectangles with perimeter 100, find the one with maximal area.

Example 3.8.2

Show that among all rectangles with a fixed area, the square has the minimum perimeter.

Example 3.8.3

One acre (43560 sq ft) is to be enclosed with a rectangular fence that costs $5.00 per ft. Additionally, the enclosed area is to be subdivided into three equal rectangles with fencing that costs $3.50 per ft. What is the minimal cost of the construction, and what are its dimensions?

Example 3.8.4

A wire of length $L$ (cm) is cut into two pieces. One piece is bent into an equilateral triangle; the other, into a circle. Find the maximum and minimum values for the sum of the areas enclosed by the two pieces of wire.

Example 3.8.5

Which point on the graph of $f\left(x\right)\=\mathrm{sinh}\left(x\right)x{e}^{3x}$, $1\le x\le 3$, is closest to the point $\left(1\,7\right)$?

Example 3.8.6

Two posts are in a line perpendicular to a straight road, one at a distance of 20 m from the road, the other at a distance of 50 m. Where on the road is the angle formed by the lines of sight to the posts a maximum?

Example 3.8.7

Find the dimensions of the largest rectangle that can be inscribed in the upper half of the ellipse $5{x}^{2}plus;12{y}^{2}equals;60$.

Example 3.8.8

A hiker in a forest is at point $A$, five miles from $B$, the nearest point along a straight road bounding the forest. The hiker wishes to get to point $C$, some eight miles down the road from $B$. If the hiker can walk at 2 mph in the woods, and 4 mph along the road, where should the hiker emerge from the woods to complete the journey in least time?

Example 3.8.9

A telecommunications company contemplates running a fiberoptic cable from point $A$ to point $C$ in the woods and along the road in Example 3.8.8. If the ratio of costs (woods/road) is $\mathrm{\λ}\>1$, determine the characteristics of a minimalcost solution to this problem of laying a communications cable.

Example 3.8.10

The strength of fields such as gravity or electromagnetism is inversely proportional to distance from the source. If two such sources are separated by a distance of $L$ units, determine where on the line connecting the sources will the minimum strength be detected? Assume that the strength of one source is $k$ times that of the other.

Example 3.8.11

A can in the shape of a rightcircular cylinder, fashioned from a rectangle rolled into a cylinder and disks welded at the top and bottom, must have volume $V$. What are the dimensions of the leastcost can if the side costs $a$ per square inch and the top and bottom cost $b$ per square inch?

Example 3.8.12

An 8 × 10 rectangular sheet of paper is oriented with the long sides vertical. The upper left corner is placed along the right edge, and the paper folded. Where should the upper left corner be located so that the crease so formed has minimal length?

Example 3.8.13

Snell's law,
$\frac{\mathrm{sin}\left({\mathrm{\θ}}_{1}\right)}{\mathrm{sin}\left({\mathrm{\θ}}_{2}\right)}\=\frac{{v}_{1}}{{v}_{2}}$
relates the speed of light in two adjacent media with the angles of incidence (${\mathrm{\θ}}_{1}$) and refraction (${\mathrm{\θ}}_{2}$). See Figure 3.8.13(a).
Derive Snell's law from Fermat's principle, namely, that light traverses the path for which travel time is least.


Figure 3.8.13(a) Light passing from point $A$ to point $B$







Example 3.8.14

Find the length of the longest ladder that can be carried horizontally around the corner of the passageway shown in Figure 3.8.14(a). (The horizontal and vertical segments, corridors of widths $b$ and $a$, respectively, are at right angles to each other.)


Figure 3.8.14(a) Ladder in rightangled corridor




Hint: The longest ladder that can be carried around the corner at point $B$ is the shortest line segment from $A$ to $C$ that also passes through $B$.
Hint: Angles $\mathrm{ABE}$ and $\mathrm{BCF}$ are equal because they are corresponding interior angles of the parallel lines $\mathrm{BE}$ and $\mathrm{CF}$.






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