College Algebra & Trigonometry, 2018a

236 CHAPTER 4. FUNCTIONS
4.9 Optimization
One of the major applications of differential calculus is optimization. This is the
process of finding maximum or minimum function values for a given relationship.
There are four typical types of problems that we will examine in this section.
a) Analytic Optimization - these problems typically use the distance formula
to determine the closest point to a particular curve.
b) Geometry/Cost Optimization - these problems generally give a box or container
of a particular shape and ask either to determine the cheapest manufacturing
cost given a particular volume or to determine the greatest volume given a
particular cost.
c) Distance Optimization - these problems generally use two objects travelling
at right angles to each other and determine the maximum or minimum distance
between the objects.
d) Distance/Cost Optimization - these problems are usually focused on a situation
in which two distances at right angles can be cut with a diagonal at a certain
point to minimize cost or time.
Analytic Optimization
d = √ (x 2 − x 1 ) 2 +(y 2 − y 1 ) 2
1) Express the distance of a point (x, y) (in the first quadrant) on the graph of
the parabola y = x 2 − 4 from the point (5, 2) as a function of x.
2) Use the graph of the distance function d(x) from Part I to determine the
point of the graph of the parabola y = x 2 − 4 that is closest to the point (5, 2).
3) How far is this point from the point (5, 2)?