Section 7.2 Optimization
Our next application concerns “optimization problems”. Suppose that you are interested in maximizing (or minimizing) a certain quantity, which may depend on a number of factors. How can you determine how these factors should be adjusted so that the quantity of interest is maximized (or minimized)? It turns out that differentiation is the tool of choice for answering questions of this type.
Objectives
You should be able to:
- Introduce notation to transform an optimization problem into a calculus question about finding the extremum of a function.
- Solve various optimization problems from physics, chemistry, economics, population dynamics, etc. by applying calculus techniques to find the extremum of a function.
Subsection 7.2.1 Instructional video
Subsection 7.2.2 Key concepts
Concept 7.2.1. Strategy to solve optimization problems.
- Read the problem carefully.
- Draw a picture if possible.
- Introduce notation. Assign a symbol (say \(Q\)) to the quantity that is to be maximized or minimized, and symbols to the other unknown quantities relevant to the problem.
- Write an equation for \(Q\) in terms of the other unknowns of the problem. If \(Q\) is expressed in terms of more than one variables, use the information of the problem to find relationships between these variables, and eliminate all but one the variables in the expression for \(Q\text{.}\) Then \(Q\) will be expressed as a function of a single variable, say \(Q = f(x)\text{.}\)
- Find the absolute maximum or minimum value of \(f(x)\) over the range of \(x\) allowed by the problem. If this is a closed interval in \(x\text{,}\) you can use the closed interval method, otherwise, you need to find the local min or max and justify why a given local min or a max is the absolute max or min of \(f(x)\text{.}\)
Further readings 7.2.3 Further readings
[1]