Businesses often want to identify a nonlinear function's maximum and minimum points, which are also known as extrema.
The maximum point on a production function will tell a manager the
point at which marginal production levels are at their highest. The
minimum point on a cost function will tell managers where marginal unit
costs are at their lowest. Extrema occur at the point at which slope is
zero. Therefore, maximum and minimum points can be found by taking the
first derivative of the function, setting it equal to zero, and solving
for x.
This use of derivatives has many important practical applications,
including identifying a project team's maximum unit output of work.
Consider a department that is assembling a project team to develop a
new web site. The company has many projects going on concurrently and,
although the web site development project is important, the company
would like to staff it optimally so that the project is completed in
the fastest manner possible and with the "right" number of people.
The manager of this department hires you to determine the level of
staffing at which each team member on the project will have the
greatest level of output in terms of the tasks completed per week. You
may have experienced this scenario on a project team for a job or
school project in the past. If the team has too few people, team
members are spread too thin across tasks and seldom get tasks
completed. A team with too many people may result in additional time
required in meetings to coordinate the activities of all the team
members, resulting in fewer tasks being completed by each individual
team member.
For this particular team, the function of productivity of an individual team member is stated as f(x) = –x^{2} + 20x – 80.
To begin to determine the optimal number of people for the team, find the first derivative of the productivity function.
When the first derivative is positive, the individual output
continues to increase as people are added to the team. When the first
derivative is negative, the individual output decreases as people are
added to the team. Individual output is maximized at the point at which
the individual output can no longer increase and has not yet begun to
decrease. This maximum point occurs at the point at which the
derivative is zero.
You find that the department should assign 10 people to the
development team in order to maximize the individual output. To
determine the individual output when 10 people are assigned to the
team, insert the quantity 10 into the original function.
The number of tasks completed when individual productivity is
maximized is 20 tasks per week. This occurs when 10 people are assigned
to the project team.
Second Derivatives
The second derivative can be written as f "(x), which can be expressed verbally as "f double prime of x."
While the first derivative measures the rate of change of a function,
the second derivative measures whether this rate of change is
increasing or decreasing.
To find the second derivative, simply take the derivative of the
first derivative. Look at the following example of a function and its
first and second derivative.
1. Consider the graph below: Is the slope of the graph at point B positive, negative, or zero?
Solution 1
2. Which point on the following graph has a slope of zero?
Solution 2
3. The two points of a secant line should be _____ to ensure a good approximation of the slope. [fill in blank]
Solution 3
4. Find the first derivative of the following function:
f(x) = 4x^{3} + 3x^{2} – 8
Solution 4
5. Find the slope of the following function at x = 2:
f(x) = 7x^{2} – 9x + 3
Solution 5
6. If f(x) = x^{2} – 4x + 8 , find the average rate of change between the points x = 2 and x = 2.1.
Solution 6
7. The amount of output a company can produce is dependent on the
number of workers. A company's production function is defined as f(x) = –x^{2} + 40x; where x is the number of workers employed, and f(x)
is the number of units produced by the company. The company currently
has six employees; find the average rate of change (slope of the secant
line) of output of adding one more employee.
Solution 7
8. A company produces both color printers and blackandwhite
printers. Because the materials used to produce the two printers are
mostly the same, the company faces a trade off between the number of
color printers produced and the number of blackandwhite printers
produced. The function that models this relationship f(x) = –.25x^{2} + 36; where x is the number of color printers, and f(x) is the number of blackandwhite printers. If
the company is currently producing 10 color printers, what is the
instantaneous rate of change (slope of the tangent) at this point?
Solution 8
9. A computer company sells a particular desktop model for $900. The company has a cost function of f(x) = 3x^{2} + 36x + 6,000, where x
is the number of units produced. To maximize profit, how many desktop
computers should be produced, and how much will the company's profit
be? (Note: Profit = Revenue – Cost)
Solution 9
