The Average Value of a Function
We can use the definite integral to find the average value of a function.
Suppose f is an integrable function on the interval [a, b]. Then the average value of f over
1 b
the interval is
f ( x) dx . This is what average value represents:
b − a ∫a
Example 7: Find the average value of f ( x ) = x over the interval [1, 16].
Example 8: Find the average value of f ( x) = x 2 − 3 x + 5 on [2, 5].
Example 9: The sales of ABC Company in the first t years of its operation is
approximated by the function S (t ) = t 0.2t 2 + 4 where S (t ) is measured in millions of
dollars. What were the company’s average annual sales over its first five years of
operation?
From this lesson, you should be able to
Use the FTOC to compute definite integrals, including problems that require
substitution
Define average value
Use the FTOC to find average value
Solve word problems using FTOC
Math 1314
Lesson 21
Area Between Two Curves
Two advertising agencies are competing for a major client. The rate of change of the
client’s revenues using Agency A’s ad campaign is approximated by f(x) below. The rate
of change of the client’s revenues using Agency B’s ad campaign is approximated by g(x)
below. In both cases, x represents the amount spent on advertising in thousands of
dollars. In each case, total revenue is the area under the curve given in thousands of
dollars.
Agency A
Agency B
This graph shows the relationship between the two revenue functions. We see that one
function is above the other. The area between the two functions represents the projected
additional revenue that would be realized by using Agency B’s ad campaign.
This is an example of the kinds of problems you will be able to solve with the techniques
you learn in this lesson.
We can compute the area between the two curves. The general “formula” is
b
∫ (top function - bottom function)dx
a
Example 1: Find the area between the two curves, f ( x ) =
g ( x) = − x − 1 from x = 0 to x = 2.
1 2
x + 2 and
5
Example 2: Find the area between the function f ( x) = x 3 − x and the x axis from x = -1
to x = 1.
Example 3: Find the area between the functions f ( x) = x 2 − 4 x and g ( x ) = 4 x .
Example 4: Find the area between the functions f ( x) = 3 x 2 + 2 and g ( x) = x − 3 and
the vertical lines x = -1 and x = 3.
Example 5: Without any effort to curb population growth, a government estimates that
its population will grow at the rate of 60e .02 t thousand people per year. However, they
believe that an education program will alter the growth rate to − t 2 + 60 thousand people
per year over the next 5 years. How many fewer people would there be in the country if
the education program is implemented and is successful?
Example 6: The management of a hotel chain expects its profits to grow at the rate of
2
1 + t 3 million dollars per year t years from now. If the renovate some of their existing
hotels and acquire some new ones, their profits would grow at the rate of t − 2 t + 4
million dollars per year. Find the additional profits the company could expect over the
next ten years if they proceed with their renovation and acquisition plans.
I’ll use a graphing utility to generate the graphs of these two functions.
From this lesson, you should be able to
Find the area between two curves
Sketch the graphs
Set up the necessary integral(s)
Find points of intersection if necessary
Integrate and evaluate
Solve word problems involving the area between two curves
Math 1314
Lesson 22
Functions of Several Variables
So far, we have looked at functions of a single variable. In this section, we will consider
functions of more than one variable. You are already familiar with some examples of
these.
P ( x, y ) = 2 x + 2 y
A( P, i, t ) = P(1 + i ) t
These formulas are functions of several variables. We have just never called them that
before. We will, for the most part, limit our discussion to functions of two variables.
Functions of Two Variables
Definition: A real valued function of two variables, f, consists of a set A of ordered
pairs of real numbers (x, y) called the domain of the function, and a rule that associates
with each ordered pair in the domain of f one and only one real number, denoted by
z = f ( x, y ).
You will need to learn several skills using functions of several variables:
1. Evaluating a function of several variables at a given point.
Example 1: Suppose f ( x, y ) = 3 x 2 y − 4 xy + 6. Compute f (0,0), f ( 2,−1) and f ( −1,−3).
Example 2: The volume of a cylindrical tank with radius r and height h is given by the
formula V = f (r , h) = πr 2 h . Find the volume of a tank with radius 6 feet and height 20
feet.
Example 3: Suppose you borrow money to buy a house. You agree to a 30-year loan
with equal payments. This formula gives the amount of the reduction in principal of the
loan (given in dollars) after making i payments on a loan of A dollars amortized over t
years.
i


r 
 1 +  − 1 
12 

B = f ( A, r , t , i ) = A  
12 t


r 
 1 +  − 1
  12 

Suppose you borrowed $180,000 at 6% annual interest. How much has the balance of the
loan gone down after making 60 payments? How much is the remaining balance of the
loan after 20 years of payments?
2. Find the domain of a function of several variables. We’ll have several kinds of
situations that are similar to finding domain of functions of a single variable. Problems
arise when we have rational functions, functions involving radicals and logarithmic
functions.
Example 4: Find the domain of the function f ( x, y ) = 2 x 2 + 3 y 2 .
Example 5: Find the domain of the function f ( x, y ) =
3x
.
2x − 5 y
Example 6: Find the domain of the function f ( x, y ) = 16 − x 2 − y 2 .
Graphing functions in space is quite difficult. You will not need to do this. Here are a
couple of examples of graphs of functions of two variables.
Hopefully, you’ll recognize how difficult it would be to graph these manually. We
usually graph function of two variables using a computer.
Here’s an example of graphing just one point in space.
Example 7: Graph the point (2, 3, 5).
From this section, you should be able to
Evaluate a function of several variables
Find the domain of a function of several variables