Download Report

The Accumulation Function - Classwork
Now that we understand that the concept of a definite integral is nothing more than an area, let us consider a very
special type of area problem. Suppose we are given a function f ( x ) # 2. We know that to be a horizontal line at
y = 2. First, realize that the equation of the graph of f ( x ) # 2 is the same as f ( t ) # 2 is the same as f ( k ) # 2.
Whether we use x, t, or k, it does not matter. The graph is still a horizontal line. We are used to using x but we
will see a good reason that we will occasionally be using another letter.
f (t ) # 2
Consider the expression
x
! f (t ) dt
0
x
Above, we have the graph f ( t ) # 2. We now want to consider the expression
! f (t ) dt . What is this? It is a
0
x
function of x. As x changes,
! f (t ) dt changes as well. Complete the chart below.
0
x
0
1
2
3
4
5
x
! f (t ) dt
0
x
It should be apparent that as x gets bigger,
! f (t ) dt increases as well. What we appear to be doing is
0
x
“accumulating area” and we call
! f (t ) dt
the accumulation function. Finally, it should be obvious why we use
0
x
f ( t ) # 2 to describe our line rather than f ( x ) # 2.
! f ( x ) dx would be confusing.
0
Let' s calculate the value of
x
! f (t ) dt for x # 0 to 4
f (t )
0
Complete the chart below
x
0
1
2
3
4
x
! f (t ) dt
0
MasterMathMentor.com
- 147 -
Stu Schwartz
x
Example 1) Let F ( x ) #
! f (t ) dt
where the graph of f ( x ) is below. Remember f ( x ) is the same thing as f ( t ) .
0
Think of f ( x ) as the rate of snowfall over a period of time. For instance, at x = 1, snow is falling
at 3 inches per hour, at x = 3, it is not snowing, and at x = 4, snow is melting at 4 inches per hour.
y # f ( x)
a. Complete the chart. In the snow analogy, F ( x ) represents the accumulation of snow over time.
x
0
1
2
3
4
5
6
6.5
7
8
F ( x)
b. Now let’s consider F "( x ) #
d
dx
x
! f (t ) dt . If we take the derivative of an integral, what would you expect to
0
d
happen? __________________ So F "( x ) #
dx
Knowing that, let’s complete the chart.
x
0
1
2
3
4
F "( x )
c. On what subintervals of [0, 8] is F increasing?
x
! f (t ) dt is the same thing as
_____________.
0
5
6
6.5
7
8
Decreasing?
d. Where in the interval [0, 8] does F achieve its minimum and maximum value? What are those values?
f. Find the concavity of F and any inflection points. Justify your answer
h. Sketch a rough graph of F ( x )
MasterMathMentor.com
- 148 -
Stu Schwartz
x
Example 2) Let F ( x ) #
! f (t ) dt
where f is the function graphed below (consisting of lines and a semi-circle)
0
y # f ( x)
Find the following:
a) F (0)
b) F (2)
c) F ( 4 )
d) F (6)
e) F (#1)
f) F (#2)
g) F (#3)
h) F (#4 )
i) F "( 4 )
j) F "(2)
k) F "(6)
l) F "(#3)
m) On what subintervals of $#4, 6% is F increasing and decreasing. Justify your answer.
n) Where in the interval $#4, 6% does F achieve its minimum value? What is the minimum value?
o) Where in the interval $#4, 6% does F achieve its maximum value? What is the minimum value?
p) Where on the interval $#4, 6% is F concave up? Concave down? Justify your answer.
q) Where does F have points of inflection?
r) Sketch the function F .
each tic mark
on y-axis is
2 units
MasterMathMentor.com
- 149 -
Stu Schwartz
The Accumulation Function - Homework
y # f ( x)
x
1. Let F ( x ) #
! f (t ) dt
where the graph of f ( x ) is above (the graph consists of lines and a quarter circle)
0
a. Complete the chart
x
0
1
F ( x)
F "( x )
2
3
b. On what subintervals of [0, 8] is F increasing?
4
5
7
8
Decreasing? Justify your answer.
c. Where in the interval [0, 8] does F achieve its minimum value? What is the minimum value? Justify answer.
d. Where in the interval [0, 8] does F achieve its maximum value? What is the maximum value? Justify answer.
e. Find the concavity of F and any inflection points. Justify answers.
f. Sketch a rough graph of F ( x )
MasterMathMentor.com
- 150 -
Stu Schwartz
y # f ( x)
x
2. Let F ( x ) #
! f (t ) dt
where the graph of f ( x ) is above (the graph consists of lines and a semi-circle)
0
a. Complete the chart
x
-4
-3
F ( x)
F "( x )
-2
-1
0
1
2
3
4
5
6
7
8
b. On what subintervals of $#4, 8% is F increasing? Decreasing?
c. Where in the interval $#4, 8% does F achieve its minimum value? What is the minimum value? Justify answer.
d. Where in the interval $#4, 8% does F achieve its maximum value? What is the maximum value? Justify answer.
e. On what subintervals of $#4, 8% is F concave up and concave down? Find its inflection points. Justify
answers.
f. Sketch a rough graph of F ( x )
MasterMathMentor.com
- 151 -
Stu Schwartz
The Accumulation Function - Life Application
You have bought 100 shares of XYZ stock and decide to keep it for 15 days. Below is a graph that represents the
change of the price of the your stock on each day. For instance, on the end of day 1, the stock has increased by $1 a
share. At the end of day 4, it has not changed. At the end of day 6, it has gone down by $3.50 a share.
We will call the graph you see below f ( t ) . made up of straight lines. It is obviously a function of time. Remember
that the function represents the change in the value of your stock, not the value of the stock.
1
2 3 4
5
6
7
8
9
10 11 12 13 14 15
x
Let F ( x ) #
! f (t ) dt . Answer the following questions:
0
1. Complete the chart below.
x
f ( x)
F ( x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2. What is the real life meaning of F ( x ) ?
3. For what intervals is F ( x ) increasing and decreasing. Justify your answer.
4. In the interval [0, 15], find the minimum value of F ( x ) and the day it is reached. Justify your answer.
5. In the interval [0, 15], find the maximum value of F ( x ) and the day it is reached. Justify your answer.
6. Determine the concavity of F ( x ) and justify your answer.
MasterMathMentor.com
- 152 -
Stu Schwartz
7. Below, sketch F ( x ) .
8) Based on your findings, find:
a) how much money you made(lost) on XYZ stock in that 15 day time period.
b) the day that the stock’s value had the biggest rise
c) the days between which the stock’s value had the steepest rise (not the same question)
d) the day that the stock’s value had the biggest decline
e) the days between which the stock’s value had the steepest decline (not the same question)
f) the day you wished you sold your stock
g) the day you are glad you didn’t sell your stock
MasterMathMentor.com
- 153 -
Stu Schwartz