Section
5.1 Introduction to Accumulation Functions
In Chapter 4 we developed the idea that knowing a function’s rate of change
at a moment of its independent variable gives information about how much the
function changes during that moment. If a function f has a momentary
rate of change of $r_f(x_0)$ at the moment that $x=x_0$, then the change in
f during that moment is $r_f(x_0)\mathrm{d}x$ as dx varies
through that moment. We do not know the function’s value at any
moment, but we have a good approximation of how much it changes
during any moment. We will return to this idea repeatedly.
The concept of an accumulation function is quite general. When the measure
of Quantity A varies at some nonzero rate of change with respect to the
measure of Quantity B, Quantity A accumulates changes at essentially a
constant rate of change in relation to sufficiently small changes in B. This
is the meaning of Quantity A having a rate of change at a moment at a value
of Quantity B. Deriving a function whose values give the net accumulation of
one quantity by knowing its rate of change at every moment of its
accumulation addresses the first foundational problem of calculus.
Namely,
You know how fast a quantity is
changing at every moment; you want to know how much of that quantity
there is at every moment.
When we know a function’s rate of change at every moment of its independent
variable, we can estimate the function’s net change as its independent
variable varies from any value in its domain to any other value in its
domain. We do this by accumulating the function’s changes as they occur
while the value of its independent variable passes through successive
moments.
An accumulation function is a function that is
generated by accumulating changes over moments of its independent
variable using information from its rate of change of accumulation
at every moment. 
The remainder of this chapter will develop the idea of approximating an
unknown accumulation function f when all we know about it is $r_f$,
f’s exact rate of change function. We will then develop a method for
creating these approximations so that GC can compute them automatically.
Finally, we will introduce a representation of exact accumulation functions
from rate of change functions that GC understands.
Reminder to Windows users: Whenever you see the
term “∆x”, “∆t” etc. in a GC display, type \Dx\,
\Dt\, etc. instead. Windows gives no easy way to type the
character "∆".
Reminder to Mac users: Whenever you see the term “∆x”,
“∆t” etc. in a GC display, type \optionj x\,
\optionj t\, etc. The keystroke optionj gives the
character "$\Delta$". 
Example: A Basketball
rocket
A team of engineering students designed a rocketpowered basketball for
their course project. They designed the motor so that within the first 90
seconds after launch, the rocket will have a velocity of $t^{1.3}$ $\mathrm
{\frac {meters}{sec}}$ t seconds after the motor starts. That is,
they know that the rocket’s velocity function v will be
$v(t)=t^{1.3}\text{, }0\le t \le 90$.
Figure 5.1.1 shows how the team anticipates the rocket’s velocity will vary
with time. However, they also need to predict the rocket’s altitude at each
moment because they have maneuvers to program that are based on the rocket’s
altitude. Unfortunately for them, their rocket will have a clock, but not an
altimeter. Their problem, then, is to develop a function h that will
give the rocket’s height at each moment after the rocket is launched when
all they can predict is the rocket’s velocity at each moment.
Figure 5.1.1. The basketball rocket. Engineers have designed the rocket
so that at t seconds after the motor starts, its velocity will be $v(t)=t^{1.3}\text{,
}0\le t \le 90$.
The solution to defining h would be simple if the rocket rose at a
constant velocity k — the rocket’s altitude would just be $h(t)=kt$.
However, the rocket’s velocity is not constant over any interval of time.
The team's actual solution draws upon the
meaning of rate of change at a moment. The function $v(t)=t^{1.3}$ gives
the rocket’s exact rate of change of altitude with respect to time at each
moment in time since launch. This means, for example, that at $t=2.1$
seconds after launch, the rocket will rise at essentially a
constant rate of $v(2.1)=2.1^{1.3} \mathrm{\frac{m}{sec}}$ for a brief
interval of time that contains $t=2.1$. As the value of t varies
through this interval, or as dt varies from 0 to the width of the
interval, dh, the change in h during this interval, will be
essentially equal to $v(2.1)\mathrm{d}t$. That is, $\mathrm{d}h=v(2.1)
\mathrm{d}t$ when $t=2.1$ sec.
The team’s dilemma at this juncture is that they do not know the widths of
the intervals over which v is essentially constant. Were they to
know these widths, they could get the total accumulated change in altitude
from $t=0$ to any current value of t by adding all the changes in h
that accumulate over these intervals of time.
However, they could experiment with values of ∆t.
Suppose that all intervals of time over which h has essentially a
constant rate of change were at least $∆t=0.1$ seconds. Then they could just
use $∆t=0.1$ seconds as the width of all the intervals. Over the first two
seconds, the rocket would rise
$$A(2)=v(0)(0.1)+v(0.1)(0.1)+v(0.2)(0.1)+...+v(1.9)(0.1)$$
where A is the accumulation function that approximates the exact
accumulation function h. We could write $A(2)$ with summation
notation as
$$A(2)=\sum_{k=0}^{19}v(k\Delta t)\Delta
t$$
Equation 5.1.1. Using summation notation to approximate the exact
accumulation function.
In principle, our approach to solving the team’s problem will start by
cutting up the time axis into intervals of width ∆t (generically, “∆tintervals”).
We will then assume that h has rates of change that are essentially
constant over each ∆tinterval. What value of v should we
choose as the constant value of v for any interval? Actually, there
is no “should” in our choice. A better question is what would be a convenient
choice for the value of v that we will take as the
essentiallyconstant rate of change of h for any interval. We choose
the value of v at the beginning of each ∆tinterval.
Figure 5.1.1 uses the ridiculously large value of 0.9 seconds as the width
of each ∆tinterval. This is just to make it easier for you to
visualize how this approach works. GC’s graph on the right is a simulation
of how the rocket’s accumulated altitude would change given that it changed
according to the approximate altitude function A with $a=0$,
$v(t)=t^{1.3}$, and $\Delta t=0.9$. Take note that even though the two
graphs look similar, it is only because the right graph has a scale on its
height axis that is more compressed than scale on the vertical axis in the
left graph.
Figure 5.1.2. The function A
approximates the exact accumulated altitude as t varies within the first
8 intervals. Partition the time axis into intervals of length 0.9
starting at 0 (right); we pretend that the rocket’s altitude changes at
essentially constant rates of change during these intervals (left). The
constant rates we assume are the values of v at the beginning of each
interval.
In Figure 5.1.2 we assume that the rocket’s altitude changes at a constant
rate throughout each ∆tinterval, starting at $t=0$. We
also assume that the constant rate over an interval of t is the
value of v at the beginning of the interval.
GC’s graph on the right shows the accumulated changes that happened by
assuming that h changes at a constant rate over each of its ∆tintervals.
Assuming that h changes at a constant rate over each ∆tinterval
forces us to assume an approximate rate function r (left graph in
Figure 5.1.2) which is constant over each ∆tinterval. All these
assumptions produce a function A (for “approximate accumulation”).
Values of A comprise an approximate altitude function that is composed
of approximate changes in altitude.
Let r be the approximate rate function illustrated in Figure 5.1.2,
the function that has a constant value within each ∆tinterval. Then,
starting at $t=0$ and letting $\Delta t=0.9$:
$r(t)=v(0)$ if $0≤t<0.9$
$r(t)=v(0.9)$ if $0.9≤t<1.8$
$r(t)=v(1.8)$ if $1.8≤t<2.7$
and so on.
Equation 5.1.2. Writing an approximate rate function.
It is worthwhile to note that there are
two types of variation happening in Figure 5.1.2: variation within a current
∆tinterval (the ∆tinterval containing the current value of t)
and variation of the completed ∆tintervals (the ∆tintervals
through which t has passed). Figure 5.1.3 illustrates the
distinction between completed and current ∆tintervals.
Figure 5.1.3. Distinction between current and complete ∆tintervals.
Using values of r to compute approximate changes in the exact
accumulated altitude function h, we get the function A, the
accumulation function that approximates the exact accumulated changes in
altitude:
Equation 5.1.3. Approximate accumulation function.
There are two things to notice about Equation 5.1.3.
 The general pattern of Equation 5.1.3 is that approximations of
accumulated altitude for any value of t are calculated in two
parts: what we might call "completed change" and "change so far".

"Completed change" is the change in altitude that happened in the
complete ∆tintervals that t passed through prior
to the ∆tinterval it is currently in. This completed
change is represented by an expression like $\left( r(0)\Delta
t+r(0.9)\Delta t\right)$. This expression represents the rocket's
approximate change in altitude according to the hypothetical
scenario that it passed through the interval $[0, 0.9]$ seconds at
$r(0)$ meters/sec and through the interval $[0.9, 1.8]$ seconds at
$r(0.9)$ meters/sec.

"Change so far" is the approximate change in altitude that has
happened from the beginning of the current ∆tinterval (the
one that contains the current value of t) to the current
value of t as t varies. This is just $dy = m
\cdot dt$, where dy is the change in altitude as dt
varies through the interval from 0 to ∆t seconds, and m
is the rocket's velocity at the beginning of the current ∆tinterval.
Keep in mind that the current value of t varies as time
passes, so both "completed change" and "current change" vary as time
passes. "Completed change" varies in chunks while "change so far"
varies continuously.

The change in accumulated altitude happens within the current ∆tinterval
and is computed by multiplying the constant velocity at which the
rocket travels within the current ∆tinterval and the amount
of time it has traveled within the current ∆tinterval.
 The constant rate at which the rocket travels is its velocity at the
beginning of the current ∆tinterval.
 The amount of time that the rocket has traveled within the current ∆tinterval
is $t\text{(the value of the left end of the }\Delta t \text{
interval containing the value of }t)$.
As the value of t varies, it is always within some ∆tinterval.
We will use "left(t)" to represent the number that begins the ∆tinterval
containing the current value of t. $$\text{left}(t)=\text{(the
value of the left end of the }\Delta t \text{interval containing the
current value of }t)$$
You might be unfamiliar with functions like $\mathrm{left}(t)$. Figure
5.1.4 illustrates the concept of left(t). It shows the independent
axis partitioned into ∆tintervals of size 0.9. As t varies
from 0, its current value is always within some ∆tinterval. The
value of left(t) is the value of the left end of the ∆tinterval
containing the current value of t. The value of dt
is the difference between the value of t and the value of left(t).
Therefore $dt = t  \mathrm{left}(t)$.
Figure 5.1.4. The meaning of the function left(t) and its relationship
with t, ∆t, and dt. In this figure,
a=0 and ∆t=0.9. The value of left(t) is the value of the left
endpoint of the
∆tinterval that contains the current value of t.
Another way to think of left(t) is given in Figure 5.1.4a. It shows
that with a = 0 and ∆t = 0.9, every real number within a
∆tinterval is mapped to the number that begins that interval.
Figure 5.1.4b shows the same thing except that it leaves a record of how
values of t are mapped to values of left(t).
Figure 5.1.4a. The top number line shows values of t
varying continuously from 0 to 3.5. The bottom number line shows values of
left(t) as t varies continuously.
Figure 5.1.4b. The top number line shows values of t
varying continuously from 0 to 3.5. The bottom number line shows values of
left(t) as t varies continuously. The video shows the
record of the mapping $t \to \mathrm{left}(t)$.
We can now summarize all the equations in Equation 5.1.2 in one:
$r(t)=v(\mathrm{left}(t))$
Equation 5.1.4. The approximate rate function in one line.
and we can summarize all the equations in Equation 5.1.3 with one equation:
Equation 5.1.5. The approximate
accumulation function in one line.
Using $\Delta t=0.9$ clearly does not give good approximations for values of
h, the exact accumulatedaltitude function that we seek to
approximate. We can get better approximations by making the value of ∆t
smaller. Figure 5.1.5 shows the same method as shown in Figure 5.1.2, but
with $ \Delta t=0.1$. GC’s graph on the right is a simulation of how the
rocket’s accumulated altitude would change given that it changed according
to the approximate altitude function A with $a=0$, $\Delta t=0.1$,
$v(t)=t^{1.3}$, and $r(t)=v(\mathrm{left}(t))$.
Figure 5.1.5: Partition the time axis
into intervals of length 0.1 starting at a=0 (right); we pretend that
the rocket’s altitude changes at essentially constant rates of change
during these intervals (left). The constant rates we assume are the
values of v at the left end of each interval.
You must understand all that is going on in Figure 5.1.5 in
order to understand how to represent what is going on symbolically.
 Partition the time axis on the right side of Figure 5.1.5 into ∆tintervals
of size 0.1 seconds.
 This imposes a partition of the time axis for the rate of change
function on the left side into ∆tintervals of size 0.1 seconds.
 As the value of t varies from $a=0$, the function r
defined as $r(t)=v(\mathrm{left}(t))$ is constant over each ∆tinterval
(see left side of Figure 5.1.5).
 The function A (right side) changes with respect to t
at the constant rate of $r(t)$ within the current ∆tinterval,
and is therefore linear over each ∆tinterval. The change in A
as t varies within the current ∆tinterval is represented
by the expression $r(t)(t\mathrm{left}(t))$.
 The change in A over completed ∆tintervals prior to
the current ∆tinterval is made by the value of t having
passed through each of them. The completed accumulation is
represented by the expression
 The total accumulation from the value of a to the current
value of t is the sum of the bits of accumulation over completed
∆tintervals plus the accumulation during the current ∆tinterval.
Equation 5.1.5, repeated below, expresses this idea symbolically.
where t is the
number of seconds elapsed since launch and A(t)
approximates the rocket's altitude at that number of seconds.
Reflection 5.1.1. The definition of A
given above defines the upper limit of the sum conceptually. It
does not use a formula for actually calculating an upper limit. Does
this matter in terms of defining the function A? Explain.
Reflection 5.1.2. The definition of A
uses the function named "left" even though left(x) is still
not defined computationally. Does this matter in terms of
understanding how the definition of A is supposed to work?
Explain.
Reflection 5.1.3. Why is it important to
use "number of complete ∆xintervals from a
to x" as the upper limit of the summation?
Reflection 5.1.4. Equation 5.1.5 defines A
in terms of t. The equation immediately above defines A
in terms of x. Does this matter? Explain.
Reflection 5.1.5. How does the term
$r(t)(t\mathrm{left}(t))$ in Equation 5.1.5 represent "accumulation
so far in the current $\Delta x$interval"?
Reflection 5.1.6. Suppose we defined a function
named "dt" as $ \mathrm{dt}(t)=t\mathrm{left}(t)$.
 What would dt(t) mean for any value of t,
given values a and $\Delta t$?
 What would $r(t)\mathrm{dt}(t)$ mean for any value of t?
 How would this change the definition of A as given in
Equation 5.1.5?
Exercise Set 5.1
For Questions 1  3. Consider the video below. It shows a ball hanging
from a board by a rubber cord. The ball hangs at rest 10 feet below
the board. At $x = 0$ seconds, the ball is set in motion with a
quick shove downward, causing the rubber cord to stretch and the ball to
bounce. The graph shows the rate of change of the ball's
displacement from rest with respect to elapsed time, which is the velocity
function $v(x) = \mathrm{cos}(x)$.
 Here is a printable
version of the ball's velocity graph. Sketch an approximate
velocity function for the bouncing ball scenario on the graph, and fill
in the 2nd column 'Approx Velocity' in the table below (click here
for a printable version). (You'll do the 3rd and 4th columns later).
Determine the approximating constant velocities using intervals of
length 0.5 seconds, with each constant velocity equal to the true
velocity at the left end of the interval.
 Fill in columns 3 and 4 of the table you printed from Exercise 5.1.1
based on your approximate velocities and the mathematics of constant
rate of change. Are these values exact or approximate?
 The graph below shows the function that gives the net accumulated
changes in the ball’s displacement from its resting position relative to
elapsed time. It is linear over each $\Delta x$interval. The graph
shows the ball's approximate displacement y, in feet, from its
resting position, x seconds after the ball was put in motion.
 How are individual values from the first few rows of the table
reflected in the portion of the graph shown above? Label each element
of the given portion of the graph with the appropriate variable and
table value. (Hint: columns 3 & 4 both apply!)
 Complete the graph in the exact same manner as the given portion,
including all elements, and labeling each element of your sketch with
variables and values from the table.
 Use the graph you made in part (c) to sketch a second graph—a graph
that shows the total accumulation of distance of the ball from the
board to the ball at each moment in time after the ball was pushed.
 A bacterial culture began with 10,000 bacteria. The graph below (click
here for a
printable version) shows the rate, measured in bacteria per hour, at
which the number of bacteria grew at each moment in time (measured in
hours) after the culture was first observed. The rate fluctuated because
the culture’s environment fluctuated.
You will use the method illustrated in this section to build an
approximate accumulation function B, which outputs the number of
bacteria in this culture t hours after the initial observation.
 Represent on the graph above a modified ROC function, for $t = 0$ to
3.2 hours, based on constant rates over 0.4 hour intervals, with each
constant rate of change being the actual rate of change at the left
side of its respective interval.
 Using your sketch of the modified rate function and estimates of
rates of change from the displayed graph, calculate the total change
in bacteria population in the first 1.6 hours. (The constant rates you
use will be estimates from the graph, since no function rule for the
rate of change is given.) Show all your work.
 Repeat question b. to estimate the total change in bacteria
population in the first 2.1 hours.
 Besides using a different number of intervals and getting a
different answer, what was different about computing the answer for
part c. compared to part b?
 Explain the phrases "completed intervals" and "current interval"?
After explaining these phrases, relate them to the calculation you
made in part c.
 Based on your estimates in parts b. and c, estimate the total
population of bacteria at 1.6 and 2.1 hours.
 Estimate the total change in bacteria population from t = 2 to 3
hours. Show your work.
 In a test drive of a new fuel efficient car, a company tracks the
car's mpg (miles per gallon) for 2 gallons of gasoline burned during a
test drive. Use the approximating method developed in this section with
intervals of 0.25 gallons of gasoline and the "left end" approach to
complete the items below. (Click here
for a printable version.)
 Based on the specifications given in the instructions above, sketch
the pretend mpg rate function on the same axes as the given mpg curve.
 Represent numerically approximations for the changes in the number
of miles driven and the cumulative number of miles driven for every
0.25 gallons of gasoline burned.
 Represent the cumulative number of miles driven graphically, also
showing the changes in distance for every 0.25 gallons on your graph.
 Approximate the total change in miles driven for the first 0.9
gallons burned. Explain your calculations in terms of "completed
intervals" and "the current interval."
 In each of (a) through (e), determine the number of completed ∆xintervals
between the given value of a and the given value of x.
Have the aim of stating a general method for finding the number of ∆xintervals
between any value of a and any value of x.

Value of a

Value of x

∆x 
No. of complete
∆xintervals between a and x

a) 
2 
17 
0.03 

b) 
3.2 
5.23 
0.0125 

c) 
3 
1.2 
0.025 

d) 
1 
5 
1.2 

e) 
5 
128.3 
0.2 

 Enter a sum into GC that will compute A(4.72) given that $a=2$,
$\Delta t=0.01$ and $r_f(x)=\sin(x)$. Type ctrlshiftS to get a
summation sign.
 Download and print this
file. Draw arrows between the symbolic statements and the part(s)
of the figure that they represent.
 Let $r_g$ be defined as $r_g(t)=e^{.5t}$ and let $a=0$. Have GC
calculate A(1.72) using:
a) $\Delta t=1$
b) $\Delta t=0.1$
c) $\Delta t=0.01$
d) $\Delta t=0.001$
e) $\Delta t=0.0001$
f) Now describe the trend in the values of A(1.72) with respect
to values of ∆t in parts (a)  (e).
 Suppose that $\Delta x=0.125$, $a=2$, and $r_f(x)=x\cos(x)$. What will
be the numerical value of the 134th term in $\sum_{k=1}^{500}
r(a+(k1)\Delta x) \Delta x$ ?
 The Basketball Rocket
engineering team was meeting one afternoon. Rachel ran into the meeting
room, exclaiming
We made a mistake on the velocity
function! It is actually $$v(t) = 15 \frac{2^{t10}}{2^{t10}+1}.$$
a) What needs to be changed in the system
of statements that calculates approximations of the rocket’s height
for values of elapsed time?
b) What would the graph of elevation
relative to elapsed time look like now?
 Each of (a) through (d) is GC’s displayed graph of an exact rate of
change function for an accumulation function. Let $a=1$ and $\Delta
x=0.001$ for each of (a)(d). For each graph, sketch a graph of $y=A(x)$
that you anticipate that GC would produce. Click the solution link on
each line after you have sketched your anticipated graph of $y=A(x),
x≥a$.
a) Solution
b) Solution
c) Solution
d) Solution
 A function f has $r(x)=x(x1.3)$ as its rate of change
function. Here
is a table of values of x, values of r(x),
approximate changes in f, and approximate net accumulation as
a function of x, all with $\Delta x=0.02$.
a) What is the value of A(2.74)?
b) Express A(2.74) using summation notation.
c) What is the value of A(2.743)?
d) Express A(2.743) using summation notation.
e) Sketch a graph of y = A(x), $2.70 \le
x \le 2.86$.
f) Sketch a graph of y = A(x), $1 \le x
\le 3$.