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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 non-zero 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 \option-j x\, \option-j t\, etc. The keystroke option-j gives the character "$\Delta$".

Example: A Basketball rocket

A team of engineering students designed a rocket-powered 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, “∆t-intervals”). We will then assume that h has rates of change that are essentially constant over each ∆t-interval. 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 essentially-constant rate of change of h for any interval. We choose the value of v at the beginning of each ∆t-interval.

Figure 5.1.1 uses the ridiculously large value of 0.9 seconds as the width of each ∆t-interval. 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 ∆t-interval, 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 ∆t-intervals. Assuming that h changes at a constant rate over each ∆t-interval forces us to assume an approximate rate function r (left graph in Figure 5.1.2) which is constant over each ∆t-interval. 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 ∆t-interval. 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 ∆t-interval (the ∆t-interval containing the current value of t) and variation of the completed ∆t-intervals (the ∆t-intervals through which t has passed). Figure 5.1.3 illustrates the distinction between completed and current ∆t-intervals.



Figure 5.1.3. Distinction between current and complete ∆t-intervals.

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.
  1. 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".
  2. The change in accumulated altitude happens within the current ∆t-interval and is computed by multiplying the constant velocity at which the rocket travels within the current ∆t-interval and the amount of time it has traveled within the current ∆t-interval.

As the value of t varies, it is always within some ∆t-interval. We will use "left(t)" to represent the number that begins the ∆t-interval 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 ∆t-intervals of size 0.9. As t varies from 0, its current value is always within some ∆t-interval. The value of left(t) is the value of the left end of the ∆t-interval 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
∆t-interval 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 ∆t-interval 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 accumulated-altitude 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.



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 completex-intervals 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)$.

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 $v(x) = -\mathrm{cos}(x)$.

  1. 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.


  2. 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?

  3. 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.


    1. 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!)

    2. 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.

    3. 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.

  4. 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.


    1. 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.

    2. 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.

    3. Repeat question b. to estimate the total change in bacteria population in the first 2.1 hours.

    4. 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?

    5. Explain the phrases "completed intervals" and "current interval"? After explaining these phrases, relate them to the calculation you made in part c.

    6. Based on your estimates in parts b. and c, estimate the total population of bacteria at 1.6 and 2.1 hours.

    7. Estimate the total change in bacteria population from t = 2 to 3 hours. Show your work.

  5. 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.)


    1. Based on the specifications given in the instructions above, sketch the pretend mpg rate function on the same axes as the given mpg curve.

    2. 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.

    3. Represent the cumulative number of miles driven graphically, also showing the changes in distance for every 0.25 gallons on your graph.

    4. 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."

  6. In each of (a) through (e), determine the number of completed ∆x-intervals 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 ∆x-intervals between any value of a and any value of x.


  7. Value of a
    Value of x
    ∆x No. of complete ∆x-intervals 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

  8. 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 ctrl-shift-S to get a summation sign.

  9. Download and print this file. Draw arrows between the symbolic statements and the part(s) of the figure that they represent.

  10. Let $r_g$ be defined as $r_g(t)=e^{-.5t}$ and let $a=0$. Have GC calculate A(1.72) using:

  11. 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).

  12. 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 $\sum_{k=1}^{500} r(a+(k-1)\Delta x) \Delta x$ ?

  13. The Basketball Rocket engineering team was meeting one afternoon. Rachel ran into the meeting room, exclaiming
  14. We made a mistake on the velocity function! It is actually $$v(t) = 15 \frac{2^{t-10}}{2^{t-10}+1}.$$

    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?


  15. 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 $y=A(x), x≥a$.
  16. a)   Solution

    b)   Solution

    c)   Solution

    d)   Solution
     
  17. A function f has $r(x)=x(x-1.3)$ 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 $\Delta x=0.02$.

  18. 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), $1 \le x \le 3$.