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Section 5.1Introduction 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 varies 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 $r_f(x_0)dx$ 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 varies 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 variations at a rate that is essentially constant in relation to sufficiently small variations 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 varying 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. We do this by accumulating variations in the function’s value 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 variations in its value over moments of its independent variable.

The accumulation occurs by the function varying at some rate at every moment of its independent variable as its independent variable varies.

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: You cannot type "$\Delta x$", "$\Delta t$" etc. in GC. Instead, type "\Dx\", "\Dt\", etc. Windows gives no easy way to type the character "∆".

Reminder to Mac users: To type “$\Delta x$”, "$\Delta t$" etc. in GC, type "\option-j x\", "\option-j t\", etc. The keystroke option-j gives the character "$\Delta$" on a Mac.

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 $t^{1.3}$ m/s t seconds after the motor starts. That is, they know that the rocket’s velocity function v will be $v(t)=t^{1.3},\, 0\le t \le 90$.

Figure 5.1.1 shows how the team anticipates the rocket’s velocity will vary with time.

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 team must also 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 such that $h(t)$ 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.

The solution to defining h would be simple if the rocket rose at a constant velocity k. The rocket’s altitude would be $h(t)=kt$. Unfortunately, the rocket’s velocity is not constant over any interval of time within 90 seconds after launch. On the other hand, its velocity is essentially constant over sufficiently small intervals of time within 90 seconds after launch.

The team cannot determine a definition of $h$. However, they can approximate values of $h(t)$ by drawing on their understanding 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 a rate of change that is essentially $v(2.1)=2.1^{1.3} \mathrm{m/s}$ 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 as t varies through this interval, will be essentially equal to $v(2.1)dt$.
• The team can approximate any value of $h(t_0)$ by adding all the variations in $h$ as as t varies from 0 to $t_0$ through intervals of time that are small enough that $v(t)$ is essentially constant through each interval.

The team’s dilemma at this juncture is that they do not know the widths of the time intervals over which $v(t)$ 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 variations in h that accumulate over these intervals of time.

However, they can experiment with values of $\Delta t$.

Suppose that h has essentially a constant rate of change over time intervals of length $$\Delta t=0.1 seconds. Let A be the function whose values A(t) approximate values of h(t). Over the first two seconds, the rocket would rise$$A(2)=v(0)(0.1)+v(0.1)(0.1)+v(0.2)(0.1)+...+v(1.9)(0.1)$$A is the accumulation function that approximates the exact accumulation function h. We could write A(2) with summation notation as$$\color{red}{\text{(Eq. 5.1.1)}}\qquad A(2)=\sum_{k=0}^{19}v(k\Delta t)\Delta t$$In principle, the team's approach to solving its problem starts by cutting up the time axis into intervals of width \Delta t (generically, “\Delta t-intervals”). They will then assume that h has rates of change that are essentially constant over each \Delta t-interval. What value of v should they choose as the constant value of v(t) as t varies through an interval? Actually, there is no “should” in their choice. A better question is what would be a convenient choice for the value of t at which to evaluate v(t) as the essentially-constant rate of change of h over the interval. They chose, for convenience, the value of v at the beginning of each \Delta t-interval. Figure 5.1.2 uses the ridiculously large value of 0.9 seconds as the width of each \Delta t-interval. This is just to make it easy for you to visualize how this approach works. Figure 5.1.2. The function A approximates the exact accumulated altitude as t varies within the first 8 intervals. Move your pointer away from the animation to hide its control bar. You must examine Figure 5.1.2 in a particular way to understand all that is going on in it. • First, partition the time axis in both graphs into intervals of length 0.9 starting at 0. It is for these intervals we pretend that the rocket’s altitude varies at essentially constant rates of change. • The constant rates we assume are the values of v at the beginning of each interval. (The value of v(t) is 0 at the beginning of the first \Delta t-interval.) • As the value of t varies in the left graph, the constant velocity we assume is the value of v(t) at the beginning of the interval. • GC’s graph on the right side of Figure 5.1.2 is a simulation of how the rocket’s accumulated altitude varies given that the velocity is constant over each \Delta t-interval. • The graph on the right of Figure 5.1.2 is of the function A, the approximate altitude function whose values are determined by out assumption of constant velocities over \Delta t-intervals. • Take note that even though the two graphs Figure 5.1.2 look similar, it is only because the vertical axis in the right graph is compressed to fit within the diagram. In Figure 5.1.2 we assume that the rocket’s altitude varies at a constant rate throughout each \Delta 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 side of Figure 5.1.2 shows the accumulated variations that happened by assuming that h varies at a constant rate over each of its \Delta t-intervals. Defining the Approximate Rate Function Our assumption that h varies at a constant rate over each \Delta t-interval forces us to assume an approximate rate function r (left graph in Figure 5.1.2) which is constant over each \Delta 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 variations in altitude. Let r be the approximate rate function illustrated in Figure 5.1.2. This is the function that has a constant value as t varies through each \Delta t-interval. We can define r piecewise, as:$$\color{red}{\text{(Eq. 5.1.2)}}\qquad r(t)=\begin{cases} v(0) &\text{if $0\le t \lt 0.9$}\\[1ex] v(0.9) &\text{if $0.9\le t \lt 1.8$}\\[1ex] v(1.8) &\text{if $1.8\le t \lt 2.7$}\\[1ex] &\text{... and so on}\end{cases}$$Reflection 5.1.1. Let r be defined as in Equation 5.1.2. Complete this table.  t r(t) 0.5 0.6 1.2 1.3 1.4 2.71 2.72 2.95 It is worthwhile to note that there are two types of variation happening in Figure 5.1.2: variation within a current \Delta t-interval (the \Delta t-interval containing the current value of t) and variation of the completed \Delta t-intervals (the \Delta t-intervals through which the value of t has passed). Figure 5.1.3 illustrates the distinction between completed and current \Delta t-intervals. Figure 5.1.3. Distinction between current and completed \Delta t-intervals. Move your pointer away from the animation to hide its control bar. Using values of r to compute approximate variations in the exact accumulated altitude function h, we get the function A, the accumulation function that approximates the exact accumulated variations in altitude:$$\color{red}{\text{(Eq. 5.1.3)}}\qquad A(t)= \begin{cases} r(t)(t-0) &\text{if $0\le t \lt 0.9$}\\[1ex] \left(r(0)\Delta t\right)+r(t)(t-0.9) &\text{if $0.9\le t \lt 1.8$}\\[1ex] \left(r(0)\Delta t+r(0.9)\Delta t\right)+r(t)(t-1.8) &\text{if $1.8\le t \lt 2.6$}\\[1ex] \left(r(0)\Delta t+r(0.9)\Delta t+r(1.8)\Delta t\right)+r(t)(t-2.7) &\text{if $2.7\le t \lt 3.6$}\\[1ex]\text{... and so on}\end{cases}$$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 "current change". • "Completed change" is the change in altitude that happened in the complete \Delta t-intervals the value of t passed through prior to the \Delta t-interval 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. • "Current change" is the approximate change in altitude that has happened from the beginning of the current \Delta t-interval (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 \Delta t seconds, and m is the rocket's velocity at the beginning of the current \Delta t-interval. 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 "current change" varies continuously. 2. The change in approximate accumulated altitude happens within the current \Delta t-interval and is computed by multiplying the constant velocity we pretend the rocket travels within the current \Delta t-interval and the amount of time it has traveled within the current \Delta t-interval. • The amount of time that the rocket has traveled within the current \Delta t-interval is t-\mathrm{left}(t). • The constant rate at which the rocket travels is r(t)=v(\mathrm{left}(t)), the ball's velocity at the left end of the current \Delta t-interval. The Concept of Differential, Said Another Way We use the letter "d" preceding a variable to mean that the variable's value "varies a little bit". The way to envision a differential is to first keep in mind that a variable represents the value of a quantity whose value varies. Variables vary, always. Another key idea in understanding differentials is that they are variables whose values vary "a little bit". To vary a little bit, they must vary from somewhere. We therefore think of differentials of an independent variable as varying through fixed intervals in the variable's domain. Finally, a differential is a variable by which another variable varies. This surely sounds confusing. Figure 4.1.1 illustrates this idea. The term "left(x)" in the animation means the left end of the interval containing the current value of x. Variables and Differentials. The value of x varies smoothly through its domain. Differentials in x (dx) are small variations in the variable's value. The symbols "dx" and "\Delta x" have differnt meanings. Defining a function conceptually that gives the left end of an interval As the value of t varies, it is always within some \Delta t-interval. We will use "left(t)" to represent the number at the beginnning of the \Delta t-interval containing the current value of t. left(t) = the value of the left end of the \Delta t-interval containing the current value of t. Figure 5.1.4 illustrates the concept of left(t). It shows the independent axis partitioned into \Delta t-intervals of size 0.9. Figure 5.1.4. The meaning of the function left(t) and its relationship with t, \Delta t, and dt. In this figure, a=0 and \Delta t=0.9. The value of left(t) is the value of the left endpoint of the \Delta t-interval that contains the current value of t. As t varies from 0, its current value is always within some \Delta t-interval. The value of left(t) is the value of the left end of the \Delta 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) as the value of t varies in the interval \mathrm{left}(t)\lt t \le \mathrm{left}(t)+\Delta t. Another way to think of left(t) is given in Figure 5.1.4a. It shows that with a = 0 and \Delta 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. Move your pointer away from the animation to hide its control bar. 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). Move your pointer away from the animation to hide its control bar. We can now summarize Equation 5.1.2 in one line:$$\color{red}{\text{(Eq. 5.1.4)}}\qquad r(t)=v(\mathrm{left}(t))$$Defining Approximate Accumulation Function Conceptually Now that we have left(x) defined, we can summarize Equation 5.1.3 with one line: 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 \Delta 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 varied 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 varies 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. Move your pointer away from the animation to hide its control bar. You must understand all that is going on in Figure 5.1.5 in order to represent what is going on symbolically. • Partition the time axis on the right side of Figure 5.1.5 into \Delta t-intervals of size 0.1 seconds. • This imposes a partition of the time axis for the rate of change function on the left side into \Delta t-intervals 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 \Delta t-interval (see left side of Figure 5.1.5). • The function A (on the right of Figure 5.1.5) varies with respect to t at the constant rate of r(t) within a current \Delta t-interval, and is therefore linear over each \Delta t-interval (whether completed or current). The change in A as t varies within the current \Delta t-interval is represented by the expression r(t)(t-\mathrm{left}(t)). • The change in A over completed \Delta t-intervals prior to the current \Delta t-interval 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 ∆ t-intervals plus the accumulation during the current \Delta t-interval. Equation 5.1.5, repeated below, expresses this idea symbolically. where t is the number of seconds elapsed since launch, r(t) is the ball's velocity at the beginning of the \Delta t-interval containing the value of t, and A(t) approximates the rocket's altitude at that number of seconds. Reflection 5.1.2. 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 5.1.3. 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 5.1.4. Why is it important to use "number of completet-intervals from a to t" as the upper limit of the summation? 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"? 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 variations 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 variations 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 variations 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 \Delta 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 exact rate of change function. Here is a table of values of x, values of r(x), approximate variations in f, and approximate net accumulation as a function of x, all with $\Delta x=0.02$.
1. What is the value of A(2.74)?
2. Express A(2.74) using summation notation.
3. What is the value of A(2.743)?
4. Express A(2.743) using summation notation.
5. Sketch a graph of y = A(x), $2.70 \le x \le 2.86$.
6. Sketch a graph of y = A(x), $1 \le x \le 3$.