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# Refresher: Approximating Exact Accumulation Functions

## Assume constant first-order rate of change of accumulation (1st derivative)

In Chapter 5 we employed the power of digital technologies to develop a method for approximating the net accumulation from a reference point of a quantity whose exact values we assumed could be represented by a yet-to-be-known function f in relation to moments of its independent variable x. The method, in outline, was this:

• Start with a known rate of change function $r_f$, defined in closed form, whose values $r_f(x)$ give the exact rate of change of $f$ at every moment $x$ in the domain of $f$.
• Assume a value $x=a$ from which net accumulation will be calculated.
• Define a function $r$ that approximates $r_f$ by assuming that $f$ changes at a constant rate with respect to $x$ as $x$ changes by the variable $dx$ over small intervals of length $\Delta x$. This assumption allows us to compute bits of accumulation as $dx$ varies through the interval $0\lt dx \le \Delta x$ and varies through successive intervals of length $\Delta x$.
• Define a function $A$ whose values $A(x)$ give the approximate net accumulation from $a$ to $x$
• Define a function $A_f$, in open form, whose values $A_f(a,x)$ give the exact net accumulation in $f$ from $a$ to $x$.

Figure 10.1.1, repeated from Chapter 5, illustrates this process.

Figure 10.1.1. Visual summary of defining an exact net accumulation function from an exact rate of change function.

The function $A$ that we defined in Chapter 5 was this.

Start with a reference value $x=a$, a standard interval length $\Delta x$, and a function $r_f$, defined in closed form, whose values give the exact rate of change of the exact accumulation function $f$ at each moment in its domain. Values of the function $A$, defined below, give the approximate accumulation in $f$ from $a$ to $x$. (See Section 4.7 for a refresher on summation notation.)

\begin{align} \color{red}{\text{(Eq. 10.1.1)}}\qquad A(x)&=\left(\sum_{k=1}^\left\lfloor \frac {x-a}{\Delta x}\right\rfloor r\left(a+(k-1)\Delta x\right)\Delta x\right)+r(x)(x-\mathrm{left}(x)) \text{, where}\\[1ex] \mathrm{left}(x)&=a+\Delta x \left\lfloor\frac{x-a}{\Delta x}\right\rfloor\\[1ex] r(x)&=r_f(\mathrm{left}(x)) \end{align}

Reflection 10.1.1. Where is dx in the definition of A in Eq. 10.1.1?

The definition of $\mathrm{left}(x)$ assigns the left end of the $\Delta x$-interval containing the value of $x$ to each value of $x$ in that interval (see the discussion of Figures 5.1.3 - 5.1.5). The definition of $r$ makes $r(x)$ have the constant value $r_f(\mathrm{left}(x))$ for every value of $x$ in any $\Delta x$-interval.

In defining $A$, we could just as well have used the value of $r_f$ at the midpoint of each $\Delta x$-interval as the constant rate of change over each $\Delta x$-interval, or the value of $r_f$ at the right end of each $\Delta x$-interval. Here is the definition of $r$ using the value of $r_f$ at the midpoint of each $\Delta x$-interval.
\begin{align} \color{red}{\text{(Eq. 10.1.2)}}\qquad \mathrm{left}(x)&=a+\Delta x\left\lfloor \frac{x-a}{\Delta x}\right\rfloor\\[1ex] \mathrm{mid}(x)&=\mathrm{left}(x)+\frac{\Delta x}{2}\\[1ex] r(x)&=r_f(\mathrm{mid}(x)) \end{align}
Examine Figures 10.1.2, 10.1.3, and 10.1.4. The definition of $r_f$ in each figure is $$r_f(x)=1.6\sin(x-2)+0.5-0.009(x-8)^2.$$
Each figure shows the graph of $y=r_f(x)$ and $y=r(x)$, with $a=-3$ and $\Delta x=0.6$, for one of the three methods of defining a constant function that approximates $r_f$. The right panes show graphs of $y=A(x)$ according to each approximation of $r_f$.

Figure 10.1.2. We approximate $r_f(x)$ over intervals $0\lt dx \le \Delta x$ with a function that is constant over each $\Delta x$-interval. The value of the constant function for each value of x in any $\Delta x$-interval is the value of $r_f$ at the left end of the interval.

Reflection 10.1.2. Where is $\Delta x$ in Figure 10.1.2? Where is dx in Figure 10.1.2? What influence does the value of $\Delta x$ have on the graph of y=A(x)?

Figure 10.1.3. We approximate $r_f$ over intervals $0\lt dx \le \Delta x$ with a function that is constant over each $\Delta x$-interval. The value of the constant function for each value of x in any $\Delta x$-interval is the value of $r_f$ at the midpoint of the interval.

Reflection 10.1.3.Where is $\Delta x$ in Figure 10.1.3? Where is dx in Figure 10.1.3? Why is the graph of y=r(x) in Figure 10.1.3 different from the graph of y=r(x) in Figure 10.1.2?

Reflection 10.1.4. Modify the definition of A in Eq. 10.1.1 using the definition of r in Eq. 10.1.2. Graph y=A(x) with $a=-3$ and $\Delta x=0.6$.

Figure 10.1.4. We approximate $r_f(x)$ over intervals $0\lt dx \le \Delta x$ with a function that is constant over each $\Delta x$-interval. The value of the constant function for each value of x in any $\Delta x$-interval is the value of $r_f$ at the right end of the interval.

Reflection 10.1.5. Modify the definition of r so that you use the value of $r_f$ at the right end of each $\Delta x$-interval. Modify the definition of A accordingly and graph y=A(x) in GC using $a=-3$ and $\Delta x=0.6$.

Reflection 10.1.6. Redefine $r_f$ as $r_f(x)=x \sin\left(2\pi \sqrt{x+3}\right)$. What would you need to change in Equations 10.1.1 or 10.1.2? Why? What would you need to change in your responses to Reflections 10.1.4 and 10.1.5? Why?

## Approximation Error

Up until now we trusted that we can make $\Delta x$ small enough so that making it smaller makes no appreciable difference in the accuracy of the approximate accumulation function. We did not, however, investigate how good an approximation is for a particular value of $\Delta x$ or for particular values of $x$. We shall do that now.

Our explorations of how well an approximation method works will follow this pattern:

• Start with an exact rate of change function that has an exact accumulation function that can be defined in closed form.
• Compare values of the approximate accumulation function with values of the exact accumulation function.
• Create a method for determining upper and lower bounds for "error of approximation" (how far from exact our approximate accumulation function can be).
• Apply our method for determining error of approximation to rate of change functions that have exact accumulation functions that cannot be defined in closed form.

There are several ways to judge how well an approximation method works for a particular rate of change function.

• Inspection: Compare graphs of approximate accumulation functions with the graph of the exact accumulation function
• over intervals that are visible on the default screen
• over larger intervals than are currently displayed
• Closer inspection: Compare the difference (absolute error) or quotient (relative error) of the exact and approximate accumulation functions over small and large intervals
• Numerically:
• Examine the distance $\left|f(x)-A(x)\right|$ between the exact and approximate accumulation functions for values across their domains (absolute error)
• Examine the relative size of the absolute error and the exact value of the accumulation function $\left|\dfrac{ f(x)-A(x)}{f(x)}\right|, \, f(x)\neq 0$ (relative error)
• Analytically: Create a formula that gives the maximum absolute or relative error of approximation for any value in the rate of change function's domain.

#### Absolute versus relative error

Whether to focus on absolute or relative error is a matter of judgment. If you are measuring an accumulation of microscopic particles, an approximation that has an absolute error of $3.7\times10^{-7}$ grams might seem very close to exact. If, however, a highly accurate measure is $5.1\times 10^{-8}$ grams, then your relative error is $$\frac{3.7\times10^{-7}}{5.1\times 10^{-8}}=7.255,$$ or over 720%.

On the other hand, if you are measuring the distance to a star, an approximation that has an absolute error of 2 billion kilometers might seem highly inaccurate. If, however, a highly accurate measure is 700.21 light years, then your relative error is $$\frac{2\times10^9}{700.21\times 9.461\times 10^{12}}=0.0000302,$$or about  3/1000 of 1%.

## Judging Approximation Error by Inspection

We know (using integration by parts) that
\begin{align} \int_0^x \left(t\sin(t)-1+0.1\cos(15t)\right)\,dt&=\left. -t+\sin(t)+\frac{2}{300}\sin(15t)-t\cos(t)\right|_{\,0}^{\,^x}\\[1ex] &=-x+\sin(x)+\frac{2}{300}\sin(15x)-x\cos(x) \end{align}
We can therefore compare our approximation methods against the exact accumulation function $f$ defined as $$f(x)=-x+\sin(x)+\frac{2}{300}\sin(15x)-x\cos(x).$$Please download this GC file to use in conjunction with the following discussion of Figures 10.1.5a through 10.1.5c.

Figure 10.1.5. Part (a) shows $r_f$ approximated by $r(\mathrm{left}(x))$ and the resulting graph of $y=A(x)$.
Part (b) shows $r_f$ approximated by $r(\mathrm{mid}(x))$ and the resulting graph of $y=A(x)$.
Part (c)
shows $r_f$ approximated by $r(\mathrm{right}(x))$ and the resulting graph of $y=A(x)$.

Part (a) of Figure 10.1.5 shows that, with this particular function $r_f$, the definition of $r$ as $r(x)=r_f(\mathrm{left}(x))$ produces a constant rate of change that is a systematic under-approximation whenever $r_f$ increases over an interval and a systematic over-approximation when $r_f$ decreases over an interval.

Part (b) of Figure 10.1.5 shows that the definition of $r$ as $r(x)=r_f(\mathrm{mid}(x))$ produces a constant rate of change that is an under-approximation for part of the interval and an over-approximation for the other when $r_f$ increases or decreases over the interval. The errors cancel each other to some extent.

Part (c) of Figure 10.1.5 shows that the definition of $r$ as $r(x)=r_f(\mathrm{right}(x))$ produces a constant rate of change that is a systematic over-approximation whenever $r_f$ increases over an interval and a systematic under-approximation when $r_f$ decreases over an interval.

Reflection 10.1.7. In the GC file for Figure 10.1.5, compare the absolute and relative error over the interval defined by the slider named p by varying p's value. Are your observations of these errors consistent with the graphs in Figure 10.1.5?

Reflection 10.1.8. In the GC file for Figure 10.1.5, reduce the value of $\Delta x$. Compare the absolute and relative error both graphically and by varying p's value. Are your observations of these errors still consistent with the graphs in Figure 10.1.5?

Reflection 10.1.9. In the GC file for Figure 10.1.5, change the definition of $r_f$ to $r_f(x)=0.5\sqrt{x}e^{\cos(x)}$. Change the definition of f to $f(x)=\int_0^x 0.5\sqrt{t}e^{\cos(t)} dt$. Set $a=0$. Explore the absolute and relative error of the three methods of approximation.

Reflection 10.1.10. Sketch a graph of a function $r_g$ over a $\Delta x$-interval for which $r_g(\mathrm{left}(x))$ produces a better approximation of $r_g$ than does $r_g(\mathrm{mid}(x)))$. Explain what you mean by "better".

Reflection 10.1.11. Sketch a graph of a function $r_g$ over a $\Delta x$-interval for which $r_g(\mathrm{right}(x))$ produces a better approximation of $r_g$ than does $r_g(\mathrm{mid}(x)))$. Explain what you mean by "better".

# Other approximation methods

## Assume constant 2nd-order rate of change of accumulation (constant acceleration--2nd derivative)

The three approximation methods discussed above were all based on the idea of pretending that the accumulation function has a constant rate of change over all $\Delta x$-intervals. The differences among those approaches arose from our choice of the place within a $\Delta x$-interval at which we take a value of $r_f$ as the constant rate of change that approximates the values of $r_f$ over that interval.

We could make other assumptions that lead to different choices of our assumed constant rate of change over $\Delta x$-intervals. For example, instead of assuming that $f$ has a constant rate of change over a $\Delta x$-interval, we could assume that $f$ has constant acceleration over each $\Delta x$-interval. This would be the same as assuming that $r_f$ changes linearly over each $\Delta x-interval$.

 Instead of approximating exact net accumulation by pretending that accumulation happens at a constant rate over each $\Delta x$-interval, we can approximate exact net accumulation by pretending that accumulation happens with constant acceleration over each $\Delta x$-interval.

Figure 10.1.6 shows what it looks like to assume that net accumulation changes with a constant acceleration over each $\Delta x$-interval. Notice that by assuming that net accumulation changes with a constant acceleration, we are thereby assuming that the accumulation's rate of change function $r_f$ is changing linearly (i.e., changes at a constant rate of change).

Figure 10.1.6. Approximate net accumulation function changes with constant acceleration over $\Delta x$-intervals, which means that approximate rate of change function changes linearly (i.e., at a constant rate of change) over each $\Delta x$-interval.

Our assumption that accumulation has a constant acceleration over small $\Delta x$-intervals means that we are also assuming that $r_f$ is essentially linear over each $\Delta x$-interval. This implies that over any interval $[a,b]$ that $r(x)$, $r_f$'s approximate rate of change function, has the form $r(x)=A(x-a)+B$ over $[a,b]$, where $A=\frac{r_f(b)-r_f(a)}{b-a}$ and $B=r_f(a)$. The approximate accumulation function evaluated over an interval $a\le x\le b$ therefore has the value

\begin{align} \color{red}{\text{(Eq. 10.1.3)}}\qquad \int_a^b r(x)dx &= \int_a^b \left( A(x-a)+B)\right) dx \\[1ex] &=A\int_a^b (x-a)dx + B\int_a^b dx\\[1ex] &= \left. A\left(\frac{x^2}{2}-ax\right)\right|_a^b+\left. Bx\right|_a^b\\[1ex] &= A\left(\left(\frac{b^2}{2}-ab\right)-\left(\frac{a^2}{2}-a^2\right)\right)+B(b-a)\\[1ex] &=\frac{1}{2}A\left(b^2-2ab+a^2\right)+B(b-a)\\[1ex] &=\frac{1}{2}A(b-a)^2+B(b-a)\\[1ex] &=\frac{1}{2}\left(A(b-a)+2B\right)(b-a)\\[1ex] &=\frac{1}{2}\left(\frac{r_f(b)-r_f(a)}{b-a}(b-a)+2r_f(a)\right)(b-a)\\[1ex] &=\frac{1}{2}\left(r_f(b)+r_f(a)\right)(b-a) \end{align}
Here is what we accomplished in Equation 10.1.3: Given an exact rate of change function $r_f$, we can approximate net accumulation in $f$ over a complete $\Delta x$-interval by computing
$$\frac{r_f(a)+r_f(b)}{2}\Delta x$$
where $a$ and $b$ are the left and right ends, respectively, of the $\Delta x$-interval.

We can compute the approximate rate function for $r_f$ over a partial $\Delta x$-interval from $\mathrm{left}(x)$ to $x$ by computing $r_\text{cacc}(x)$ as
$$r_\text{cacc}(x)=\frac{r_f(\mathrm{left}(x))+r_f(x)}{2}$$where "cacc" is short for "constant acceleration", the assumption we made about how the approximate accumulation function changes. We can then compute the net accumulation in the current $\Delta x$-interval by $$r_\text{cacc}(x)\left(x-\mathrm{left}(x)\right)$$
In short, we can approximate exact net accumulation in $f$ from $a$ to $x$ by computing accumulation over the completed $\Delta x$-intervals included between $a$ and $x$, plus accumulation over the current, partial $\Delta x$-interval (see Equation 10.1.4).
$$\color{red}{\text{(Eq. 10.1.4)}}\qquad A_\text{cacc}(x)=\left(\sum_{k=1}^\left\lfloor \frac {x-a}{\Delta x}\right\rfloor \left(\frac{r_f(a+(k-1)\Delta x)+r_f(a+k\Delta x)}{2}\right)\Delta x\right)+r_\text{cacc}(x)(x-\mathrm{left}(x))$$

Figure 10.1.9 shows linear approximation of $r_f$ and the resulting approximation of $f$ by $A_\text{cacc}$ (assuming constant acceleration) with $\Delta x=0.6$. Figure 10.1.10 shows linear approximation of $r_f$ and the resulting approximation of $f$ by $A_\text{cacc}$ (assuming constant acceleration) with $\Delta x=0.2$.

 Figure 10.1.9. The graph of $y=A_\text{cacc}(x)$ with $\Delta x$=0.6, where $A_\text{cacc}$ is the approximate net accumulation function for f gotten by using $r_\mathrm{cacc}(x)$ as the approximate rate of change of f with respect to x for every value of x in a $\Delta x$-interval.

Reflection 10.1.11a. Explain why the graph of $y=A_\text{cacc}(x)$ must be the graph of a quadratic function over every $\Delta x$-interval.

 Figure 10.1.10. The graph of $y=A_\text{cacc}(x)$ with $\Delta x$=0.2, where $A_\text{cacc}$ is the approximate net accumulation function for f gotten by using $r_\mathrm{cacc}(x)$ as the approximate rate of change of f with respect to x over each $\Delta x$-interval.

Reflection 10.1.12. Use GC to compare two methods of approximating exact accumulation, one using $A_\text{mid}(x)$ (assuming constant rate of change over intervals using mid-method) and one using using $A_\text{cacc}(x)$ (assuming accumulation with constant acceleration over intervals).

• Use subscripts to differentiate between your two accumulation functions and your two approximate rate of change functions.

• Refer to Equations 10.1.1, 10.1.2, and 10.1.4 for reminders.

• Compare the accuracy of your new definitions graphically and numerically, using $r_f$ and f as shown below.\begin{align}r_f(x)&=\dfrac{e^{\cos x}}{1+e^{-x}}\\[1ex]f(x)&=\int_0^xr_f(t)dt\end{align}
• Write a summary of your observations.

## Quantifying Approximation Error

Remind yourself of the definitions of absolute and relative approximation errors. Given a reference value of $x=a$, an exact rate of change function $r_f$ for an exact net accumulation function f, and a value of $A(a,x)$ that approximates $f(a,x)$, the exact net accumulation from $a$ to $x$, then:
\begin{align} \text{Absolute approximation error}&= \left|f(a,x)-A(a,x)\right|\\[3ex] \text{Relative approximation error}&=\left|\frac{f(a,x)-A(a,x)}{f(a,x)}\right|\qquad f(a,x)\neq 0 \end{align}
In your investigations, you probably noticed that the midpoint method produced approximations to values of an exact accumulation function that were at least as accurate as the left, right, and constant acceleration methods, and often produced approximations that were noticeably better than the others.

The error in an approximation of an accumulation function's value is due not only to the method we use. It is also influenced by the size of $\Delta x$ in conjunction with the behavior of the accumulation function's rate of change function.

For example, the left side of Figure 10.1.11 shows (on the left) graphs of $y=r_f(x)$, $\color{red}{y=r_\mathrm{mid}(x)}$, and $\color{blue}{y=r_\mathrm{cacc}(x)}$ and (on the right) their respective accumulation functions as the value of $\Delta x$ goes from $\Delta x=1$ to $\Delta x=0.02$.

We recommend that you watch the animation in Figure 10.1.11 several times, focusing first on the left pane, then on the right pane in relation to the left pane. Move your cursor away from the animation to make its scroll bar disappear.

Figure 10.1.11. Graphs of $y=r_f(x)$, $y=r_{mid}(x)$, and $y=r_{cacc}(x)$ and their respective accumulation functions as the value of $\Delta x$ goes from $\Delta x=1$ to $\Delta x=0.02$.

There is a lot to see in Figure 10.1.11. Drag the scroll bar to appropriate places in the animation as we focus on different aspects of what the animation illustrates.

Regarding approximations of $r_f(x)$:

• Values of $r_\mathrm{mid}(x)$ and $r_\mathrm{cacc}(x)$, shown in the left pane, become closer to each other as the value of $\Delta x$ becomes smaller.
• As the value of $\Delta x$ becomes smaller, both methods produce constant rates of change over $\Delta x$-intervals that become closer to the values of $r_f(x)$ over the same intervals.

• For any given value of $\Delta x$, the faster that $r_f(x)$ changes over that interval the farther apart are values of $r_\mathrm{mid}(x)$ and $r_\mathrm{cacc}(x)$. Using differential notation, the greater the value of $\dfrac{d^2}{dx^2}r_f(x)$ over an interval, the farther apart are values of $r_\mathrm{mid}(x)$ and $r_\mathrm{cacc}(x)$.

Regarding approximations of $f(x)$:

• Values of $A_\mathrm{mid}(a, x)$ and $A_\mathrm{cacc}(a, x)$, shown in the right pane, become closer to each other as the value of $\Delta x$ becomes smaller.
• As the value of $\Delta x$ becomes smaller, both methods produce approximate accumulations $\Delta x$-intervals that become closer to the values of $f(a, x)$ over the same intervals.

• For any given value of $\Delta x$, the faster that the rate of change of $r_f(x)$ changes with respect to $x$ over an interval, the farther apart are values of $A_\mathrm{mid}(a, x)$ and $A_\mathrm{cacc}(a, x)$. Using differential notation, the greater the value of $\dfrac{d^2}{dx^2}r_f(x)$ over an interval, the farther apart are values of $A_\mathrm{mid}(a, x)$ and $A_\mathrm{cacc}(a, x)$.

• As the value of $\Delta x$ becomes smaller, values of $A_\mathrm{mid}(a, x)$ become closer to values of $f(a, x)$ faster than do values of $A_\mathrm{cacc}(a, x)$.

Put another way, if we halve the value of $\Delta x$, the improvement in approximations of $f(a, x)$ over an interval by $A_\mathrm{mid}(a, x)$ tends to be greater than the improvement in the approximations of $f(x)$ by $A_\mathrm{cacc}(a, x)$.

Another way to look at these same observations is by building a table of values of $A_\mathrm{mid}(a, x)$ and $A_\mathrm{cacc}(a, x)$ for particular values of x and diminishing values of $\Delta x$.

The table in Figure 10.1.12 uses definitions of $A_\mathrm{mid}$ and $A_\mathrm{cacc}$ that are modified to accept the accumulation's reference value and the value of $\Delta x$ as arguments. We did this so that we could hold the accumulation's reference value and end value constant while varying the value of $\Delta x$.

The GC file that produced Figure 10.1.12 is at this link.

Figure 10.1.12. Values of $A_\mathrm{mid}(c.x,\Delta x)$ and $A_\mathrm{cacc}(c,x,\Delta x)$ for
initial value of $c=-1$, $x=b$, and various values of $\Delta x$.

You should inspect Figure 10.1.12 closely. Approximations of $f(-1,15)$ become better with smaller values of $\Delta x$. But there are two more things to see:

• The relative error of either method improves methodically. Relative error, the magnitude of the absolute error of approximation in relation to the magnitude of $f(-1,15,\Delta x)$, becomes 1/4 as large each time the value of $\Delta x$ is made 1/2 as large.
• This suggests that relative error of either method is proportional to $(\Delta x)^2$.

• In a course on numerical analysis you will derive the error bound for each method. These upper bounds are:
$$\left|\int_a^xr_f(t)dt-A_\mathrm{mid}(a,x,\Delta x)\right|\le \left(\left(\frac{x-a}{24}\right) \left({\mathrm{max}\atop{a≤t≤x}} \left|\dfrac{d^2}{dt^2}r_f(t)\right|\right)\right)(\Delta x)^2$$
and
$$\left|\int_a^xr_f(t)dt-A_\mathrm{cacc}(a,x,\Delta x)\right|\le \left(\left(\frac{x-a}{12}\right) \left({\mathrm{max}\atop{a≤t≤x}} \left|\dfrac{d^2}{dt^2}r_f(t)\right|\right)\right)(\Delta x)^2$$
• Notice that the two error bounds are identical, except for the quotients $\frac{x-a}{12}$ and $\frac{x-a}{24}$. The value of $\frac{x-a}{24}$ is half the value of $\frac{x-a}{12}$. This means that approximations of $f(c,x)$ by $A_\mathrm{cacc}(c,x)$ are, in a general sense, half as accurate as approximations of $f(c,x)$ by $A_\mathrm{mid}(c,x)$.
• Entries in the last column of Figure 10.1.12 show the relative size of the two relative errors. All entries are very close to 2, and get closer to 2 as $\Delta x$ becomes smaller. This means that the relative error for $A_\mathrm{cacc}(-1,15,\Delta x)$ is always approximately twice as large as the relative error for $A_\mathrm{mid}(-1,15,\Delta x)$.
• This is consistent with the error bounds for $A_\mathrm{mid}$ and $A_\mathrm{cacc}$ given immediately above. The bound for $A_\mathrm{mid}$ over an interval involves the quotient $\frac{x-a}{24}$ while the bound for $A_\mathrm{mid}$ involves the quotient $\frac{x-a}{12}$, and everything else is the same in both expressions. In other words, the maximum error bound for $A_\mathrm{mid}$ is half as large as the maximum error bound for $A_\mathrm{cacc}$.

 It is important to understand that a bound on absolute error of approximation is not the same as actual absolute approximation error. An error bound says that the absolute approximation error cannot be any more than a certain value. It does not say what the actual error is.

Reflection 10.1.13. The error bounds for both $A_\mathrm{mid}(a,x,\Delta x)$ and $A_\mathrm{cacc}(a,x,\Delta x)$ contain the term $$\left({\mathrm{max}\atop{a≤t≤x}} \left|\dfrac{d^2}{dt^2}r_f(t)\right|\right).$$Try to explain two things: (1) How this term means, "the largest value of the second derivative of $r_f(t)$ over the interval $a \le t \le x$", and (2) why the second derivative of $r_f$ matters in determining maximum absolute error.

Reflection 10.1.14. Use techniques from Chapter, 7, Section 3 (Optimization) to compute $\left({\mathrm{max}\atop{a≤t≤x}} \left|\dfrac{d^2}{dt^2}r_f(t)\right|\right)$ for $r_f$ as defined in Figure 10.1.12. Use this value to compute the maximum relative error of $A_\mathrm{mid}(-1,15,0.25)$ as an approximation of $f(-1,15)$, where $f(a,x)$ is defined as in Figure 10.1.12. Compare your maximum relative error with the appropriate entry in Figure 10.1.12.

Reflection 10.1.15. Is the maximum relative error that you computed in Reflection 10.1.14 for $A_\mathrm{mid}(-1,15,\Delta x)$ valid for all values of x from -1 to 15, or only for $x=15$? Explain.

## Assume constant 3rd-order rate of change of accumulation (constant jerk, constant 3rd derivative)

In the prior section we developed the method $A_\mathrm{cacc}$, which used the rate of change that is half way between $r_f$'s values at each end of a $\Delta x$-interval as the constant rate of change that we assumed for accumulation as $dx$ varies through each $\Delta x$-interval.

We developed the constant acceleration method by assuming that $f$, $r_f's$ exact accumulation function, has a constant acceleration (2nd-order rate of change) over $\Delta x$-intervals. This assumption produced a method that was better than either the left(x) or right(x) methods, but not as good as taking $r_f(\mathrm{mid}(x))$ as the constant rate of change over any $\Delta x$-interval containing the value of $x$.

Another method for approximating net accumulation from exact rate of change is to assume that $f$, the exact accumulation function for $r_f$, has a 3rd-order rate of change function (3rd derivative) that is essentially constant over each $\Delta x$-interval.

You would be right to ask, "What good does it do to assume that $f$ has a 3rd-order rate of change function that is essentially constant over sufficiently small $\Delta x$-intervals?" This assumption implies that $r_f^{(1)}$, f 's 2nd-order rate of change function, is essentially linear over each $\Delta x$-intervals and that $r_f$ is essentially quadratic over each $\Delta x$-interval. And we know how to compute exact accumulation of a quadratic rate of change function!

Reflection 10.1.16. Explain how assuming that an exact accumulation f has a 3rd-order rate of change that is essentially constant over $\Delta x$-intervals implies that its 1st-order rate of change function is essentially quadratic over the same $\Delta x$-intervals.

Put another way, a quadratic approximation to $r_f$ over an interval is more sensitive to how $r_f$ changes over that interval than is a linear approximation to $r_f$. A quadratic approximation to $r_f$ should therefore give us better approximations to the net change in accumulation over an interval than would linear approximations to $r_f$ over that interval.

Another advantage of using a quadratic function to approximate $r_f$ over an interval is that we can approximate exact accumulation over complete $\Delta x$-intervals by computing the exact accumulation of the quadratic rate of change function that approximates $r_f$ over that $\Delta x$-interval.

Figure 10.1.13 illustrates the idea of approximating exact rate of change over an interval with a quadratic function over that same interval. The animations in Figure 10.1.13 use a sliding interval instead of a fixed interval to illustrate this approximation technique. This is only to show that the technique works for any arbitrary interval. Move your cursor away from an animation to make its scroll bar disappear.

 (a) Quadratic approximation to $r_f$ over intervals of length 2.0. (b) Quadratic approximation to $r_f$ over intervals of length 1.0. (c) Quadratic approximation to $r_f$ over intervals of length 0.5.

Figure10.1.13. Three approximations to an exact rate of change function by a quadratic function over an interval of length $\Delta x$, with $\Delta x=2.0$, $\Delta x=1.0$, and $\Delta x=0.5$. The intervals slide along the x-axis to illustrate quadratic approximation over any interval.

 (a) Left. Graph of $\color{red}{y=r_\mathrm{quad}(x)}$, the quadratic approximation to $y=r_f(x)$ over intervals of length $\Delta x=1.0$. Heavy gray lines show left and right ends of successive $\Delta x$-intervals. Light gray lines show middles of $\Delta x$-intervals. Graph of $\color{red}{y=r_\mathrm{quad}(x)}$ passes through graph of $y=r_f(x)$ at left end, middle, and right end of each $\Delta x$-interval. Right. The graph, $\color{green}{\text{in green}}$, of the exact accumulation function from $r_f$ along with the graph of $\color{red}{A_\mathrm{quad}}$, the approximate accumulation function. (b) Left. Graph of $\color{red}{y=r_\mathrm{quad}(x)}$, the quadratic approximation to $y=r_f(x)$ over intervals of length $\Delta x=0.4$. Heavy gray lines show left and right ends of successive $\Delta x$-intervals. Light gray lines show middles of $\Delta x$-intervals. Graph of $\color{red}{y=r_\mathrm{quad}(x)}$ passes through graph of $y=r_f(x)$ at left end, middle, and right end of each $\Delta x$-interval. Right. The graph, $\color{green}{\text{in green}}$, of the exact accumulation function from $r_f$ along with the graph of $\color{red}{A_\mathrm{quad}}$, the approximate accumulation function.