Section 6.2
Closed Form Rate of Change from Closed Form Accumulation

6.2.1 Deriving Rate of Change Functions in Closed Form

Section 6.1 introduced you to a method for defining an approximate rate of change function for an exact accumulation function given in closed form.

The method was to allow an interval of length h to vary through f’s domain. Each value of the function r defined as $$r(x)=\frac{f(x+h)-f(x)}{h}$$ gives the constant rate of change of the linear function whose graph passes through the points $(x,f(x))$ and $(x+h, f(x+h))$ on the graph of f.

We spoke of $r_f$, the (unknown) exact rate of change function for f, in terms of approximations to it by using very small values of h in the definition of r.

We can do more than write the definition of r in open form. If we start with a specific exact accumulation function in closed form, we can derive its approximate rate of change function in closed form.

Suppose $f(x)=(x+1)^2$ is an exact accumulation function.

For $h≠0$, we have $$\color{red}{\text{(Eq. 6.2.1)}}\qquad \begin{align} r(x) &= \frac{{f(x + h) - f(x)}}{(x+h)-x}\\[1ex] &= \frac{{f(x + h) - f(x)}}{h}\\[1ex] &= \frac{{{{\left( {(x + h) + 1} \right)}^2} - {{\left( {x + 1} \right)}^2}}}{h}\\[1ex] &= \frac{{\left( {{x^2} + 2xh + 2h + {h^2} + 1} \right) - \left( {{x^2} + 2x + 1} \right)}}{h}\\[1ex] &= \frac{{2xh + 2h + {h^2}}}{h}\\[1ex] &= 2x + 2 + h{\text{ for }}h \ne 0 \end{align}$$

Equation 6.2.1. Using the definition of r to find an approximate rate of change function for $f(x)=(x+1)^2$.

So, $r(x)=2x+2+h$ for any value of x and any value of $h≠0$.

Therefore, for $h\doteq 0,\, h\neq 0$ (i.e., for sufficiently small values of h), $r(x) \doteq 2x+2$ and $r_f$, f’s exact rate of change function, is $r_f(x)=2x+2$.

If this analysis is valid, then, given $f(x)=(x+1)^2$, two things should be true:

• The graph of $r(x)=\dfrac{f(x+0.001)-f(x)}{0.001}$ should essentially match the graph of $y=2x+2$.
• The graph of $y=f(x)$ should coincide with the graph of $y=g(x)$ where g is defined as $g(x) = f(a) + \displaystyle{\int_a^x (2t+2)dt}$ for any value of a.

Exercise Set 6.2.1

1. In Equation 6.2.1 we used the specific definition of f to derive an explicit definition of $r_f$. We simply used algebra and we used the fact that $h \doteq 0$, where "$h \doteq 0$" means that we are talking about values of h that are larger than 0 but exceedingly small.

Do the same as in Equation 6.2.1 in each of the following functions. Part (a) is completed as an example. Check your derivations using GC. If your derivation is correct, then

• The graph of f should coincide with the graph of $y=f(a)+\int_a^x r_f(t)dt$ after you substitute your derived formula for $r_f$ in place of the expression "$r_f(t)$".
• The graph of $r_f$ should coincide with the graph of r for suitably small values of h.

	$f(x)=$	$r(x)=\dfrac{f(x+h)-f(x)}{h}$	Derived $r_f$	Graphs coincide?
a)	$(x+1)^2$	$\dfrac{((x+h)+1)^2-(x+1)^2}{h}$	$2x+2$	Yes
b)	$(x-1)^3$	?	?	?
c)	$(2x+4)^2$	?	?	?



2. Derive exact rate of change functions $r_f$, $r_g$, and $r_h$ in closed form for the exact accumulation functions f, g, and h. Why are all the exact rate of change functions the same?
   1. $f(x)=x^2-2x-1$
   2. $g(x)=x^2-2x+2$
   3. $h(x)=x^2-2x+7$

6.2.2 Holy cow! A connection!

In Chapter 5, we derived exact accumulation functions in open form from exact rate of change functions in closed form. We started with exact rate of change functions like $r_f(x)=2x+2$ and we ended with the open-form exact net accumulation function $A_f(x)=\int_a^x (2t+2)dt$.

In Equation 6.2.1 we started with the exact accumulation function f defined as $f(x)=(x+1)^2$ and we ended with the exact rate of change function $r_f$ defined as $r_f(x)=2x+2$. We came full circle! $\int_a^x (2t+2)dt$ is somehow the same as $(x+1)^2$ !!

That is, by finding the exact rate of change function for $f(x)=(x+1)^2$ in closed form, we learned about the closed-form representation of that rate of change function’s accumulation function!

Suppose we have an exact accumulation function f represented in closed form. Is it true that f is the closed-form representation of $r_f$’s exact net accumulation function $A_f(x)=\int_a^x r_f(t)dt$?

Well, the general statement above is almost true.

From Exercise 6.4.2, you know that had we started with any of $f(x)=(x+1)^2+1$, $f(x)=(x+1)^2+7.3$, or $f(x)=(x+1)^2+c$ for any constant $c$, we would end with $r_f(x)=2x+2$ as f’s exact rate of change function.

Therefore, starting with $r_f(x)=2x+2$ does not automatically point to $f(x)=(x+1)^2$ as the closed-form representation of $\int_a^x (2t+2)dt$.

Rather, starting with $r_f(x)=2x+2$ points to a whole class of functions that differ only by a constant, each of which has $r_f(x)=2x+2$ as its exact rate of change function.

In Chapter 5, we always had an initial reference point at which accumulation started. However, an exact accumulation function in closed form does not have a natural starting point for the accumulation.

In Section 6.1.5 we started with an accumulation function f and recovered net variation in f by integrating $r_f$ from some reference value a.

To get the entire accumulation up to any value of x we must adjust the integral by adding $f(a)$ to the integral in order to recover f exactly.

Therefore, $$f(x)=f(a)+\int_a^x r_f(t)dt.$$

A little algebra then gives us $$\int_a^x r_f(t)dt=f(x)-f(a)$$ where f is any function having $r_f$ as its exact rate of change function.

The last statement bears repeating. It is so important in the historical development of calculus that it was given a special name:

6.2.3 Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) reveals how to compute values of an exact accumulation from values of an exact rate of change function.

The importance of the FTC is this:

Put another way, we can compute the value of $\int_a^x r_f(t)dt$ by computing $f(x)-f(a)$.

As a concrete example, we learned that $r_f(x)=2x+2$ when $f(x)=(x+1)^2$. By the FTC, we now can say that $$\int_a^x (2t+2)dt = (x+1)^2-(a+1)^2.$$That is to say, we can use $(x+1)^2-(a+1)^2$ to compute the value of $\int_a^x (2t+2)dt$ for any value of x and any value of a.

Closed form representations of integrals allow us to compute integrals directly, without having to approximate them by adding bits of accumulation.

The FTC tells us that we have actually accomplished two things whenever we derive an exact rate of change function $r$ in closed form from an exact accumulation function $f$ in closed form: (1) we derived f’s rate of change function, and (2) we found a closed form representation of $\int_a^x r_f(t)dt$.

The second accomplishment motivates us to find exact rate of change functions for as many types of accumulation functions as we can. Finding exact rate of change functions from exact accumulation functions will be helpful in evaluating exact values of integrals.

We shall do this in Section 6.3.

Historical Note on Derivatives and Antiderivatives

There are customary names for two functions f and $r_f$ when f is an exact accumulation function for $r_f$ and $r_f$ is f’s exact rate of change function.

The custom is that $r_f$ is called a derived function (or derivative) of f and that f is called an antiderivative of $r_f$.

We are compelled to point out, however, that being called an antiderivative or derivative confers no additional meaning to the fact that f is an exact accumulation function for $r_f$ and that $r_f$ is the exact rate of change function for $f$.

Exercise Set 6.2.3

These exercises are designed to help you solidify your understanding of the FTC.

1. State the FTC in your own words.
2. Let $r(x)=2x+2$. Use the FTC to define F in closed form when it is also defined as $\displaystyle{F(x)=\int_{-1}^x r(t)dt}$ in open form. What is $F(10)$? What does $F(10)$ mean?
3. Let $r(u)=2u+2$. Use the FTC to define G in closed form when it is also defined as $\displaystyle{G(s)=\int_{3}^s r(t)dt}$ in open form. What is $G(-1)$? What does $G(-1)$ mean?
4. Let $r(t)=3(t-1)^2$. Use the FTC to define H in closed form when it is also defined as $\displaystyle{H(u)=\int_2^u r(p)dp}$ in open form. What is $H(12)$? What does $H(12)$ mean?
5. Let $r(t)=3(t-1)^2$. Use the FTC to define I in closed when it is defined as $\displaystyle{I(u)=\int_a^u r(p)dp}$ in open form. Define $J$ as $\displaystyle{J(u)=\int_u^a r(p)dp}$. Show that $J(u) = -I(u)$.
6. The FTC says that $\int_a^x r_f(t)dt=f(x)-f(a)$ for any function f that has $r_f$ as its exact rate of change function. Illustrate this statement by letting $r_f(x)=3(x-1)^2$ and by using three different definitions of f, each of which has $r_f(x)=3(x-1)^2$ as its exact rate of change function.