A Modeling Approach to Calculus: MA2C

Subsection 5.1.1 Derivatives and Integrals

For sequences, we learned to think of complementary ideas of accumulation sequences and increments. With a sequence \(x\text{,}\) we had a forward difference

and a backward difference

These differences measure the change in the sequence \(x\) for consecutive values of the index, which plays the role of the independent variable.

For functions defined on intervals, there is no meaning to consecutive values of the independent variable. Near a point of interest \(x=c\text{,}\) there are infinitely many other values close to \(c\text{.}\) Consequently, when measuring the change of a function \(\Delta f\text{,}\) we must also specify the change in the independent variable \(\Delta x\text{.}\) Consider two values for the independent variable, say \(x=a\) and \(x=b\text{,}\) and we define \(\Delta x = b-a\) and \(\Delta f = f(b) - f(a)\text{.}\)

Different increments \(\Delta x\) will usually result in different function increments \(\Delta f\text{.}\) However, for many functions, the ratio \(\Delta f / \Delta x\text{,}\) called the average rate of change, has a limit as \(\Delta x \to 0\text{.}\) This limiting rate of change is called the instantaneous rate of change and in calculus is named the derivative.

The following example illustrates the role of the instantaneous rate of change to relate the increments of the independent variable with the increments of the dependent variable.

At this point, our emphasis is understanding that the rate of change or derivative measures the limiting ratio for increments of change in the value of the function to corresponding increments of change in the independent variable. Not every function has a derivative. We will study the calculation of the derivative in more depth in later chapters.

Computing a derivative for a given function is analogous to computing the increments of a sequence. The complementary calculation for sequences is to compute the accumulation sequence for given increments. That is, if \(x=(x_n)_{n=1}^{\infty}\) is a sequence of increments, then the accumulation sequence \(u\) with increments \(\nabla u_n=u_n - u_{n-1} = x_n\) and initial value \(u_0\) was written

The calculus analogue is to be given a function that represents a rate of change and use it to find a new function, the accumulation function, that has that rate of change as its derivative. Suppose \(f(x)\) is the rate of change or derivative of a quantity \(Q\) with respect to \(x\text{,}\)

We are then interested in finding \(Q\) as a function of \(x\) if we know an initial value \(Q(x_0) = Q_0\text{.}\) The analogue of summation of increments is the definite integral, and we will write

The rest of this chapter is focused on bringing meaning to the idea of the definite integral. We study definite integrals before derivatives because we have just studied summation and sequences. The calculations involved in developing the ideas of definite integrals apply these concepts. Ultimately, the Fundamental Theorem of Calculus will provide a connection between the definite integral and the derivative, showing that our two ideas of rate of change represent the same thing.

Subsection 5.1.2 A Technological Aside

Computational tools play an important role in the real-world application of mathematics. It is increasingly common to have a tool perform actual computations with the user responsible to formulate the appropriate problem.

For example, you may have heard of the website WolframAlpha. This site acts like a search engine for mathematical content, and you can enter queries like “factor x^2+3x”. The ability extends to calculus tools as well. We might have asked for our earlier example “derivative of 40-16t^2 at t=1”.

Disadvantages of a site like WolframAlpha is that you are limited to a single query at a time and it can sometimes be hard to state precisely what you want. More powerful tools are available, including advanced programmable calculators and commercial software tools like Wolfram's Mathematica and MapleSoft's Maple programs.

A free, but similarly powerful tool is SageMath. A calculation in SageMath uses a script based on the Python programming language. Comments in the scripts follow the # symbol and are ignored by the computer but are useful to understand what is happening.

Subsection 5.1.3 Summary

• Calculus is developed using ideas similar to those for sequences—limits, increments, and accumulation— to limits of functions, derivatives, and integrals.
• The derivative \(\displaystyle \frac{dQ}{dx}\) measures the instantaneous rate of change of a quantity \(Q\) with respect to the independent variable \(x\text{,}\) represented by a limit,
  \begin{equation*} \frac{dQ}{dx} = \lim_{\Delta x\to 0} \frac{\Delta Q}{\Delta x}. \end{equation*}
  Consequently, increments of change in \(Q\text{,}\) \(\Delta Q\text{,}\) will be approximately proportional to the increment in \(x\text{,}\)
  \begin{equation*} \Delta Q \approx \frac{dQ}{dx} \cdot \Delta x, \end{equation*}
  for sufficiently small \(\Delta x\text{.}\)
• Given a function \(f'\) for the rate of change of a quantity \(Q\text{,}\) \(x \overset{f'}\mapsto \frac{dQ}{dx}\text{,}\) and an initial value \(Q_0\) when \(x=x_0\text{,}\) the accumulation function will be that function with derivative \(\frac{dQ}{dx} = f'(x)\text{,}\) represented by the integral
  \begin{equation*} Q = Q_0 + \int_{x_0}^{x} f'(z) \, dz\text{.} \end{equation*}
• Computational tools, such as WolframAlpha and SageMath, are available to perform these calculations, leaving us the responsibility of formulating problems and interpreting the results.