When we studied sequences, we learned that the behavior of the sequence could be described in terms of its increments. (Recall 3.4.) The following summarizes what we learned about sequences.

Behavior of a Sequence Using Increments

Every sequence \((x_k)\) has a corresponding increment sequence, \(\Delta x_k = x_{k} - x_{k-1}\). The dynamic behavior of the sequence \(x\) can be described using the attributes of the increments as summarized below.

If the increments are positive on a range, \(\Delta x_k \gt 0\) for all \(k=m,\ldots,n\), then the sequence \(x\) is increasing on the index range \(k=m-1,\ldots,n\).

If the increments are negative on a range, \(\Delta x_k \lt 0\) for all \(k=m,\ldots,n\), then the sequence \(x\) is decreasing on the index range \(k=m-1,\ldots,n\).

If the increments are increasing on a range, \(\Delta x_k \gt 0\) for all \(k=m,\ldots,n\), then the sequence \(x\) is concave up on the index range \(k=m-1,\ldots,n\).

If the increments are decreasing on a range, \(\Delta x_k \lt 0\) for all \(k=m,\ldots,n\), then the sequence \(x\) is concave down on the index range \(k=m-1,\ldots,n\).

Note5.1.1

The range of index values for the sequence always starts one index earlier than the corresponding range of index values describing the increments because each increment is computed as a backward different. The increment \(\Delta x_m = x_m - x_{m-1}\) characterizes the behavior of the sequence \(x\) going from index \(m-1\) to index \(m\). So a range of index values \(k=m,\ldots,n\) for the increments characterizes the change of the sequence \(x_{m-1}\) to \(x_{m}\), and ultimately to \(x_n\).

We now attempt to use this understanding by way of analogy to describe functions. Functions are more general than sequences, although sequences are a special subcollection of functions. A function is a general rule mapping values in a domain set to a co-domain set. For a sequence, the domain is a collection of integers. More generally, functions might have any valid set as the domain. We will usually be working with functions whose domains are intervals or unions of intervals.

When we learned about definite integrals, for a function \(f\) that is integrable (which includes all continuous functions), we can define an accumulation function \begin{equation*}A(x)=\int_{a}^{x} f(z) \, dz.\end{equation*} (See 4.4,) The accumulation function measures the accumulated increments of change using \(f(x)\) as the rate of change as \(x\) goes from \(x=a\) to the present value. The Riemann sum approximation reinforces this interpretation, \begin{equation*} \int_{a}^{x} f(z) \, dz \approx \sum_{k=1}^{n} f(z_k^*) \Delta z \end{equation*} where \(\displaystyle \Delta z = \frac{x-a}{n}\), \(z_k=a+k \Delta z\) and \(z_k^*\) is any value in the subinterval \([z_{k-1}, z_k]\). The increments \(f(z_k^*) \Delta z\) represent the product of a rate \(f(z_k^*)\) and an increment of the variable \(\Delta z\).

The rate function for an accumulation function is analogous to the increments for a sequence. The features of the rate function informs us about the behavior of the accumulation function \(A(x)\). The rate function \(f(x)\) is called the derivative of the accumulation function \(A(x)\), and we write \(f(x)=A'(x)\) or \(\displaystyle f(x)=\frac{dA}{dx}\). This statement, known as the Fundamental Theorem of Calculus, serves as the central result of calculus. One of our goals is to understand this result at a level where we know not only what it says but why it is true.

Behavior of a Function Using Derivatives

Given an accumulation function \(A(x)\) that has a derivative \(A'(x)=f(x)\) defined on an interval \(I\), then the behavior of \(A\) is determined by the behavior of \(A'=f\) as given below.

If \(A'(x) \gt 0\) for all \(x \in I\), then \(A(x)\) is increasing on the interval \(I\).

If \(A'(x) \lt 0\) for all \(x \in I\), then \(A(x)\) is decreasing on the interval \(I\).

If \(A'(x)\) is increasing on the interval \(I\), then \(A(x)\) is concave up on \(I\).

If \(A'(x)\) is decreasing on the interval \(I\), then \(A(x)\) is concave down on \(I\).

When a function is defined as an accumulation function, identifying the derivative (or rate function) is as easy as identifying the function that appears inside the integral operation. But what about other functions? Do they have derivatives as well? Unfortunately, the answer is not always. We will be studying this question throughout the course as well as learn rules for how to find the derivatives when they exist.

Note that because \(A'(x)=f(x)\) is also a function, it too might have a derivative, represented by \(A''(x)=f'(x)\). If so, the sign of \(A''\) can tell us whether \(A'\) is increasing or decreasing, and therefore gives us the concavity of the original function \(A\). Also note that we will later learn some conditions in which we will be able to add the end-points of the intervals. Also, we will not generally use the name \(A\) for the function of interest.

In order to use these results about the behavior of a function in terms of its derivative, we will need to have methods of determining derivatives. For now, we will let a computer assist us. Online tools, such as WolframAlpha.com or SageMathCell, are convenient enough. Because Sage allows more flexibility, some guidance is provided below.

Suppose you have a function \(F(x) = x^3+4x^2\). Sage requires that you explicitly indicate multiplication. In order to work with our function, we will save it with a label y. To verify our work, we will have Sage display its results.

In Sage, once we have a label (a Sage variable), we can apply Sage operations. In this example, we want to find the derivative. We will save this with a new label as well, say dy. So long as you evaluated the above results already, you can evaluate the next step below.

If your problem involves an independent variable other than \(x\), you need to let Sage know what symbols represent mathematical variables. The following script uses the same function but with \(t\) as the independent variable. It also finds \(f''(t)\) as the derivative of \(f'(t)\).

Knowing formulas for the derivatives allows us to interpret the behavior of the original function. The following example works with the function we just differentiated with Sage.

Example5.1.2

Given \(f(t) = t^3+4t^2\), describe the behavior of \(f\) giving intervals of monotonicity and of concavity.

Using a computer, we found \(f'(t) = 3t^2+8t\) and \(f''(t)=6t+8\). Both of these derivatives are defined for every value of \(t\). So we are looking for intervals on which \(f'(t)\) and \(f''(t)\) are positive or negative.

To address these questions, we will use factoring. The sign of a product can be determined from the signs of factors. When there is an odd number of negative factors, the product is negative. When there is an even number of negative factors (including zero negative factors), the product is positive. Simple factors can change signs where the factor equals zero or is not defined.

First, we consider \(f'(t)=3t^2+8t = t(3t+8)\) The factor \(t\) changes sign at \(t=0\), is positive when \(t \gt 0\) and negative when \(t \lt 0\). The factor \(3t+8\) changes sign when \(3t+8=0\) or \(t = - \frac{8}{3}\). It is positive when \(t \gt -\frac{8}{3}\) and negative when \(t \lt -\frac{8}{3}\). These results can be conveniently summarized with a number line. Below the line, we indicate the values of the independent variable \(t\). Above the line, we indicate the sign of the function under consideration, in this case \(f'(t)\) by showing the signs of the factors.

Our sign analysis of the factors shows us that \(f'(t) \gt 0\) on the intervals \((-\infty,-\frac{8}{3})\) and \((0,\infty)\) and that \(f'(t) \lt 0\) on the interval \((-\frac{8}{3},0)\). We interpret these results as behaviors for the function \(f(t)=t^3+4t^2\).

\(f(t)\) is increasing for \(t \lt -\frac{8}{3}\), that is, on the interval \((-\infty,-\frac{8}{3})\).

\(f(t)\) is decreasing for \(-\frac{8}{3} \lt t \lt 0\), that is, on the interval \((-\frac{8}{3},0)\).

\(f(t)\) is increasing for \(t \gt 0\), that is, on the interval \((0,\infty)\).

The second derivative \(f''(t)=6t+8=2(3t+4)\) has only one factor that changes sign at \(t=-\frac{4}{3}\). The sign analysis of \(f''(t)\) is summarized by another number line summary.

We interpret the sign analysis as telling us that \(f''(t) \gt 0\) on the interval \((-\frac{4}{3},\infty)\) and \(f''(t) \lt 0\) on the interval \((-\infty,-\frac{4}{3})\). Because \(f''(t)\) is the derivative of \(f'(t)\), this implies that \(f'(t)\) is increasing on the interval \((-\frac{4}{3},\infty)\) and is decreasing on the interval \((-\infty,-\frac{4}{3})\). Knowing where \(f'(t)\) is increasing and decreasing allows us, in turn, to know where \(f(t)\) is concave up and concave down.

\(f(t)\) is concave down on the interval \((-\infty,-\frac{4}{3})\).

\(f(t)\) is concave up on the interval \((-\frac{4}{3},\infty)\).

Compare these results to the graph \(y=f(t)\), shown below. Pay particular attention to how the intervals we found above relate to the features of the graph. Look for how the graph is increasing for \(t \lt -\frac{8}{3}\), decreasing for \(-\frac{8}{3} \lt t \lt 0\) and increasing for \(t \gt 0\). Notice how the graph is concave down for \(t \lt -\frac{4}{3}\) but concave up for \(t \gt -\frac{4}{3}\). Many students mistakenly think that we should have said the graph is concave down on the interval \((-\infty,0)\) (\(t \lt 0\)) and concave up on the interval \((-\frac{8}{3},\infty)\) (\(t \gt -\frac{8}{3}\)). Why would they say that? What would you say to help them understand why concavity changes at \(t=-\frac{4}{3}\)?

Key Questions Still Needing Answers

This section introduced a number of things that we will study as the course progresses. It leaves a number of questions unanswered for now.

The derivative was introduced as the rate function in a definite integral. What does the derivative measure?

How does one mathematically define a derivative?

How does one calculate a derivative?

What functions even have a derivative?

What is the precise relationship between definite integrals and derivatives (i.e., the fundamental theorem of calculus)?

Concavity is defined by where a derivative is increasing or decreasing. What does that really mean?