A Modeling Approach to Calculus: MA2C

Subsection 3.1.1 Functions Have Shapes

Subsubsection 3.1.1.2 Concavity

Concavity describes how the graph of a function in the \((x,y)\) plane bends. If the graph bends upward, we say the function is concave up. If the graph bends downward, we say the function is concave down.

As with monotonicity, these attributes of functions apply over intervals rather than at individual points. When a graph changes concavity at a point, for example switching from bending up to bending down, the function has an inflection point. A technical definition of concavity that depends on the concept of a derivative will be provided later.

However, we can capture the essential idea by thinking about how the slope is changing between points. A function that has an increasing slope or rate of change over an interval is concave up on the interval. A function that has decreasing slope or rate of change is concave down.

Definition 3.1.13. Concavity.

A function \(f\) is concave up on a subset \(S\) of the domain (usually an interval) if for every \(x_1,x_2,x_3 \in S\text{,}\) the slope or rate of change is increasing,

\begin{equation*} x_1 \lt x_2 \lt x_3 \qquad \text{implies} \qquad \frac{f(x_2)-f(x_1)}{x_2-x_1} \lt \frac{f(x_3)-f(x_2)}{x_3-x_2}\text{.} \end{equation*}

A function \(f\) is concave down on a subset \(S\) of the domain (usually an interval) if for every \(x_1,x_2,x_3 \in S\text{,}\) the slope or rate of change is decreasing,

\begin{equation*} x_1 \lt x_2 \lt x_3 \qquad \text{implies} \qquad \frac{f(x_2)-f(x_1)}{x_2-x_1} \gt \frac{f(x_3)-f(x_2)}{x_3-x_2}\text{.} \end{equation*}

This definition is not very easy to use directly. When we learn more about derivatives to describe the slope at individual points, we will have a much better method known as the second derivative test for concavity. However, the following examples will illustrate what is happening.

Example 3.1.14.

The function \(f(x)=x^2\) is concave up on \((-\infty,\infty)\) (the entire domain). We will not prove that this is true because this is too difficult without derivatives. But we can illustrate the idea.

Consider the graph \(y=f(x)=x^2\) and the particular values \(x_1=-4\text{,}\) \(x_2=-2\text{,}\) and \(x_3=-1\text{.}\) We will calculate the slope or rate of change between \((x_1,y_1)=(-4,16)\) and \((x_2,y_2)=(-2,4)\) and between \((x_2,y_2)\) and \((x_3,y_3)=(-1,1)\text{.}\)

\begin{gather*} m_{12} = \frac{y_2-y_1}{x_2-x_1} = \frac{4-16}{-2--4} = \frac{-12}{2} = -6 \\ m_{23} = \frac{y_3-y_2}{x_3-x_2} = \frac{1-4}{-1--2} = \frac{-3}{1} = -3 \end{gather*}

We can see that the slope or rate of change is increasing, \(m_{12} \lt m_{23}\text{.}\) These slopes are illustrated in the following figure.

This is not a proof of concavity because we only illustrated the order for three specific points. Use the following dynamic figure to convince yourself that for any three points we might choose, the slopes increase from left to right.

The reason that \(f\) has an inflection point at \(x=0\) is that point is where \(f\) has the steepest negative slope. To the left, \(x \lt 0\text{,}\) the slope decreases; to the right, \(x \gt 0\text{,}\) the slope increases.

Example 3.1.16.

The function \(f(x)=x^3-3x\) changes concavity at \(x=0\text{.}\) \(f\) is concave down on \((-\infty,0]\) and concave up on \([0,\infty)\text{.}\) When three points are chosen with \(x \in (-\infty,0]\text{,}\) the slope is decreasing. When the three points are chosen with \(x \in [0,\infty)\text{,}\) the slope is increasing. This can be verified in the following dynamic figure. However, the three points must all be in either \((-\infty,0]\) or in \([0,\infty)\text{.}\)

Subsubsection 3.1.1.3 Combining Monotonicity and Concavity

The shape of a graph of a function is often defined in terms of the monotonicity and concavity combined. There are four basic shapes that correspond to the four quadrants of a circle, illustrated in the figure below. A curve that has a positive and increasing slope is increasing and concave up. A curve that has a positive but decreasing slope is increasing and concave down. A curve that has a negative but increasing (becoming less negative) slope is decreasing and concave up. A curve that has a negative and decreasing (becoming more negative) slope is decreasing and concave down.

We can describe the shape of a graph by stating intervals on which the function satisfies each of the possible behaviors. The intervals are separated by points where the graph reaches either a maximum or minimum value (changes in monotonicity) or where the slope of the graph reaches an extreme and begins to bend the other direction (changes in concavity or points of inflection).

Example 3.1.19.

The graph of a function \(y=f(x)\) is shown below, with labeled extreme points and inflection points. Describe the shape of the graph by giving intervals of monotonicity and concavity.

Solution

Intervals for monotonicity are based on the function increasing or decreasing. The end-points of these intervals are the extreme points for the function. When the graph extends beyond the frame of the figure, we assume the function behavior continues as shown. Intervals always are read from left to right. The end-point of an interval is included (closed) if the behavior extends up to and including that point.

The function \(f\) is decreasing on \((-\infty,-3]\text{,}\) increasing on \([-3,3]\text{,}\) and decreasing on \([3,\infty)\text{.}\) Notice that the extremes at \(x=-3\) and \(x=3\) are included in two intervals. The continuous function is decreasing on \((-\infty,-3)\) as an open interval. Because \(f\) decreases up to and including \(x=-3\text{,}\) we include the end-point.

Intervals for concavity are based on where the slope is increasing or decreasing. Intervals on which the graph bends upward, \(f\) is concave up. Intervals on which the graph bends downward, \(f\) is concave down. Notice our graph has inflection points (where the concavity changes) at \(x=-1\text{,}\) \(x=0\text{,}\) and \(x=1\text{.}\) At these points, the graph starts to bend in the opposite direction.

The function \(f\) is concave up on \((-\infty,-1]\text{,}\) concave down on \([-1,0]\text{,}\) concave up on \([0,1]\text{,}\) and concave down on \([1,\infty)\text{.}\) We include the inflection points as the end points of the intervals (closed) because the slope is increasing or decreasing up to and including those points.

Subsection 3.1.2 End Behavior

End-behavior of a function describes what happens to a function as the size of the input grows. Consider the possibilities of a linear function, \(y=f(x)=mx+b\text{.}\) So long as the slope is non-zero, the function is unbounded, meaning that the graph eventually goes above every level and eventually goes below every level (on opposite sides of the graph).

If the slope is positive, \(m \gt 0\text{,}\) then the function is increasing. We say \(f(x) \to +\infty\) as \(x \to +\infty\text{,}\) which we read as “the value of \(f(x)\) tends to positive infinity as the value of \(x\) goes to positive infinity”. This is because the \(y\)-values will eventually rise above any level on the right side of the graph (for sufficiently large positive values \(x\)). We also say \(f(x) \to -\infty\) as \(x \to -\infty\) because the \(y\)-values are below any specified value on the left side of the graph (for sufficiently large negative values \(x\)). When the slope is negative, \(m \lt 0\text{,}\) the unbounded behavior is reversed.

For consistency in notation to describe the tendency of a function (as opposed to the value of a function), we use limits to describe unbounded behavior.

When the graph of a function \(f\) behaves more and more like a constant function (horizontal line) for larger and larger values of the independent variable, we say \(f\) has a horizontal asymptote. A horizontal asymptote \(y=L\) on the right side (large, positive values for \(x\)) uses the limit statement

which means that the value of \(f(x)\) approaches the constant value \(L\) as \(x \to +\infty\text{.}\) When \(f\) has a horizontal asymptote \(y=L\) on the left side (large, negative values of \(x\)), we use the limit statement

Example 3.1.21.

The graph of a function \(y=f(x)\) is shown below. This function has two horizontal asymptotes: \(y=-2\) as \(x \to -\infty\) and \(y=1\) as \(x \to +\infty\text{.}\) We write

\begin{gather*} \lim_{x \to -\infty} f(x) = -2,\\ \lim_{x \to +\infty} f(x) = 1. \end{gather*}

A function can also have unbounded behavior near a particular input value, say at \(x=a\text{.}\) Using limit notation, this means that at least one of the following must be true.

The graph has a vertical asymptote at \(x=a\text{,}\) meaning that the graph of the function approaches closer and closer to this vertical line.

Example 3.1.22.

The graph of a function \(y=f(x)\) is shown below with two vertical asymptotes. The vertical asymptote at \(x=0\) corresponds to left- and right-limits

\begin{align*} \lim_{x \to 0^-} f(x) &= -\infty,\\ \lim_{x \to 0^+} f(x) &= +\infty, \end{align*}

The vertical asymptote at \(x=2\) only corresponds to the right-limit

\begin{equation*} \lim_{x \to 2^+} f(x) = +\infty. \end{equation*}

It is hard to tell from a graph alone where a vertical asymptote occurs. Using only the limited graph window, it is not obvious that the vertical asymptote is at exactly \(x=0\) since the graph is still fairly far away from that vertical line from this perspective.

When a function physically relates two variables, \(x \mapsto y\text{,}\) a horizontal asymptote indicates that for sufficiently large values of the independent variable, the dependent variable is essentially a constant. A common description in physical settings for this constant is a saturation value. We think of the quantity measured by the independent variable as a control variable. The dependent variable can be thought of as a response. As the control variable is increased, the response will pass through some of its range of values. However, there will come a point where even though you continue to increase the control variable, the response is no longer able to change very much at all. That is, the response has saturated.

Example 3.1.24.

An enzyme is a protein that helps catalyze a chemical reaction. The rate or velocity of reaction \(V\) depends on the concentration of the reactant \(C\text{.}\) Commonly, the function \(C \mapsto V\) is increasing, concave down, and has a horizontal asymptote, known as Michaelis–Menten reaction kinetics. The physical domain is \(C \in [0, \infty)\text{.}\) Because the relation is increasing, we know that adding more reactants will raise the reaction rate. Because the relation is concave down, we know that the degree to which the rate increases slows down as more reactants are added. The horizontal asymptote means that this increase in the reaction rate saturates to some maximum rate \(V_{\text{max}}\text{,}\)

\begin{equation*} \lim_{C \to \infty} V = V_{\text{max}}. \end{equation*}

The reactant concentration where the reaction rate is halfway to the maximum value is called the half-saturation value, and is usually represented with a constant \(K\text{.}\)

Example 3.1.26.

Imagine a crop of plants growing in a field. The total biomass harvested \(B\) depends on the number of seeds \(S\) that are sown. If very few seeds are sown, the biomass harvested will be small. For more seeds sown, we expect the biomass would increase. However, if too many seeds are sown, then the crop will be overcrowded, resulting in a lower harvest. We expect that there might be an optimal number of seeds \(S^*\) for which the biomass is at a maximum.

Describe the behavior of the function \(S \mapsto B\) and sketch a possible graph.

Solution

The function \(S \mapsto B\) will have a physical domain of \(S \in [0,\infty)\text{.}\) Because \(B\) is a maximum at \(S=S^*\text{,}\) the function is increasing on \([0,S^*]\) and decreasing on \([S^*,\infty)\text{.}\) The simplest assumption for concavity would be that the function starts concave down. However, a concave down and decreasing function will eventually approach \(-\infty\text{,}\) which is not physically possible for our physical scenario. Therefore, the function must change concavity at some inflection point after \(S^*\text{,}\) say at \(S=S^\dagger\text{.}\) Our function would be concave down on \([0,S^\dagger]\) and concave up on \([S^\dagger,\infty)\text{.}\) Continuing to increase the number of seeds will result in ever smaller biomass due to overcrowding until it approaches some saturating biomass \(B_\infty\text{,}\)

\begin{equation*} \lim_{S \to \infty} B = B_\infty. \end{equation*}

Note: The asterisk and dagger are decorations so that the symbols \(S^*\) and \(S^\dagger\) represent general constants. We don't know actual values for the maximum and inflection point, so we can't use numbers. The symbols are place-holders for values that would be determined experimentally. Similarly, the symbol \(B_{\infty}\) represents the value for the biomass harvested when the number of seeds sown saturates the system.

Subsection 3.1.3 Summary

• Describing the monotonicity of a function is determining intervals on which the function is increasing or decreasing.

• A function \(f\) is increasing on a set \(S\) if the function is order preserving: For all \(x_1,x_2 \in S\text{,}\) we must have

  \begin{equation*} x_1 \lt x_2 \quad \Rightarrow \quad f(x_1) \lt f(x_2)\text{.} \end{equation*}

  This corresponds to a graph that is rising left to right (positive slopes).

  A function \(f\) is decreasing on a set \(S\) if the function is order reversing: For all \(x_1,x_2 \in S\text{,}\) we must have

  \begin{equation*} x_1 \lt x_2 \quad \Rightarrow \quad f(x_1) \gt f(x_2)\text{.} \end{equation*}

  This corresponds to a graph that is falling left to right (negative slopes).

• Describing the concavity of a function is determining intervals on which the function is concave up or concave down.

• A function \(f\) is concave up on a set \(S\) if the slope or rate of change is increasing on \(S\text{:}\) For all \(x_1,x_2,x_3 \in S\text{,}\) we must have

  \begin{equation*} x_1 \lt x_2 \lt x_2 \quad \Rightarrow \quad \frac{f(x_2)-f(x_1)}{x_2-x_1} \lt \frac{f(x_3)-f(x_2)}{x_3-x_2}\text{.} \end{equation*}

  The graph will be bending upward.

  A function \(f\) is concave down on a set \(S\) if the slope or rate of change is decreasing on \(S\text{:}\) For all \(x_1,x_2,x_3 \in S\text{,}\) we must have

  \begin{equation*} x_1 \lt x_2 \lt x_2 \quad \Rightarrow \quad \frac{f(x_2)-f(x_1)}{x_2-x_1} \gt \frac{f(x_3)-f(x_2)}{x_3-x_2}\text{.} \end{equation*}

  The graph will be bending downward.

• A point of inflection is a point where a function is continuous and changes concavity.

• Lists of intervals of monotonicity and concavity should be separated by commas and not joined by unions.

• Limits as \(x \to \pm\infty\) describe end behavior.

  • To say \(f(x) \to +\infty\) means values of \(f(x)\) eventually rise above any possible value.
  • To say \(f(x) \to -\infty\) means values of \(f(x)\) eventually fall below any possible value.
  • To say \(f(x) \to L\) means values of \(f(x)\) eventually approaches a horizontal asymptote \(y=L\text{.}\)