Function rate of change
The function rate of change is a fundamental concept in calculus, particularly in differential calculus, which deals with derivatives and rates of change. It offers a mathematical framework to describe how a function changes over time or with respect to another variable. For instance, when analyzing the motion of a ball thrown into the air, one can calculate the average rate of change of its height over specific time intervals, revealing how quickly the ball rises or falls. This analysis leads to the distinction between average rates of change—which can be calculated over intervals—and instantaneous rates of change, which defines the speed at a precise moment.
Historically, discussions around change date back to ancient philosophers, with different views on the nature of change influencing the development of calculus. The transition from average to instantaneous rates of change is crucial, as it underpins the foundation of differential calculus. Real-world applications of these concepts are vast, impacting fields from physics to economics. Understanding rates of change is essential for interpreting various phenomena, such as economic indicators or ecological trends, providing insights into whether certain changes are beneficial or detrimental. This comprehension not only enhances mathematical literacy but also fosters informed decision-making about critical issues facing society today.
Function rate of change
Summary: The rate of change of a function is a key focus of differential calculus.
Though calculus has a reputation for impenetrability compared to algebra and geometry, one of its two main branches, differential calculus, is concerned with derivatives, or rates of change. Rate of change is intuitively understood: the stock market is falling, but how fast? The rolling ball is slowing down, but when will it stop? A derivative is the rate of change of a mathematical function, discovered through a process called differentiation.
History and Language of the Study of the Rate of Change
The ancient Greeks wrestled with the concept of change. Parmenides of Elea asserted that change is impossible, while Heraclitus of Ephesus believed that everything changes and nothing remains still. Aristotle accepted some forms of change but he denied a “change of change” related to motion. Historians have commented that a lack of recognition of a rate of change of a velocity was the major stumbling block to the development of calculus by Archimedes of Syracuse almost 2000 years before Sir Isaac Newton.
The language that describes changing quantities can still be confusing. A politician announces that a proposed spending bill will “cut the deficit” because it will “lower the rate at which the deficit is growing.” Geologists announce that global oil production continues to increase and that “the rate at which production is growing is decreasing.” Does this mean that the rate of global oil production itself may soon also decrease? Reflecting on a particular twenty-first century recession, economists shared that “the rate at which jobs are being lost is decreasing.” Are these announcements good news or bad news?
Average and Instantaneous Rates of Change
In order to understand these statements, several key ideas about functions must be investigated; in particular, it is important to know what it means to say that a function is “increasing” or “decreasing,” as well as whether a function’s rate of change is increasing or decreasing. A familiar physical situation is helpful to consider. Let a ball be tossed into the air and follow its height above the ground at time t.
In the late 1600s, Sir Isaac Newton correctly conjectured that the ball’s height can be modeled by a certain quadratic function. Consider the function y=h(t)=-16t2+32t+4, where h is measured in feet and t is measured in seconds, and observe its graph below. The goal is to study how the function changes as time moves forward and, hence, to understand what is meant formally by “the rate of change of the function.”
From our understanding of quadratic functions, we can observe that at time t = 0, when the ball is tossed, its height is h(0)=4 feet. The ball lands when its height is h = 0, which occurs for the positive value of t that satisfies -16t2+32t+4=0; the quadratic formula indicates that the positive t that satisfies this equation is t=1+√5/2=2.118 seconds.
Finally, since the vertex of the parabola occurs at t=-32/(2•(-16))=1, it follows that the maximum height the ball reaches is h(1)=-16•12+32•1+4=20 feet. Clearly the ball is going up on the interval from t = 0 until t = 1, and the ball is going down thereafter. Perhaps a more interesting question is “how is the ball going up and going down?” Or, “how fast is the ball rising or falling at a particular moment?” For instance, consider the interval [½, 1].
On that interval, the ball rose 4 feet, since h(1)-h(1/2)=20-16=4. In addition, half a second of time elapsed. This knowledge shows that the function’s “average rate of change” on the time interval [½, 1]is
The units on this quantity are important: the numerator is measured in feet, while the denominator is in seconds, so the overall units are “feet per second,” reflecting the rate of change of height with respect to time. The algebraic form of the average velocity on the time interval [a, b],
is reminiscent of another familiar quantity: the slope of a line that passes through the points x1,y1 and x2,y2 is given by
Straight line segments are used to model and approximate the parabolic function, for example, from point B to point C in the graph. Hence, it is seen that the average rate of change of the function h on a given interval is understood visually to be the slope of a line that passes through two points on the graph of h.
The average rate of change of the function quantifies how fast the ball is rising or falling and will vary on different intervals. This accurately reflects that the ball is “falling faster,” since its average rate of change is more negative than on the preceding interval. For instance, on the interval [0,½], the average rate of change of h is
while on the interval [½,1], the average rate of change is
These quantities confirm numerically what we can see visually from the graph: the ball is rising faster during the first half-second than it is during the second half-second. What happens in the subsequent half-second?
Similar computations, shows that on the time interval [1,3/2], the average rate of change is
Here, for the first time, a negative average rate of change is encountered; the minus sign is extremely important, as it is the numerical indicator that the ball is falling. From the symmetry of the parabola, one can expect (and can calculate to check) that on [3/2m2], the average rate of change of h is -24 feet per second. This result accurately reflects that the ball is “falling faster,” since its average rate of change is more negative than on the preceding interval.
It is next natural to seek to understand the difference between the ball’s average rate of change on a time interval and its “instantaneous” rate of change at a single value of t. By taking average rates of change on smaller and smaller time intervals, one encounters a remarkable phenomenon. For instance, consider the average rates of change on [0.5, 1], [0.5, 0.6], [0.5, 0.51], and [0.5, 0.501]. The average rate is 8 feet per second on the first interval; on the next interval, the function’s average rate of change is
On [0.5,0.51], similar computations reveal that the average rate of change is 15.84 feet per second, while on the final interval, [0.5, 0.501], the rate is 15.984. Here, despite the fact that one is dividing by numbers that are getting closer and closer to zero (0.5, 0.1, 0.01, 0.001), it can be seen that the resulting quantities themselves seem to be settling down, nearer and nearer a single number. Calculus is the mathematics that allows these ideas to be made precise. The notion of limits and other key related ideas allow mathematicians to move from the notion of average rate of change to instantaneous rate of change and indeed the instantaneous rate of change of the ball’s height with respect to time at the time t=0.5 is 16 feet per second. By considering the corresponding line segments that pass through two points on the curve, the so-called secant lines actually approach a single line that is “tangent” to the curve at the point (0.5, 16), as pictured in the graph as point B. The red line touches the curve only at (0.5, 16), has slope 16, and represents the instantaneous rate of change of the ball’s height with respect to time at the moment t=0.5.
The Beginnings of Calculus
This idea of moving from average rates of change to instantaneous ones is the starting point for the entire subject of differential calculus. Abu Arrayhan Muhammad ibn Ahmad al-Biruni investigated instantaneous velocity and acceleration approximately 1000 years ago, and Isaac Barrow may have been the first to draw tangents to curves in 1670. The development of calculus led to a rich collection of concepts centered on the idea of a rate of change, many of which were introduced by Isaac Newton in his attempts to develop a universal theory of gravitation. For instance, Newton attempted to avoid the use of the infinitesimal by forming calculations based on ratios of changes and he determined the area under a curve by extrapolating the rate of change. In fact, Newton’s second law states that the rate of change of momentum of a body is equal to the force acting on the body in the same direction. Gottfried Leibniz also investigated concepts related to rate of change. He explored maxima, minima, and tangents in 1684, but mathematicians had difficulty understanding his six-page work.
Other notions of rate of change are also important in mathematics and in real-life applications. For example, in 1847, Jean Frédéric Frenet assigned a frame of vectors to each point on a curve and described the twists and turns of the curve by the rate of change of the frame. Joseph Alfred Serret independently considered similar ideas in 1851. The Frenet–Serret frame continues to be useful in the early twenty-first century when it is impossible to assign a natural coordinate system.
Many real-life problems, such as population growth, can be expressed and modeled as an equation involving a quantity and its rate of change. All of these ideas rest in some way on the fundamental concept of slope, which is investigated beginning in the middle grades, while the notion of rate of change is first developed in high school. Other methods to solve these types of problems are studied in the field of differential equations, which is usually introduced in college.
Applications of Rates of Change
Returning to two of the original questions about the meaning of certain statements and whether they are good news or bad news: a proposed spending bill will “cut the deficit” because it will “lower the rate at which the deficit is growing.” This is not great news, since the deficit is still growing, but a deficit growing at a decreasing rate is better than one growing at an increasing rate. It would be much better to hear that the budget deficit itself was decreasing. Next, the information that “the rate at which oil production is growing is decreasing” may likely mean that the rate of oil production could be leveling off and soon start to decrease—the concept of “peak oil” (when the rate of daily global oil production reaches its maximum)—and many analysts believe humans have just passed this peak and that the rate of oil production will only continue to fall from here.
With the Earth’s human population growing at a present rate of 83 million people per year, as well as so many other changing quantities, collective efforts to understand resource allocation and management require sound understanding of rates of change and trends in data. Calculus and its language of change are a key tool in building a sustainable future for humanity and the planet.
Bibliography
Cohen, David, and James Henle. Calculus: The Language of Change. Sudbury, MA: Jones and Bartlett, 2005.
Connally, Eric, et al. Functions Modeling Change: A Preparation for Calculus. 2nd ed. Hoboken, NJ: Wiley 2007.
Dunham, William. The Calculus Gallery. Princeton, NJ: Princeton University Press, 2005.
Strogatz, Steven. “Change We Can Believe In.” New York Times (April 11, 2010). http://opinionator.blogs.nytimes.com/2010/04/11/change-we-can-believe-in.