Average Rate of Change
The average rate of change measures the average amount of change in a quantity over a specific interval. It is calculated by finding the difference in the values of the quantity at the endpoints of the interval and dividing by the length of the interval. In mathematics, it is commonly used to analyze the average speed or average growth rate of a function.
4 Key excerpts on "Average Rate of Change"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 3 Concepts of the Derivative CHAPTER 3 CONCEPTS OF THE DERIVATIVE 3.1 INTRODUCTION With the establishment of limits and continuity, a foundation has been laid for studying the rest of calculus. Calculus is the study of change—both large and infinitesimally small change. In previous courses, you studied the slope of linear functions, but the use of limits in calculus enables mathematicians to study curves with constantly changing slopes. With calculus, one can also apply varying slope to position, velocity, acceleration, and anything else that is in a state of change. 3.2 RATES OF CHANGE Average Rate of Change Suppose 120 miles is covered in 3 hours on a car trip. It is common to speak of the average speed for the trip as 40 miles per hour. Without question, during the trip the vehicle traveled at speeds other than 40 miles per hour, but based on just the distance covered and the time it took to cover that distance, regardless of what happened during the trip, an average rate can be determined. Anything that changes over time can have an average rate calculated. For example, if the outside temperature increases 12 degrees in 6 hours, the average rate of increase is 2 degrees per hour. Average Rate of Change If a quantity Q changes as a function of t on the interval [ t 1, t 2 ], then the Average Rate of Change of Q with respect to t is The formula for the Average Rate of Change should look familiar. It is simply the slope between two points of a function. In previous courses, the majority of functions were of the form y = f (x), and the slope was calculated by using It is important to remember that determining the Average Rate of Change takes no calculus, and that it always measures change over an interval. EXAMPLE 3.1 Find the Average Rate of Change of f (x) = 9 – x 2 on the interval [–2,1]. SOLUTION The Average Rate of Change of a function must be understood from a graphical standpoint as well...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...While the x values are typically units of time, they do not have to be. EXAMPLE 1: Given f (x) = x 2 − 2 x − ln x, find the Average Rate of Change of f from x = 1 to x = e. SOLUTION: Avg. rate of change EXAMPLE 2: A particle is moving such that its position on the x -axis is given by x (t) = 2cos t + 3sin t + 2, where t is measured in seconds and x (t) is measured in meters. a) Find the Average Rate of Change of its position on the interval. Specify units. b) What is the special name of the quantity you found? SOLUTION: Avg. rate of change of position a) b) This is the average velocity for. Average Rate of Change can be important but doesn’t necessarily tell us much, especially if the gap in time is large. If you were told that the stock market had an Average Rate of Change of 200 points/month, that doesn’t tell you how it fared on a particular day. You can have an average of 80% in calculus over a semester, but if you failed the last two exams, at this particular moment, you are doing failing work. So we are interested in the instantaneous rate of change or how a quantity is changing at a particular instant. To find it, we need calculus. But for now, we will find an approximation for the instantaneous rate of change. It is possible to approximate instantaneous rates of change if you are given a table of n values. To find the approximate instantaneous rate of change at x = i, you can use any of these formulas: Again, while the x values are typically units of time, they do not have to be. EXAMPLE 3: Values of f (x) are given for selected values of x in the table below. Find a) the Average Rate of Change of f on the interval [0, 20]. b) the approximate rate of change of f at x = 12. c) the approximate rate of change of f at x = 20. SOLUTIONS: a) Average Rate of Change b) Approximate rate of change at time x = 12 is either c) Approximate rate of change at time x = 20 is DID YOU KNOW? A jiffy is actually a unit of time...
- Magno Urbano(Author)
- 2019(Publication Date)
- Wiley(Publisher)
...2 Infinitesimal Calculus : A Brief Introduction 2.1 Introduction In this chapter we will do a brief introduction to infinitesimal calculus or differential and integral, known as simply, calculus. Wikipedia has as good definition about calculus: Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves) and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of Calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well‐defined limit. Generally, modern calculus is considered to have been developed, independently, in the 17 th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, Calculus has widespread uses in science, engineering, and economics. Calculus is a part of modern mathematics education. 2.2 The Concept Behind Calculus Suppose we have a car traveling in a straight line for 1 h, at a constant speed of 100 km/h, and later reducing the speed in half and traveling for another hour. What is the average speed of that car? The answer is Now let us see a more complex problem. The car travels 15 min at a nonconstant speed of 15 km/h, stops for 5 min, travels for 5 km at a nonconstant speed of 8 km/h, stops again, and then travels a distance of 2 km at 15 km/h. What is the average speed now? The answer is not evident because the method we have cannot deal with variable entities...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...(y 2 – y 1)/(x 2 – x 1), the increase in y values divided by the increase in x values. Figure 5.7 The position of an apple at 0, 20, 40, and 60 ms after it begins its descent toward the head of Isaac Newton. Figure 5.8 A plot of y against x for y (x) = 0.05 x 2. The gradient of the straight line between points at x = 40 and x = 60 gives a value for the average velocity between 40 and 60 ms. 5.2 Average and instantaneous rates of change Moving on now to consider functions more complex than simple straight lines, imagine that we are watching a slow motion movie of Isaac Newton sitting in the orchard behind his family home in Woolsthorpe, Lincolnshire, UK. If a ripe apple falls off a branch above him, it will approach his head by y millimeters in x milliseconds, given approximately by the formula: y (x) = 0.05 x 2. (EQ5.3) Note that this is another example of a numerical value equation ; that is, y is the distance traveled divided by a reference value of 1 mm and x is the time divided by a reference value of 1 ms. Figure 5.7 uses a coordinate line to show the position of the apple at 0, 20, 40, and 60 ms after it begins its descent toward the cranium of the soon-to-be Lucasian Professor of Mathematics. The plot of y against x in Figure 5.8 shows that the apple travels ever-larger distances during consecutive 20 ms time intervals. A single gradient value can therefore never provide enough information to describe the complete shape of the curve. Because the rate of change of distance with respect to time is another way of talking about velocity, this is equivalent to saying that a single velocity would not be able to give a full account of the trajectory of the apple as it accelerates downward under the influence of gravity. We can, however, work out the average velocity of the apple over a given time interval...
