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Motivating Questions
- What do we mean by the average rate of change of a function over an interval?
- What does the average rate of change of a function measure? How do we interpret its meaning in context?
- How is the average rate of change of a function related to a line passing through two points on the curve?
Given a function that models a particular phenomenon, it is natural to ask questions like "how does the function change at a given interval" or "at which interval does the function change faster?". The concept ofaverage rate of changeallows us to make these questions mathematically precise. First, let's focus on the average rate of change of an object moving along a rectilinear trajectory.
For a function \(s\) giving the location of a moving object on a straight path at time \(t\text{,}\), we define the average rate of change of \(s\) in the interval \( [ a,b]\) as a set
\[ AV_{[a,b]} = \frac{s(b)-s(a)}{b-a}\text{.} \nonumber \]
In particular, note that the average rate of change from \(s\) to \([a,b]\) is measuredchange of positiondivided by thechange time.
Activity Preview \(\PageIndex{1}\)
The height function for a ball thrown vertically is given by \(s(t) = 64 - 16(t-1)^2\text{,}\) where \(t\) is in seconds and \(s \) is in measured feet above the ground.
- Calculate the value of \(AV_{[1.5,2.5]}\text{.}\)
- What are the units of \(AV_{[1.5,2.5]}\text{?}\) What does this number mean in connection with the rising/falling ball?
- InDesmos, plot the function \(s(t) = 64 - 16(t-1)^2\) along with the points \((1.5,s(1.5))\) and \((2.5, s(2.5)) \text{.}\) Make a copy of your chart on the axes in Figure 1.3.1 and label key points and the scale on your axes. What is the domain of the model? The range? Why?
4. Find the equation of the straight line by hand through the points \((1.5,s(1.5))\) and \((2.5, s(2.5))\text{.}\) Write the straight line in the bilde \( y = mt + b\) and draw the straight lineDesmos, as well as on the axes above.
5. Given your work in the previous questions, what is a geometric interpretation of the value \(AV_{[1.5,2.5]}\)?
6. How do your answers in the previous questions change if we use the interval \([0.25, 0.75]\text{?}\) \([0.5, 1.5]\text{?}\) \([1, 3]\text{?}\)
Define and interpret the average rate of change of a function
In the context of a function that measures the height or position of a moving object at a given point in time, the meaning of the function's average rate of change over a given interval is theAverage speedof the moving object because it is the ratio ofchange of positionTochange time. For example inactivity preview \(\PageIndex{1}\), the units on \(AV_{[1.5,2.5]} = -32\) are "feet per second" since the units on the numerator are "feet" and on the denominator are "seconds". Furthermore, \(-32\) is numerically the same value as the slope of the line connecting the two corresponding points in the position function plot, as seen inFigure \(\PageIndex{2}\)The fact that the average rate of change is negative in this example indicates that the ball is falling.
While the average rate of change of a position function tells us the average speed of the moving object, the average rate of change of a function can be defined similarly in other contexts and has a related interpretation. We make the following formal definition.
Definition \(\PageIndex{4}\)
For a function \(f\) defined on an interval \([a,b]\text{,}\), theaverage rate of change from \(f\) to \([a,b]\)is the crowd
\[ AV_{[a,b]} = \frac{f(b) - f(a)}{b-a}\text{.} \nonumber \]
In any situation, the units of average rate of change help us interpret its meaning, and these units are always "units of output per unit of input." Also, the average rate of change from \(f\) to \([a,b]\) is always equal to the slope of the line between the points \((a,f(a))\) and \(( b,f(b ))\text{,}\) as seen inFigure \(\PageIndex{3}\)
Activity \(\PageIndex{2}\)
The population of Kent and Ottawa counties, Michigan, in which GVSU is located, according to the United States Census1from 1960 to 2010 measured in \(10\) year intervals are given in the following tables.
| 1960 | 1970 | 1980 | 1990 | 2000 | 2010 |
| 363.187 | 411.044 | 444.506 | 500.631 | 574.336 | 602.622 |
| 1960 | 1970 | 1980 | 1990 | 2000 | 2010 |
| 98.719 | 128.181 | 157.174 | 187.768 | 238.313 | 263.801 |
Let \(K(Y)\) be the population of Kent County in year \(Y\) and \(W(Y)\) be the population of Ottawa County in year Y.
- Calculate \(AV_{[1990,2010]}\) for \(K\) and \(W\text{.}\)
- What are the units for each of the quantities you calculated in (a.)?
- Write a careful sentence that explains the meaning of the average rate of change of the Ottawa County population over the time interval \([1990,2010]\text{.}\). Your sentence should begin something like this: "In an average year between 1990 and 2010, the population of Ottawa County was \(\ldots\)"
- Which county had a greater average rate of change during the time interval \([2000,2010]\text{?}\) Were there any intervals where any of the counties had a negative average rate of change?
- Given the data, what is your estimate of the population of Ottawa County in 2018? Why?
The average rate of change of a function over an interval gives us an excellent way of describing how the function behaves on average. For example, if we calculate \(AV_{[1970,2000]}\) for Kent County, we find this
\[ AV_{[1970,2000]} = \frac{574,336 - 411,044}{30} = 5443,07\text{,} \nonumber \]
which tells us that in an average year from 1970 to 2000, the population of Kent County increased by approximately \(5443\) people. In other words, we could also say that Kent County grew at an average rate of \(5443\) people per year from 1970 to 2000. These ideas also provide an opportunity to make comparisons over time. Since
\[ AV_{[1990,2000]} = \frac{574,336 - 500,631}{10} = 7370,5\text{,} \nonumber \]
Not only can we say that Kent County's population grew by about \(7370\) in an average year between 1990 and 2000, but also that the population grew faster from 1990 to 2000 than from 1970 to 2000.
Finally, we can even use the average rate of change of a function to predict future behavior. Since the population changed by \(7370.5\) people per year on average from 1990 to 2000, we can estimate that the population in 2002 is like this
\[ K(2002) \approx K(2000) + 2 \cdot 7370,5 = 574.336 + 14.741 = 589.077 \text{.} \nonumber \]
How the average rate of change indicates feature trends
We've already seen that it's natural to use words like "increasing" and "decreasing" to describe a function's behavior. For example, for the tennis ball, whose height is modeled by \(s(t) = 64 - 16(t-1)^2\text{,}\), we calculated that \(AV_{[1.5,2.5] } = -32\text{,}\), indicating that the height of the tennis ball is decreasing in the interval \([1.5,2.5]\text{,}\) at an average rate of \(32\) feet per second . Similarly, for the Kent County population, since \(AV_{[1990,2000]} = 7370.5\text{,}\), we know that the population in the interval \([1990,2000]\) with a increases average rate of \(7370.5\) people per year.
We make the following formal definitions to clarify what it means to say that a function increases or decreases.
Definition \(\PageIndex{7}\)
Let \(f\) be a function defined on an interval \((a,b)\) (i.e. on the set of all \(x\) for which \(a \lt x \lt b\)) . We say that \(f\) isincreasingly on\((a,b)\)provided that the function is always increasing as we move from left to right. That is, for each \(x\) and \(y\) in \((a,b)\text{,}\) if \(x \lt y\text{,}\) then \(f( x) \lt f(y)\text{.}\)
Likewise we say that \(f\) isdecreasing to \((a,b)\) provided that the function always falls as we move from left to right. That is, for each \(x\) and \(y\) in \((a,b)\text{,}\) if \(x \lt y\text{,}\) then \(f( x) \gt f(y)\text{.}\)
If we calculate the average rate of change of a function over an interval, we can decide whether the function is increasing or decreasingon averagein the interval, but it requires more work2to decide whether the function is increasing or decreasingalwayson the interval.
Activity \(\PageIndex{3}\)
Let's consider two different functions and see how different calculations of their average rate of change tell us about their respective behaviors. Diagrams of \(q\) and \(h\) are shown in Figures 1.3.8 and 1.3.9.
1. Consider the function \(q(x) = 4-(x-2)^2\text{.}\) Calculate \(AV_{[0,1]}\text{,}\) \( AV_{ [1,2]}\text{,}\) \(AV_{[2,3]}\text{,}\) and \(AV_{[3,4]}\text{.}\) What do your last two calculations tell you about the behavior of the function \(q\) on \([2,4]\text{?}\)
2. Consider the function \(h(t) = 3 - 2(0.5)^t\text{.}\) Calculate \(AV_{[-1,1]}\text{,}\) \( AV_{ [1,3]}\text{,}\) and \(AV_{[3,5]}\text{.}\) What do your calculations tell you about the behavior of the function \(h\) on \ ([-1,5]\text{?}\)
3. On the plots in Figures 1.3.8 and 1.3.9, draw the line segments whose respective slopes are the average rates of change you calculated in (a) and (b).
True or false: Since \(AV_{[0,3]} = 1\text{,}\), the function \(q\) increases on the interval \((0,3)\text{.}\ ) Justify your decision.
5. Give an example of a function that has the same average rate of change regardless of the interval chosen. You can provide your example with a table, chart, or formula; Regardless of your choice, write a sentence to explain.
It's useful to be able to link information about the average rate of change of a function and its graph. For example, if we found that \(AV_{[-3,2]} = 1.75\) for a function \(f\text{,}\), this tells us that the function averages between increases between the points \(x = -3\) and \(x = 2\) and does so at an average rate of \(1.75\) vertical units for each horizontal unit. Moreover, we can even find that the difference between \(f(2)\) and \(f(-3)\) is
\[ f(2)-f(-3) = 1,75 \cdot 5 = 8,75 \nonumber \]
da \(\frac{f(2)-f(-3)}{2-(-3)} = 1,75\text{.}\)
Activity \(\PageIndex{4}\)
Sketch at least two different possible graphs that satisfy the criteria for the function given in each part. Make your charts as different as possible. If a graphic is unable to meet the criteria, explain why.
- \(f\) is a function defined on \([-1,7]\) such that \(f(1) = 4\) and \(AV_{[1,3]} = -2\text{ . }\)
2. \(g\) is a function defined on \([-1,7]\) such that \(g(4) = 3\text{,}\) \(AV_{[0,4]} = 0.5\text{,}\) and \(g\) does not always increase to \((0,4)\text{.}\)
3. \(h\) is a function defined on \([-1,7]\) such that \(h(2) = 5\text{,}\) \(h(4) = 3\) and \(AV_{[2,4]} = -2\text{.}\)
Summary
- For a function \(f\) defined on an interval \([a,b]\text{,}\), the average rate of change of \(f\) on \([a,b]\) is the magnitude
\[ AV_{[a,b]} = \frac{f(b) - f(a)}{b-a}\text{.} \nonumber \]
- The value of \(AV_{[a,b]} = \frac{f(b) - f(a)}{b-a}\) tells us how much the function increases or decreases on average for each additional unit we move move to the right in the graphic. For example, if \(AV_{[3,7]} = 0.75\text{,}\), this means that for an additional \(1\) unit increase in the value of \(x\) in the interval \ ( [3,7]\text{,}\) the function increases on average by \(0.75\) units. In applied settings, the units of \(AV_{[a,b]}\) are "units of output per unit of input".
- The value of \(AV_{[a,b]} = \frac{f(b) - f(a)}{b-a}\) is also the slope of the straight line passing through the points \((a,f ( a))\) and \((b,f(b))\) on the graph of \(f\text{,}\) as shown in Figure 1.3.3.
exercises
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9.
A cold can of soda is taken out of a refrigerator. Its temperature \(F\) in degrees Fahrenheit is measured in \(5\) minute intervals as shown in the table below.
| \(t\) (minutes) | \(0\) | \(5\) | \(10\) | \(15\) | \(20\) | \(25\) | \(30\) | \(35\) |
| \(F\) (Fahrenheit temp) | \(37,00\) | \(44.74\) | \(50.77\) | \(55.47\) | \(59.12\) | \(61,97\) | \(64.19\) | \(65.92\) |
- Determine \(AV_{[0,5]}\text{,}\) \(AV_{[5,10]}\text{,}\) and \(AV_{[10,15]}\text{, }\) including the appropriate units. Choose one of these sizes and write a careful sentence to explain its meaning. Your sentence might look something like this: “In the interval \(\ldots\text{,}\), the temperature of the soda averages \(\ldots\) by \(\ldots\) for every \(1\) unit increase in \(\ldots\)".
- In what interval does the total soda temperature change more: \([10,20]\) or \([25,35]\text{?}\)
- What can you observe when the temperature of the lemonade seems to be changing the fastest?
- Estimate the temperature of the lemonade at \(t = 37\) minutes. Write at least one sentence to explain your thoughts.
10.
The position of a car driving on a straight road at time \(t\) in minutes is given by the function \(y = s(t)\) shown inFigure \(\PageIndex{11}\)The car's position function has units measured in thousands of feet. For example, the point \((2,4)\) on the chart indicates that the car has traveled 4000 feet after 2 minutes.
In everyday language, describe the behavior of the car over the given time interval. In particular, carefully discuss what is in each of the time intervals \([0,1]\text{,}\) \([1,2]\text{,}\) \([2,3]\ text{,} \) \([3,4]\text{,}\) and \([4,5]\text{,}\) plus provide a general comment on what the car does in the interval \( [0,12 ]\text{.}\)
Calculate the average rate of change of \(s\) on the intervals \([3,4]\text{,}\) \([4,6]\text{,}\) and \([5,8 ] \text{.}\) Label your results appropriately using the notation "\(AV_{[a,b]}\)" and provide units for each quantity.
On the graph of \(s\text{,}\), sketch the three straight lines whose slope corresponds to the values of \(AV_{[3,4]}\text{,}\) \(AV_{[4,6 ]}\text{,}\) and \(AV_{[5,8]}\) that you calculated in (b).
Is there a time interval where the car's average speed is \(5000\) feet per minute? Why or why not?
Is there ever a time interval when the car reverses? Why or why not?
11.
Consider an inverted conical tank (point down) whose top is \(3\) feet radius and \(2\) feet deep. The tank is initially empty and is then filled at a constant rate of \(0.75\) cubic feet per minute. Let \(V=f(t)\) be the water volume (in cubic feet) at time \(t\) in minutes and \(h= g(t)\) be the water depth ( in feet) at time \(t\text {.}\) It turns out that the formula for the function \(g\) \(g(t) = \left( \frac{t}{\pi} \ right)^{1/3}\ text{.}\)
- Describe in everyday language how the height function \(h = g(t)\) is expected to behave with increasing time.
- For the height function \(h = g(t) = \left( \frac{t}{\pi} \right)^{1/3}\text{,}\) calculate \(AV_{[0,2 ] }\text{,}\) \(AV_{[2,4]}\text{,}\) and \(AV_{[4,6]}\text{.}\) Include units in your results .
- Again using the altitude function, can you find an interval \([a,b]\) where \(AV_{[a,b]} = 2\) feet per minute? If yes, indicate the interval; If not, explain why there is no such interval.
- Now consider the volume function \(V = f(t)\text{.}\) Even though we don't have a formula for \(f\text{,}\), it is possible to determine the average rate of change in the Volume function on the intervals \([0,2]\text{,}\) \([2,4]\text{,}\) and \([4,6]\text{?}\ ) Why or why not?
<1> Grand Rapids liegt in Kent, Allendale in Ottawa.
<2> Calculus provides a way to justify that a function is always increasing or always decreasing in an interval.
FAQs
How do you find the average rate of change of a function? ›
The average rate of change formula is used to find the slope of a graphed function. To find the average rate of change, divide the change in y-values by the change in x-values. Finding the average rate of change is particularly useful for determining changes in measurable values like average speed or average velocity.
What is the average rate of change? ›What is average rate of change? It is a measure of how much the function changed per unit, on average, over that interval. It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.
What is the average rate of change over the interval (- 1 5? ›What is the average rate of change in f(x) over the interval [1, 5]? Therefore, the average rate of change of the given function over the interval (1, 5) is -6.
How do you calculate rate of change? ›Rate of change problems can generally be approached using the formula R = D/T, or rate of change equals the distance traveled divided by the time it takes to do so.
How do I find the rate of change? ›To find the average rate of change, we divide the change in the output value by the change in the input value.
How is average rate calculated? ›Average rate of reaction = Change in concentration Time rate ( r ) = Δx Δt. Sign of average rate of reaction: When the rate of concentration of reactant decreases then the average rate of reaction will be negative. When the rate of concentration of product increases then the average rate of reaction will be positive.
How do I find the average? ›Average This is the arithmetic mean, and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
How do you find the average rate of change in a table? ›- Find the coordinates of the given points. It's optional to label them.
- Find the difference between the output values.
- Find the difference between the input values.
- Calculate the ratio between the change in y to the change in x: ΔyΔx.
The average rate of change between two points is calculated as the slope of the straight line which connects the two points. To find the average rate of change of f(x) between x=a and x=b , use the formula f(b)−f(a)b−a f ( b ) − f ( a ) b − a .
What is the average rate of change over the interval 1 3? ›The average rate of change is 1 over 3, or just 1/3. The y-values change 1 unit every time the x-values change 3 units, on this interval.
What is the rate of change on a graph? ›
The rate of change for a line is the slope, the rise over run, or the change in over the change in. The slope can be calculated from two points in a table or from the slope triangle in a graph.
What is the rate of change at a point? ›The instantaneous rate of change is the slope of the tangent line at a point. The slope at a point (the slope of the tangent line) can be approximated by the slope of secant lines as the "run" of each secant line approaches zero.
How do you solve rate of change problems? ›How do you find the average rate of change? We use the slope formula! To find the average rate of change, we divide the change in y (output) by the change in x (input). And visually, all we are doing is calculating the slope of the secant line passing between two points.
What is a rate of change in math? ›A rate of change is a rate that describes how one quantity changes in relation to another quantity. If x is the independent variable and y is the dependent variable, then. rate of change=change in ychange in x. Rates of change can be positive or negative.
What is rate of change 7th grade math? ›In seventh grade, students must use their knowledge to represent constant rates of change, which is the predictable rate at which a given variable alters over a certain period of time by representing and identifying this change when given pictorial, vertical or horizontal tables, verbal, numeric, graphical, and ...
How do you find the rate of change between two numbers? ›First: work out the difference (increase) between the two numbers you are comparing. Then: divide the increase by the original number and multiply the answer by 100. % increase = Increase ÷ Original Number × 100.
What is the change of the function? ›There are two types of rates of change of a function. An exponential rate of change increases or decreases more and more quickly, while the linear rate of change increases or decreases very steadily. An exponential rate of change is represented by an equation involving an exponent.
What is the percentage rate of change of a function? ›The percentage rate of change for the function is the value of the derivative (rate of change) at 1 over the value of the function at 1 . Substitute the functions into the formula to find the function for the percentage rate of change.
What is an example of average rate? ›Some examples of the average rate of change are: A bus travels at a speed of 80 km per hour. The number of fish in a lake increases at the rate of 100 per week. The current in an electrical circuit decreases 0.2 amperes for a decrease of 1-volt voltage.
What is an average rate? ›Average rate is a single rate applying to property at more than one location that is a weighted average of the individual rates applicable to each location.
What is mean by average rate of reaction? ›
The change in concentration of any of the reactants or any of the product per unit time over a specified interval of time is called the average rate of reaction.
What are the 3 ways to calculate average? ›There are three main types of average: mean, median and mode. Each of these techniques works slightly differently and often results in slightly different typical values. The mean is the most commonly used average. To get the mean value, you add up all the values and divide this total by the number of values.
Why do we calculate average? ›Finding an average gives us an idea as to an overall behaviour or trend – Mrs Mansell's average spend on shopping gives us an idea as to whether she usually spends a lot or a little money and Keiran's average spelling score gives us an idea as to how good he usually is at spelling.
How do you find the rate of change of a given point and equation? ›The instantaneous rate of change at a point is equal to the function's derivative evaluated at that point. In other words, it is equal to the slope of the line tangent to the curve at that point. For example, let's say we have a function f(x)=x2 . So, the instantaneous rate of change, in this case, would be 4 .
What is an example of rate of change? ›Other examples of rates of change include: A population of rats increasing by 40 rats per week. A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes) A car driving 27 miles per gallon of gasoline (distance traveled changes by 27 miles for each gallon)
What is the average rate of change over the interval (- 2 4? ›To calculate the average rate of change between the 2 points use. This means that the average of all the slopes of lines tangent to the graph of f(x) in the interval [-2 ,4] is 41.
What is an interval of 3? ›As mentioned in the opening video of this series, an interval of a Major third occurs when two notes are separated by the distance of four consecutive half steps.
What is rate of change and Y intercept? ›The slope and y-intercept values indicate characteristics of the relationship between the two variables x and y. The slope indicates the rate of change in y per unit change in x. The y-intercept indicates the y-value when the x-value is 0.
Is rate of change a slope? ›The rate of change is a ratio that compares the change in values of the y variables to the change in values of the x variables. If the rate of change is constant and linear, the rate of change is the slope of the line. The slope of a line may be positive, negative, zero, or undefined.
Can average rate of change be negative? ›It is possible for the value of the average rate of change to be negative since the average rate of change dictates the slope of the graph. Hence, if the average rate of change is negative, the slope of the graph is negative.
What is the best formula to calculate average percent change? ›
First: work out the difference (increase) between the two numbers you are comparing. Then: divide the increase by the original number and multiply the answer by 100. % increase = Increase ÷ Original Number × 100.
How do you calculate average percentage rate? ›To calculate average percentage, add all percentages together as numbered values and divide by the sum of all the sets. Then multiply by 100.
What is the best way to calculate average? ›Average This is the arithmetic mean, and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
Is the average rate of change a percentage? ›You can convert the average rate of change to a percent by multiplying your final result by 100 which can tell you the average percent of change.
How do you write the rate in an exponential function? ›The exponential growth function can be written as f ( x ) = a ( 1 + r ) x , where is the growth rate. The function f ( x ) = e x can be used to model continuous growth with. The function f ( t ) = a ⋅ e r t can be used to model continuous growth as a function of time.