Average rate of change of a function
A car travels 150 km in 2 hours: its average speed is 75 km/h, but that says nothing about how the journey went at each moment. The same happens with any function: the average rate of change measures how much it changes, on average, between two points. In this topic you will formalise it and get it ready for the next step: shrinking the interval down to a single point.
Average rate of change: definition and calculation
Calculate the average rate of change of the function $f(x) = 3x - 5$ on the interval $[1, 4]$. Does this value match the gradient of the line representing the function?
Calculate the average rate of change of the function $f(x) = 3x - 5$ on the interval $[1, 4]$. Does this value match the gradient of the line representing the function?
Find the average rate of change of the quadratic function $g(x) = x^2 - 4x + 1$ on the intervals $[0, 2]$ and $[2, 4]$.
Find the average rate of change of the quadratic function $g(x) = x^2 - 4x + 1$ on the intervals $[0, 2]$ and $[2, 4]$.
Geometric interpretation: the secant line
This works the same way for any type of function: polynomial, rational or with radicals. Only the expression you substitute changes.
Calculate the average rate of change of the rational function $f(x) = \frac{2x + 1}{x - 1}$ on the interval $[2, 5]$.
Calculate the average rate of change of the rational function $f(x) = \frac{2x + 1}{x - 1}$ on the interval $[2, 5]$.
Find the average rate of change of the radical function $p(x) = \sqrt{x + 3}$ on the interval $[1, 6]$.
Find the average rate of change of the radical function $p(x) = \sqrt{x + 3}$ on the interval $[1, 6]$.
Look at the graph of the function $f(x)$ shown in the figure. Calculate the average rate of change on the interval $[-2, 2]$. What geometrical interpretation does this…
Look at the graph of the function $f(x)$ shown in the figure. Calculate the average rate of change on the interval $[-2, 2]$. What geometrical interpretation does this value have with respect to the secant line drawn?
The ARC on a general interval $[a, a+h]$
To study what happens right at a point $x=a$ it is convenient to write the other endpoint as $a+h$, where $h$ is the step size. That way the ARC becomes a function of $h$, and we can see what happens as we shrink it.
Calculate the average rate of change of the function $f(x) = \frac{1}{x}$ on the general interval $[a, a+h]$, simplifying the resulting expression as much as possible.
Calculate the average rate of change of the function $f(x) = \frac{1}{x}$ on the general interval $[a, a+h]$, simplifying the resulting expression as much as possible.
Calculate the average rate of change of the function $f(x) = \sqrt{x+2}$ on the interval $[2, 2+h]$. Then, determine which value this rate of change approaches as $h$…
Calculate the average rate of change of the function $f(x) = \sqrt{x+2}$ on the interval $[2, 2+h]$. Then, determine which value this rate of change approaches as $h$ tends to $0$.
The ARC as average velocity and other quantities
When $f$ measures position as a function of time, the ARC is the average velocity. The same idea works for any other quantity: temperature, the volume of a tank, a company's profit.
The distance in metres travelled by a cyclist as a function of time in seconds is given by the expression $d(t) = 2t^2 + 5t$. What was their average speed between the…
The distance in metres travelled by a cyclist as a function of time in seconds is given by the expression $d(t) = 2t^2 + 5t$. What was their average speed between the times $t = 2$ and $t = 5$?
The monthly profits of a company, in thousands of euros, are given by the function $B(x) = -2x^2 + 24x - 10$, where $x$ is the amount spent on advertising in thousands…
The monthly profits of a company, in thousands of euros, are given by the function $B(x) = -2x^2 + 24x - 10$, where $x$ is the amount spent on advertising in thousands of euros. Calculate the average rate of change of profit if the advertising investment increases from $3$ to $6$ thousand euros, and interpret the result.
A distress flare is launched vertically upwards from the top of a cliff. Its height in metres above sea level after $t$ seconds is given by $h(t) = 80 + 30t - 5t^2$.…
A distress flare is launched vertically upwards from the top of a cliff. Its height in metres above sea level after $t$ seconds is given by $h(t) = 80 + 30t - 5t^2$. Determine the average velocity of the flare on the intervals $[1, 3]$ and $[3, 5]$. What does the sign of the results obtained indicate?
Comparing rates of change and special cases
Not all functions grow in the same way. Comparing the ARC of two functions on the same interval tells us which one changes faster on average over that stretch, although the comparison may be reversed on another interval.
Compare the average rate of change of the functions $f(x) = 2^x$ and $g(x) = x^3$ on the interval $[2, 4]$. Which of the two functions experiences greater average growth…
Compare the average rate of change of the functions $f(x) = 2^x$ and $g(x) = x^3$ on the interval $[2, 4]$. Which of the two functions experiences greater average growth on that interval?
Prove analytically that the average rate of change of any linear function $f(x) = mx + n$ on an arbitrary interval $[a, b]$ is always constant and equal to $m$.
Prove analytically that the average rate of change of any linear function $f(x) = mx + n$ on an arbitrary interval $[a, b]$ is always constant and equal to $m$.