Average rate of change of a 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 $[2, 4]$.

Determine the value of the parameter $k$ so that the average rate of change of the function $f(x) = kx^2 - 3x + 2$ on the interval $[1, 3]$ is equal to $5$.

Calculate the average rate of change of the rational function $f(x) = \frac{2x + 1}{x - 1}$ on the interval $[2, 5]$.

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?

Given the function $f(x) = -x^2 + 6x - 5$, find the equation of the secant line passing through the points on the graph with x-coordinates $x = 1$ and $x = 4$. How is the gradient of this line related to the average rate of change on that interval?

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.

We want to study the average rate of change of the quadratic function $f(x) = 2x^2 - 3x + 1$ on three intervals starting at the same point $a = 1$: $[1, 2]$, $[1; 1.5]$, and $[1; 1.1]$. Two classmates propose different strategies:

• Clara's method: calculate $f(b)$ and $f(1)$ separately for each endpoint and apply the formula $\text{ARC} = \frac{f(b) - f(1)}{b - 1}$ three times independently.

• Daniel's method: express the upper endpoint as $1+h$, deduce a simplified general algebraic formula for $\text{ARC}[1, 1+h] = \frac{f(1+h) - f(1)}{h}$ in terms of the increment $h$, and then specialise by substituting $h = 1$, $h = 0.5$, and $h = 0.1$.

1. a) Apply Daniel's method: expand and simplify as much as possible the algebraic expression for $\text{ARC}[1, 1+h]$, and find the three required values.
2. b) Apply Clara's method only for the interval $[1; 1.1]$ and verify that both methods lead to the exact same result.
3. c) Argue which of the two methods is more advantageous when many intervals sharing the origin $a = 1$ must be evaluated, and explain how the expression obtained by Daniel allows you to deduce the trend of the rate as the increment $h$ approaches zero.

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 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 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 $h(t) = 80 + 30t - 5t^2$. Determine the average velocity of the flare on the intervals $[1, 3]$ and $[3, 5]$. What does the sign of the results obtained indicate?

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?

Consider the exponential function $f(x) = 2^x$.

1. a) Before carrying out any numerical calculations, observe the general behaviour of an exponential function and estimate, with reasoning, which of the two rates will be greater: $\text{ARC}[0, 2]$ or $\text{ARC}[2, 4]$. Do you expect both to be positive?
2. b) Calculate $\text{ARC}[0, 2]$ and $\text{ARC}[2, 4]$ exactly. Check whether the numerical result confirms your previous estimation.
3. c) Can there be any non-constant polynomial function for which $\text{ARC}[0, 2] = \text{ARC}[2, 4]$ holds? Justify what type of function would satisfy this equality and why this does not happen for $f(x) = 2^x$.

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 $m$.

On a summer day in Cork, weather records indicate that the temperature $T$ (in degrees Celsius) between $10:00$ ($t = 0$) and $16:00$ ($t = 6$) fits a piecewise linear model:

1. a) Calculate the average rate of change of the temperature on the interval $[0, 4]$ and on the interval $[4, 6]$. Do not forget to state the units of measurement for the results.
2. b) Without performing any additional numerical calculations, deduce the value of $\text{ARC}[1, 3]$. Justify your answer using the properties of the average rate of change in affine functions.
3. c) Calculate $\text{ARC}[0, 6]$ over the entire period and interpret its physical meaning.

Given the piecewise function:

$f(x) = \begin{cases} x^2 - 2x & \text{if } x \le 2 \\ \frac{4}{x} - 2 & \text{if } x > 2 \end{cases}$

Calculate the average rate of change on the interval $[0, 4]$. Check whether the function is continuous at $x=2$ and explain whether this affects the calculation of the average rate of change.