Question

current attempt in progress (a) using the table below, find the average rate of change in the worlds population, p, between 2006 and 2020. give units. world population, in billions of people

year	2006	2008	2010	2012	2014	2016	2018	2020
population	6.53	6.71	6.77	7.02	7.17	7.32	7.50	7.68

round your answer to three decimal places. rate of change of population = (b) if p = f(t) with t in years, estimate f(2012) and give units. use a left - hand interval and round your answer to three decimal places. f(2012) =

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2006$, $b = 2020$, $f(2006)=6.53$ and $f(2020)=7.68$.

Step2: Calculate average rate of change

$\frac{7.68 - 6.53}{2020 - 2006}=\frac{1.15}{14}\approx0.082$ billion people per year.

Step3: Recall left - hand derivative approximation

The left - hand derivative of $y = f(x)$ at $x = x_0$ is approximated as $\frac{f(x_0)-f(x_0 - h)}{h}$. For $f'(2012)$ with a left - hand interval and $h = 2$ (since the data is given every 2 years), $x_0=2012$, $f(2012) = 7.02$ and $f(2010)=6.77$.

Step4: Calculate $f'(2012)$

$\frac{7.02-6.77}{2}=\frac{0.25}{2}=0.125$ billion people per year.

Answer:

(a) The average rate of change is approximately $0.082$ billion people per year.
(b) $f'(2012)=0.125$ billion people per year.