Answer:
[tex][3,4][/tex] : [tex]r = 1[/tex], [tex][4,5][/tex] : [tex]r = 0.59[/tex], [tex][5,6][/tex] : [tex]r = 0.42[/tex]
Step-by-step explanation:
Given a function [tex]f(x)[/tex] that is continuous on [tex][a,b][/tex], the average rate of change is:
[tex]r = \frac{f(b)-f(a)}{b-a}[/tex]
[tex]a[/tex], [tex]b[/tex] - Lower and upper bounds, dimensionless.
[tex]f(a)[/tex], [tex]f(b)[/tex] - Images of the function evaluated at lower and upper bounds, dimensionless.
Let be [tex]f(x) = \log_{2}(3\cdot x - 6)[/tex], then:
[tex][3,4][/tex]
[tex]a = 3[/tex] and [tex]b = 4[/tex]
[tex]f(3) = \log_{2}[3\cdot (3) - 6][/tex]
[tex]f(3) \approx 1.585[/tex]
[tex]f(4) = \log_{2}[3\cdot (4) - 6][/tex]
[tex]f(4) \approx 2.585[/tex]
[tex]r = \frac{2.585-1.585}{4-3}[/tex]
[tex]r = 1[/tex]
[tex][4,5][/tex]
[tex]a = 4[/tex] and [tex]b = 5[/tex]
[tex]f(4) = \log_{2}[3\cdot (4) - 6][/tex]
[tex]f(4) \approx 2.585[/tex]
[tex]f(5) = \log_{2}[3\cdot (5) - 6][/tex]
[tex]f(5) \approx 3.170[/tex]
[tex]r = \frac{3.170-2.585}{5-4}[/tex]
[tex]r = 0.59[/tex]
[tex][5,6][/tex]
[tex]a = 5[/tex] and [tex]b = 6[/tex]
[tex]f(5) = \log_{2}[3\cdot (5) - 6][/tex]
[tex]f(5) \approx 3.170[/tex]
[tex]f(6) = \log_{2}[3\cdot (6) - 6][/tex]
[tex]f(6) \approx 3.585[/tex]
[tex]r = \frac{3.585-3.170}{6-5}[/tex]
[tex]r = 0.42[/tex]
