Answer:
[tex]\dfrac{26}{3}[/tex]
Step-by-step explanation:
The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:
[tex]\dfrac{f(b)-f(a)}{b-a}[/tex]
Given function: [tex]f(x)=3^{x-1}+2[/tex]
Given interval: 1 ≤ x ≤ 4
Therefore, a = 1 and b = 4
Therefore, find the value of the given function when x = 1 and x = 4:
[tex]\begin{aligned}\implies f(1) & =3^{1-1}+2\\ & = 3^0+2\\ & = 1+2\\ & = 3 \end{aligned}[/tex]
[tex]\begin{aligned} \implies f(4) & = 3^{4-1}+2 \\ & = 3^3+2\\ & = 27+2\\ & = 29 \end{aligned}[/tex]
Substitute the found values into the formula:
[tex]\begin{aligned}\implies \dfrac{f(b)-f(a)}{b-a} & =\dfrac{f(4)-f(1)}{4-1}\\\\ & =\dfrac{29-3}{4-1}\\\\ & = \dfrac{26}{3} \end{aligned}[/tex]
