Answer :
Question:
Find the average rate of change of f(x) = 3(2)^x between x = 1 and x = 4
Answer:
The average rate of change is 14
Solution:
The rate of change is given by formula:
[tex]\text { Average Rate }=\frac{f(b)-f(a)}{b-a}[/tex]
Here the given interval is x = 1 and x = 4
Therefore, the formula becomes,
[tex]\text { Average Rate }=\frac{f(4)-f(1)}{4-1}\\\\\text { Average Rate }=\frac{f(4)-f(1)}{3}[/tex]
Let us find f(4) and f(1)
Given function is:
[tex]f(x)=3(2)^x[/tex]
Substitute x = 4 in function
[tex]f(4)=3(2)^4\\\\f(4) = 3 \times 16 = 48[/tex]
Thus f(4) = 48
Substitute x = 1 in function
[tex]f(1) = 3(2)^1\\\\f(1) = 6[/tex]
Now substitute the values back into formula
[tex]\text { Average Rate }=\frac{f(4)-f(1)}{3}\\\\\text { Average Rate }= \frac{48-6}{3}\\\\\text { Average Rate }= 14[/tex]
Thus average rate of change is 14