find the average rate of change of the function ( f(x)=x^{2}+3x ) from ( x_{1}=1 ) to ( x_{2}=2 ).\nthe…

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function (y = f(x)) from (x_1) to (x_2) is (\frac{f(x_2)-f(x_1)}{x_2 - x_1}).

Step2: Calculate (f(x_1)) and (f(x_2))

Given (f(x)=x^{2}+3x), when (x_1 = 1), (f(1)=1^{2}+3\times1=1 + 3=4). When (x_2=2), (f(2)=2^{2}+3\times2=4 + 6=10).

Step3: Substitute into the formula

Substitute (f(x_1) = 4), (f(x_2)=10), (x_1 = 1), (x_2 = 2) into (\frac{f(x_2)-f(x_1)}{x_2 - x_1}). We get (\frac{10 - 4}{2-1}=\frac{6}{1}=6).

Answer:

(6)