Question

given the function ( f(x)=x^{2}+3x - 1 ), determine the average rate of change of the function over the interval ( -3leq xleq4 ).

Explanation:

Step1: Recall the formula for average rate of change

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 \(a=-3\) and \(b = 4\).

Step2: Calculate \(f(-3)\)

Substitute \(x=-3\) into \(f(x)=x^{2}+3x - 1\):
\(f(-3)=(-3)^{2}+3\times(-3)-1=9-9 - 1=-1\)

Step3: Calculate \(f(4)\)

Substitute \(x = 4\) into \(f(x)=x^{2}+3x - 1\):
\(f(4)=4^{2}+3\times4-1=16 + 12-1=27\)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), with \(a=-3\), \(b = 4\), \(f(a)=-1\), \(f(b)=27\)
\(\frac{f(4)-f(-3)}{4-(-3)}=\frac{27-(-1)}{4 + 3}=\frac{27 + 1}{7}=\frac{28}{7}=4\)

Answer:

\(4\)