Question

given the function ( h(x) = x^{2}+5x - 1 ), determine the average rate of change of the function over the interval ( -6leq xleq0 ).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = h(x)\) over the interval \([a,b]\) is \(\frac{h(b)-h(a)}{b - a}\). Here, \(a=-6\) and \(b = 0\).

Step2: Calculate \(h(-6)\)

Substitute \(x=-6\) into \(h(x)=x^{2}+5x - 1\):
\(h(-6)=(-6)^{2}+5\times(-6)-1=36-30 - 1=5\)

Step3: Calculate \(h(0)\)

Substitute \(x = 0\) into \(h(x)=x^{2}+5x - 1\):
\(h(0)=0^{2}+5\times0-1=-1\)

Step4: Compute the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\), with \(a=-6\), \(b = 0\), \(h(-6) = 5\) and \(h(0)=-1\):
\(\frac{h(0)-h(-6)}{0-(-6)}=\frac{-1 - 5}{6}=\frac{-6}{6}=-1\)

Answer:

\(-1\)