The average rate of change is 2
Solution:
Given that we have to find the rate of change
The average rate of change is given as:
[tex]Rate\ of\ change = \frac{f(b)-f(a)}{b-a}[/tex]
Given function is:
[tex]h(x) = \frac{1}{2}x^3 -x^2[/tex]
Given interval is:
[tex]-2\leq x\leq 2[/tex]
Therefore,
a = -2 and b = 2
Thus formula becomes:
[tex]Rate\ of\ change = \frac{h(2)-h(-2)}{2-(-2)}\\\\Rate\ of\ change = \frac{h(2)-h(-2)}{4} ----- eqn 1[/tex]
Find h(2)
Substitute x = 2 in given function
[tex]h(2) = \frac{1}{2} \times 2^3 - 2^2\\\\h(2) = 2^2-2^2 = 0[/tex]
Find h(-2)
Substitute x = -2 in given function
[tex]h(-2) = \frac{1}{2} \times (-2)^3 -(-2)^2\\\\h(-2) = -4 -4 = -8[/tex]
Substitute the values in eqn 1
[tex]Rate\ of\ change = \frac{0-(-8)}{4}\\\\Rate\ of\ change =\frac{8}{4} = 2[/tex]
Thus average rate of change is 2