what is the average rate of change for the function ( f(x) = -2x^{3}+x - 1 ) over the interval from ( x = 1…
what is the average rate of change for the function ( f(x) = -2x^{3}+x - 1 ) over the interval from ( x = 1 ) to ( x = 3 )? -27 -25 25 27
Answer
Explanation:
Step1: Recall the average rate of change formula
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 = 1), (b=3).
Step2: Calculate (f(1))
Substitute (x = 1) into (f(x)=-2x^{3}+x - 1). (f(1)=-2(1)^{3}+1 - 1=-2).
Step3: Calculate (f(3))
Substitute (x = 3) into (f(x)=-2x^{3}+x - 1). (f(3)=-2(3)^{3}+3 - 1=-2\times27 + 2=-54 + 2=-52).
Step4: Compute the average rate of change
Using the formula (\frac{f(3)-f(1)}{3 - 1}), substitute (f(1)=-2) and (f(3)=-52). (\frac{-52-(-2)}{3 - 1}=\frac{-52 + 2}{2}=\frac{-50}{2}，=-25，).