for ( f(x)=4 x^{2}-5 x + 1 ), what is the average rate of change over (0,3)?
for ( f(x)=4 x^{2}-5 x + 1 ), what is the average rate of change over (0,3)?
Answer
Explanation:
Step1: Recall the formula for average rate of change
The formula for 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 = 0), (b=3), and (f(x)=4x^{2}-5x + 1).
Step2: Calculate (f(0)) and (f(3))
- For (x = 0): Substitute (x = 0) into (f(x)): (f(0)=4\times0^{2}-5\times0 + 1=1).
- For (x = 3): Substitute (x = 3) into (f(x)): (f(3)=4\times3^{2}-5\times3 + 1=4\times9-15 + 1=36-15 + 1=22).
Step3: Apply the average - rate - of - change formula
Substitute (f(0)=1), (f(3)=22), (a = 0), and (b = 3) into (\frac{f(b)-f(a)}{b - a}). We get (\frac{f(3)-f(0)}{3-0}=\frac{22 - 1}{3}=\frac{21}{3}=7).
Answer:
d. 7