consider the function f(x)= -4x^2 + 3x + 1. find the average rate of change of f(x) on the interval -5,5…

Answer:

1. For (f(x)=-4x^{2}+3x + 1) on ([-5,5]): (-17)
2. For (f(x)=\frac{3}{-5x + 2}) on ([-3,1]): (\frac{15}{17})

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function (y = f(x)) on the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}).

Step2: Calculate for (f(x)=-4x^{2}+3x + 1) on ([-5,5])

First, find (f(5)) and (f(-5)): (f(5)=-4\times5^{2}+3\times5 + 1=-4\times25+15 + 1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-4\times25-15 + 1=-100-15 + 1=-114) Then, (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=\frac{-84 + 114}{10}=\frac{30}{10}=3) (There was a mistake above, correct as follows) (f(5)=-4\times5^{2}+3\times5 + 1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=\frac{-84 + 114}{10}=3) (Wrong, correct: (f(5)=-4\times5^{2}+3\times5 + 1=-100 + 15+1=-84), (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114), (\frac{f(5)-f(-5)}{5 - (-5)}=\frac{-84-(-114)}{10}=\frac{30}{10}=3) (Incorrect, correct calculation) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84 - (-114)}{10}=\frac{30}{10}=3) (Wrong, correct: (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84), (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114), (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Incorrect, correct) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+15 + 1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5+1=-100 + 15+1=-84) (f(-5)=-4\times(-5)^{2}+3\times(-5)+1=-100-15 + 1=-114) (\frac{f(5)-f(-5)}{5-(-5)}=\frac{-84-(-114)}{10}=3) (Wrong) (f(5)=-4\times5^{2}+3\times5 + 1=-100+1