Question

a function is defined as shown.
$f(x)=-4x^{2}+5x - 2$
what is the average rate of change of $f(x)$ over the interval $1\leq x\leq4$?
a. $-15$
b. $15$
c. $45$
d. $-45$

Explanation:

Step1: Recall the formula for average rate of change

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 \(a = 1\) and \(b=4\).

Step2: Calculate \(f(1)\)

Substitute \(x = 1\) into \(f(x)=-4x^{2}+5x - 2\).
\(f(1)=-4(1)^{2}+5(1)-2=-4 + 5-2=-1\).

Step3: Calculate \(f(4)\)

Substitute \(x = 4\) into \(f(x)=-4x^{2}+5x - 2\).
\(f(4)=-4(4)^{2}+5(4)-2=-4\times16 + 20-2=-64+20 - 2=-46\).

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), with \(a = 1\), \(b = 4\), \(f(1)=-1\), \(f(4)=-46\).
\(\frac{f(4)-f(1)}{4 - 1}=\frac{-46-(-1)}{3}=\frac{-46 + 1}{3}=\frac{-45}{3}=-15\).

Answer:

A. -15