a function is defined as shown.\n$f(x)=-4x^{2}+5x - 2$\nwhat is the average rate of change of $f(x)$ over…

a function is defined as shown.\n$f(x)=-4x^{2}+5x - 2$\nwhat is the average rate of change of $f(x)$ over the interval $1\\leq x\\leq4$?\na. $-15$\nb. $15$\nc. $45$\nd. $-45$

a function is defined as shown.\n$f(x)=-4x^{2}+5x - 2$\nwhat is the average rate of change of $f(x)$ over the interval $1\\leq x\\leq4$?\na. $-15$\nb. $15$\nc. $45$\nd. $-45$

Answer

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