Question

1. find the average rate of change of $f(x) = 4x^2 - 5x$ over the interval $-2, 0$

working:
answer: __________________

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = -2 \) and \( b = 0 \), and \( f(x)=4x^{2}-5x \).

Step2: Calculate \( f(-2) \)

Substitute \( x = -2 \) into \( f(x) \):
\[

$$\begin{align*} f(-2)&=4(-2)^{2}-5(-2)\\ &=4\times4 + 10\\ &=16 + 10\\ &=26 \end{align*}$$

\]

Step3: Calculate \( f(0) \)

Substitute \( x = 0 \) into \( f(x) \):
\[

$$\begin{align*} f(0)&=4(0)^{2}-5(0)\\ &=0 - 0\\ &=0 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\) with \( a=-2 \), \( b = 0 \), \( f(-2)=26 \) and \( f(0)=0 \):
\[

$$\begin{align*} \text{Average rate of change}&=\frac{f(0)-f(-2)}{0-(-2)}\\ &=\frac{0 - 26}{0 + 2}\\ &=\frac{-26}{2}\\ &=-13 \end{align*}$$

\]

Answer:

\(-13\)