Question

describe and correct the error a student made in finding the average rate of change for $f(x) = 0.5x^2$ over the interval $-4 \leq x \leq -2$
find the slope of the
line that passes through
$(-4, -8)$ and $(-2, -2)$.
$\frac{-2 - (-8)}{-2 - (-4)} = \frac{6}{2} = 3$

a. the points were correct, but the student made an error in calculating the
denominator. the correct slope is
(type an integer or a simplified fraction.)
b. the points were correct and the student correctly simplified the numerator
and denominator, but they did not simplify the fraction correctly. the correctly
simplified fraction is
(type an integer or a simplified fraction.)
c. the points used to find the slope were incorrect. the correct points are
$(-4,8),(-2,2)$
(type an ordered pair. use a comma to separate answers as needed.)
d. the points were correct, but the student made an error in calculating the numerator
the correct slope is
(type an integer or a simplified fraction.)
correct the students error
the correct average rate of change is \square

Explanation:

Step1: Find correct function values

For \( f(x) = 0.5x^2 \), at \( x = -4 \):
\( f(-4) = 0.5\times(-4)^2 = 0.5\times16 = 8 \)
At \( x = -2 \):
\( f(-2) = 0.5\times(-2)^2 = 0.5\times4 = 2 \)

Step2: Calculate average rate of change

Average rate of change formula: \( \frac{f(b) - f(a)}{b - a} \) for interval \( [a, b] \).
Here, \( a = -4 \), \( b = -2 \), \( f(a) = 8 \), \( f(b) = 2 \).
So, \( \frac{2 - 8}{-2 - (-4)} = \frac{-6}{2} = -3 \).

Answer:

First, the student used incorrect \( y \)-values (points \((-4, -8)\) and \((-2, -2)\) are wrong; correct points are \((-4, 8)\) and \((-2, 2)\)). Then, using the correct points, the average rate of change is calculated as \( \frac{2 - 8}{-2 - (-4)} = \frac{-6}{2} = -3 \).

The correct average rate of change is \(\boxed{-3}\).