Question

find the average rate of change of the function on the interval specified for real number c in simplest form. enter the missing part of the answer.

$f(x)=2x^{2}-15$ on $-4,c$

$\frac{\delta y}{\delta x}=2c-$ _

answer

Explanation:

Step1: Find \( f(-4) \) and \( f(c) \)

For \( f(x)=2x^{2}-15 \), when \( x = - 4 \), \( f(-4)=2\times(-4)^{2}-15=2\times16 - 15=32 - 15 = 17 \).
When \( x = c \), \( f(c)=2c^{2}-15 \).

Step2: Use the average - rate - of - change formula

The average rate of change formula is \(\frac{\Delta y}{\Delta x}=\frac{f(c)-f(-4)}{c - (-4)}=\frac{(2c^{2}-15)-17}{c + 4}\).
Simplify the numerator: \((2c^{2}-15)-17=2c^{2}-32\).
So \(\frac{\Delta y}{\Delta x}=\frac{2c^{2}-32}{c + 4}\).
Factor the numerator: \(2c^{2}-32 = 2(c^{2}-16)=2(c + 4)(c - 4)\).
Then \(\frac{\Delta y}{\Delta x}=\frac{2(c + 4)(c - 4)}{c + 4}\).
Cancel out the common factor \(c + 4\) (assuming \(c
eq - 4\)), we get \(\frac{\Delta y}{\Delta x}=2(c - 4)=2c-8\).

Answer:

\(8\)