Answer :
Sure! Let's find the average rate of change of the function [tex]\( f(x) = -5x^2 - 3x - 4 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex].
### Step-by-step Solution:
1. Evaluate the function at [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -5(2)^2 - 3(2) - 4 \][/tex]
[tex]\[ f(2) = -5(4) - 6 - 4 \][/tex]
[tex]\[ f(2) = -20 - 6 - 4 \][/tex]
[tex]\[ f(2) = -30 \][/tex]
2. Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = -5(4)^2 - 3(4) - 4 \][/tex]
[tex]\[ f(4) = -5(16) - 12 - 4 \][/tex]
[tex]\[ f(4) = -80 - 12 - 4 \][/tex]
[tex]\[ f(4) = -96 \][/tex]
3. Calculate the average rate of change:
The average rate of change of the function [tex]\( f(x) \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} \][/tex]
Substituting the values we found:
[tex]\[ \text{Average Rate of Change} = \frac{-96 - (-30)}{4 - 2} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{-96 + 30}{2} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{-66}{2} \][/tex]
[tex]\[ \text{Average Rate of Change} = -33.0 \][/tex]
Thus, the average rate of change of the function [tex]\( f(x) = -5x^2 - 3x - 4 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex] is [tex]\( -33.0 \)[/tex].
### Step-by-step Solution:
1. Evaluate the function at [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -5(2)^2 - 3(2) - 4 \][/tex]
[tex]\[ f(2) = -5(4) - 6 - 4 \][/tex]
[tex]\[ f(2) = -20 - 6 - 4 \][/tex]
[tex]\[ f(2) = -30 \][/tex]
2. Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = -5(4)^2 - 3(4) - 4 \][/tex]
[tex]\[ f(4) = -5(16) - 12 - 4 \][/tex]
[tex]\[ f(4) = -80 - 12 - 4 \][/tex]
[tex]\[ f(4) = -96 \][/tex]
3. Calculate the average rate of change:
The average rate of change of the function [tex]\( f(x) \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} \][/tex]
Substituting the values we found:
[tex]\[ \text{Average Rate of Change} = \frac{-96 - (-30)}{4 - 2} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{-96 + 30}{2} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{-66}{2} \][/tex]
[tex]\[ \text{Average Rate of Change} = -33.0 \][/tex]
Thus, the average rate of change of the function [tex]\( f(x) = -5x^2 - 3x - 4 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex] is [tex]\( -33.0 \)[/tex].
