To find the average rate of change of the function [tex]\( f(x) = 3x^4 - x^3 + 3x^2 + x - 3 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex], follow these steps:
1. Evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3(0)^4 - (0)^3 + 3(0)^2 + 0 - 3 = -3
\][/tex]
2. Evaluate the function at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3(1)^4 - (1)^3 + 3(1)^2 + 1 - 3 = 3 - 1 + 3 + 1 - 3 = 3
\][/tex]
3. Calculate the average rate of change:
The average rate of change of a function from [tex]\( x = a \)[/tex] to [tex]\( x = b \)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
In this case, [tex]\( a = 0 \)[/tex] and [tex]\( b = 1 \)[/tex]:
[tex]\[
\frac{f(1) - f(0)}{1 - 0} = \frac{3 - (-3)}{1 - 0} = \frac{3 + 3}{1} = 6
\][/tex]
Hence, the average rate of change of the function [tex]\( f(x) \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex] is [tex]\( 6 \)[/tex].
