Calculate the average rate of change for the function $f(x) = -x^4 + 4x^3 - 2x^2 + x + 1$, from $x = 0$ to $x = 1$.
The objective is to find the average rate of change of the function \(f(x) = -x^4 + 4x^3 - 2x^2 + x + 1\) on the interval \([0, 1]\). The average rate of change of a function \(f(x)\) over an interval \([a, b]\) is calculated using the formula:
\[\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}\]
Here, the interval is \([0, 1]\), so \(a = 0\) and \(b = 1\). We need to calculate \(f(0)\) and \(f(1)\) to apply this formula.
First, we evaluate the function at the start of the interval, \(x = 0\):
\[f(0) = -(0)^4 + 4(0)^3 - 2(0)^2 + (0) + 1 = 1\]
Next, we evaluate the function at the end of the interval, \(x = 1\):
\[f(1) = -(1)^4 + 4(1)^3 - 2(1)^2 + (1) + 1 = -1 + 4 - 2 + 1 + 1 = 3\]
Thus, we have \(f(0) = 1\) and \(f(1) = 3\).
Now, we substitute these values into the average rate of change formula:
\[\text{Average Rate of Change} = \frac{f(1) - f(0)}{1 - 0} = \frac{3 - 1}{1} = 2\]
Comparing this result to the given options:
The calculated value of 2 matches option (c).
In engineering, the average rate of change is used to determine the average velocity of a vehicle over a specific time period. If the position of a car is modeled by a function \(s(t)\), calculating the average rate of change between two times \(t_1\) and \(t_2\) gives the car's average speed. For example, if a car travels according to a complex polynomial position function, knowing the average speed helps in optimizing travel routes and fuel efficiency.