Calculate the average rate of change for the function $f ( x )=- x ^4+4 x ^3-2 x ^2+ x +1$, from $x =0$ to $x =1$.
To find the average rate of change of the function \(f(x) = -x^4 + 4x^3 - 2x^2 + x + 1\) on the interval \([0, 1]\), we use the formula for the average rate of change over an interval \([x_1, x_2]\), which is given by:
\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
Here, the starting point is \(x_1 = 0\) and the ending point is \(x_2 = 1\).
First, we evaluate the function at \(x = 0\):
\[f(0) = -(0)^4 + 4(0)^3 - 2(0)^2 + (0) + 1 = 1\]
Next, we evaluate the function at \(x = 1\):
\[f(1) = -(1)^4 + 4(1)^3 - 2(1)^2 + (1) + 1 = -1 + 4 - 2 + 1 + 1 = 3\]
Now, substitute these values into the average rate of change formula:
\[\text{Average Rate of Change} = \frac{f(1) - f(0)}{1 - 0} = \frac{3 - 1}{1} = \frac{2}{1} = 2\]
Thus, the average rate of change for the function on the interval \([0, 1]\) is \(2\). Comparing this to the given choices, option c is the correct one.
In fields like economics, calculating the average rate of change is essential for understanding trends. For example, if a company's profit over a set period is represented by a function of time, calculating the average rate of change tells the business owners how, on average, their profit per unit of time changed during that interval, helping them make informed strategic decisions about future investments.