Question

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

Answer 1

The average rate of change is given by the following formula:
\[\frac{\int\limits_a^b f(x) \ dx}{b-a}\]
To find the average rate of change of your function, simply integrate, then divide by the interval that you are integrating over (b-a, which, in this case, is 1-0 = 1).
\[\frac{\int\limits_a^b -x^4 + 4x^3 -2x^2 + x + 1 \ dx}{b-a}\]
Remember that (simple) integration of exponents follows this rule:
\[\int\limits_a^b x^n \ dx = \frac{x^{n+1}}{n+1}\]
Then, following this rule, you should be able to integrate the second equation above to solve for the rate of change of f(x).

Answer 2

You don't have to do any integrals here...the average rate of change for \(f(x)\) from \(x=a\) to \(x=b\) is simply
\[\frac{f(b)-f(a)}{b-a}\]

Answer 3

@DisplayError I think you are thinking of the average value of the function over that interval, but the problem asks for the average rate of change, which is simply the slope of a line that connects the two endpoints \((a,f(a))\) and \((b,f(b))\)

Answer 4

@whpalmer4 Oh, I see what you mean. Silly me for mixing up average rate of change with average value.