Determine the average rate of change of the function between \(x = -1\) and \(x = 1\) where the function is defined by:
\[ f(x) = x^2-x^4+\frac{{1}}{{2}}x \]Solution
Net Change: \[ \begin{{array}}{{rcl}} f(1)-f(-1) &=& 1^2-1^4+\frac{{1}}{{2}}(1)-\left((-1)^2-(-1)^4+\frac{{1}}{{2}}(-1)\right)\\ f(1)-f(-1) &=& 1-1+\frac{{1}}{{2}}-\left(1-1-\frac{{1}}{{2}}\right)\\ f(1)-f(-1) &=&\frac{{1}}{{2}}+\frac{{1}}{{2}}\\ f(1)-f(-1) &=&1 \end{{array}}\]
Horizontal Change: \(1-(-1))=2\)
Average Rate of Change between \(x=-1\) and \(x=1\): \(\frac{{1}}{{2}}\)
We can check our work by graphing the function along with the line that has a slope of 1/2 and passes through one of the points. If that line also passes through the other point, then we have the right value for the average rate of change. Since \(f(1) = \frac{{1}}{{2}}\), we test to see if the line \(y=\frac{{1}}{{2}}(x-1)+\frac{{1}}{{2}}\) passes through \(f(-1) = -\frac{{1}}{{2}}\).