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By definition of average acceleration, the ave. acc. for this particle between 1.0 s and 2.0 s is
[tex]a_{\mathrm{ave}[1.0\,\mathrm s,2.0\,\mathrm s]} = \dfrac{v(2.0\,\mathrm s) - v(1.0\,\mathrm s)}{2.0\,\mathrm s - 1.0\,\mathrm s}[/tex]
At the endpoints of the the interval, we have
[tex]v(2.0\,\mathrm s) = 2.3\dfrac{\rm m}{\rm s}+\left(4.1\dfrac{\rm m}{\mathrm s^2}\right)(2.0\,\mathrm s)-\left(6.2\dfrac{\rm m}{\mathrm s^3}\right)(2.0\,\mathrm s)^2 = -14.3\dfrac{\rm m}{\rm s}[/tex]
[tex]v(1.0\,\mathrm s) = 2.3\dfrac{\rm m}{\rm s}+\left(4.1\dfrac{\rm m}{\mathrm s^2}\right)(1.0\,\mathrm s)-\left(6.2\dfrac{\rm m}{\mathrm s^3}\right)(1.0\,\mathrm s)^2 = 0.2\dfrac{\rm m}{\rm s}[/tex]
Then the average acceleration is
[tex]a_{\mathrm{ave}[1.0\,\mathrm s,2.0\,\mathrm s]} = \dfrac{-14.3\frac{\rm m}{\rm s}-0.2\frac{\rm m}{\rm s}}{1.0\,\mathrm s} = \boxed{-14.5\dfrac{\rm m}{\mathrm s^2}}[/tex]