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Section 6.1 : Average Function Value

For problems 1 – 4 determine \({f_{{\rm{avg}}}}\) for the function on the given interval.

1. \(f\left( x \right) = 8{x^4} - 7{x^3} + 2\) on \(\left[ { - 2,1} \right]\)
2. \(f\left( x \right) = \left( {4 - x} \right){{\bf{e}}^{{x^{\,2}} - 8x}}\) on \(\left[ {1,4} \right]\)
3. \(f\left( x \right) = 6x - \frac{{4x}}{{{x^2} + 1}}\) on \(\left[ { - 3,0} \right]\)
4. \(f\left( x \right) = \cos \left( {3x} \right){\left[ {2 + \sin \left( {3x} \right)} \right]^4}\) on \(\left[ {0,\frac{\pi }{6}} \right]\)

For problems 5 – 8 find \({f_{{\rm{avg}}}}\) for the function on the given interval and determine the value of c in the given interval for which \(f\left( c \right) = {f_{{\rm{avg}}}}\).

1. \(f\left( x \right) = 10 - 4x - 6{x^2}\) on \(\left[ {2,6} \right]\)
2. \(f\left( x \right) = 7{x^2} + 2x - 3\) on \(\left[ { - 1,1} \right]\)
3. \(f\left( x \right) = 9 - 2{{\bf{e}}^{4x + 1}}\) on \(\left[ { - 1,2} \right]\)
4. \(f\left( x \right) = 8 - \cos \left( {\frac{x}{4}} \right)\) on \(\left[ {0,4\pi } \right]\)