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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 $$\left[ {0,4\pi } \right]$$