**Problem : **
Find the average value of *f* for *f* (*x*) = 3*x*^{2} - 6*x* on [- 1, 1].

f_{avg} | = f (x)dx | ||

= x^{3}-3x^{2}_{-1}^{1} | |||

= 1-3-(-1-3) | |||

= 1 |

**Problem : **
Find the average value of *f* for *f* (*x*) = *sin*(*x*) on [- *Π*, *Π*].

f_{avg} | = f (x)dx | ||

= -cos(x)_{-Π}^{Π} | |||

= --1-(-(-1)) | |||

= 0 |

**Problem : **
For the function above, find the *c* on [- *Π*, *Π*] such that *f* (*c*) = *f*_{avg}

**Problem : **
Use the second fundamental theorem of calculus to find *F'*.

F(x) = (9t+9)dt |

*F'*(*x*) = 9*x* + 9

**Problem : **
Now integrate the above expression for *F* and then take the derivative to find *F'*(*x*) and verify the result from
the previous question.

F(x) = (9t+9)dt |

9t+9dt = t^{2}+9t_{2}^{x} | |||||

x^{2}+9x - (18 + 18) | |||||

F(x) = x^{2} +9x - 36 | |||||

F'(x) = 9x + 9 |

This verifies the result given by the second fundamental theorem.

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