$\"\"$

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The normal monthly precipitation model is $\"\"$ ,

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where $\"\"$ is measured in inches and $\"\"$ is the time in months.

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$\"\"$ corresponding to January $\"\"$ .

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(a) Determine the extreme of the function over a one-year period.

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Equating first derivative to zero.

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$\"\"$

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$\"\"$

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$\"\"$

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The solutions of $\"\"$ are $\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The value of $\"\"$ is out of the interval $\"\"$ .

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The other values are also out of the interval $\"\"$ .

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The values of $\"\"$ are $\"\"$ and $\"\"$ .

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Find the minimum and maximum values.

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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The relative maximum value of $\"\"$ is at $\"\"$ .

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The relative maximum at February.

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The relative maximum value is $\"\"$ .

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The minimum value of $\"\"$ at $\"\"$ .

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The relative minimum at July..

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The minimum value is $\"\"$ .

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$\"\"$

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(b) Use integration to approximate the normal annual precipitation.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ inches.

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$\"\"$

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(c) Approximate the average monthly precipitation during the months of

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October, November and December.

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The limits of the integration are $\"\"$ to $\"\"$ .

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The average value of $\"\"$ on the interval $\"\"$ is $\"\"$ .

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The average value is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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The average monthly precipitation during the months of october, november

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and december is $\"\"$ inches.

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$\"\"$

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(a) The relative maximum at February.

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The relative maximum value is $\"\"$ .

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The relative minimum at July..

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The minimum value is $\"\"$ .

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(b) $\"\"$ inches.

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(c) The average monthly precipitation during the months of october, november

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and december is $\"\"$ inches.