1. Temperature

The temperature on New Year's Day in Hinterland was given by T(h) = -a - bcos(PiH/12), where T is the temperature in degreen Fahrenheit and h is the number of hours from midnight from 0 to 24.

The initial temperature at midnight T(0) was -15F and at noon, T(12) was 5F. Find a and b.

-What do I do?

Find the average temperature for the first 10 hours.
-DO I just use the Average value theorem?

Use the trapezoid Rule with 4 subdivisions to estimate integral from 0 to 8 of T(h)dh.

Find an expression for the rate that the temperature is changing with respect to h

2. The initial temperature at midnight T(0) was -15F and at noon, T(12) was 5F. Find a and b.
$\displaystyle T(h) = -a - b\cos\left(\frac{\pi h}{12}\right)$

$\displaystyle T(0) = -15$

$\displaystyle -15 = -a - b\cos(0)$

$\displaystyle -15 = -a - b$

$\displaystyle T(12) = 5$

$\displaystyle 5 = -a - b\cos(\pi)$

$\displaystyle 5 = -a + b$

$\displaystyle a = 5$ ... $\displaystyle b = 10$

Find the average temperature for the first 10 hours.
$\displaystyle T_{avg} = \frac{1}{10-0} \int_0^{10} T(h) \, dh$

Use the trapezoid Rule with 4 subdivisions to estimate integral from 0 to 8 of T(h)dh.
$\displaystyle \int_0^8 T(h) \, dh \approx 1[T(0) + 2T(2) + 2T(4) + 2T(6) + T(8)]$

Find an expression for the rate that the temperature is changing with respect to h
$\displaystyle \frac{d}{dh} \left[-5 -10\cos\left(\frac{\pi h}{12}\right)\right]$