# Trig Identity Homework due 5-7-14 (MRS22-3) or 5-8-14 (MRS22-1)

Prove each identity.

1. $\frac{\cot\theta}{\csc\theta}=\cos\theta$

2. $\cos^{2}x\tan^{2}x=\sin^{2}x$

3. $\frac{1-\sin^{2}\beta}{\cos\beta}=\cos\beta$

4. $\frac{\tan^{2}\alpha+1}{\sec\alpha}=\sec\alpha$

5. $1-\csc{x}\sin^{3}x=\cos^{2}x$

6. $\cos^{2}\theta(\tan^{2}\theta+1)=1$

7. $\sin^{2}\beta(1+\cot^{2}\beta)=1$

8. $\cot\theta+\tan\theta=\csc\theta\sec\theta$

9. $\sin^{2}\alpha+\tan^{2}\alpha+\cos^{2}\alpha=\sec^2\alpha$

10. $1+\csc^{2}x\cos^{2}x=\csc^{2}x$

# Calculus Homework Due 4-24-14

1. Find the exact global minimum value of $f(x)=x+\frac{1}{x}$ on the domain $\{x|x>0\}$

2. Find the exact global minimum and maximum values of $g(t)=te^{-t}$ on the domain $\{t|t\geq0\}$

3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant $k$, its pressure, $P$, and volume, $V$, satisfy the equation $PV^{1.4}=k$. (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?

4. (a) A hemispherical bowl of radius 10cm contains water to a depth of $h$cm. Find the radius of the surface of the water as a function of $h$. (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?