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

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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 hcm. 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?