For the given functions f and g, find: $g \circ g$, $f(x)=x^{2} ; \quad g(x)=x^{2}+4$

Carlo said a population can have more than one sample associated with it. Do you agree or disagree with his statement? Justify your answer.

Show that the equation $\sin \theta \frac{d^{2} y}{d \theta^{2}}+\cos \theta \frac{d y}{d \theta}+n(n+1)(\sin \theta) y=0$ can be transformed into Legendre's equation by means of the substitution $x=\cos \theta.$

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph.

$$r=6$$

Use the value of the first integral I to evaluate the two given integrals.

$$I=\int_{0}^{\pi / 2}(\cos \theta-2 \sin \theta) d \theta=-1$$

$$(a) \int_{0}^{\pi / 2}(2 \sin \theta-\cos \theta) d \theta \quad \text { b. } \int_{\pi/ 2}^{0}(4 \cos \theta-8 \sin \theta) d \theta$$

Use the information given about the angle $\theta, 0 \leq \theta<2 \pi,$ to find the exact value of $(a) \sin (2 \theta)\quad (b) \cos (2 \theta)\quad (c) \sin \frac{\theta}{2}\quad (d) \cos \frac{\theta}{2}.$ $\cot \theta=-2, \quad \sec \theta<0$

Use the information given about the angle $\theta, 0 \leq \theta<2 \pi,$ to find the exact value of $(a) \sin (2 \theta)\quad (b) \cos (2 \theta)\quad (c) \sin \frac{\theta}{2}\quad (d) \cos \frac{\theta}{2}.$ $\sec \theta=3, \quad \sin \theta>0$

Let G be a group acting on a set A and fix some $a \in A$. Show that the following sets are subgroups of G: (a) the kernel of the action, (b) $\{g \in G | g a=a\}-$ this subgroup is called the stabilizer of a in G.

Find the eigenvalues and associated eigenvectors for A.

$$\mathrm{A}=\left[\begin{array}{rrr} 0 & -1 & 0 \\ 0 & 1 & 0 \\ -1 & -1 & 1 \end{array}\right]$$

Find the exact values of $\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2}$, and $\cos \frac{\theta}{2}$ for the of the following.

$$\sin \theta=-\frac{2}{3}, 180^{\circ}<\theta<270^{\circ}$$

Find all solutions of the equation. Check your solutions in the original equation.

$$\sqrt{5 x-26}+4=x$$

An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; and during the fourth second, it falls 112 feet. Assume this pattern continues. How many feet will the object fall in 8 seconds?

Show that if f, g, and h are differentiable, then [f(g(h(x)))]' = f'(g(h(x)))g'(h(x))h'(x)

Write the system of equations associated with each augmented matrix.

$$\left[\begin{array}{ll|l} 1 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right]$$