Assume $A>0$. If $A \sin (\omega t+\phi)=C_1 \cos \omega t+$ $C_2 \sin \omega t$, show that we must have

$$A=\sqrt{C_1^2+C_2^2} \text { and } \tan \phi=C_1 / C_2 \text {. }$$

(a) If $\phi(n) | n-1,$ prove that $n$ is a square-free integer. [Hint: Assume that $n$ has the prime factorization $n=p_{1}^{k_{1}} p_{2}^{k_{2}} \ldots p_{r}^{k_{r}},$ where $k_{1} \geq 2$ Then $p_{1} | \phi(n),$ whence $p_{1} | n-1,$ which leads to a contradiction.] (b) Show that if $n=2^{k}$ or $2^{k} 3^{j},$ with $k$ and $j$ positive integers, then $\phi(n) | n$ .

Identify the surfaces whose equations are given.

(a) $\theta=\pi / 4$ (b) $\phi=\pi / 4$

Give an example to show that the assumption that $\phi$ is onto is needed in part (a) of Proposition $3.7.6$.

Given that $V=\rho^{2} z \cos \phi$, find $\nabla V \text { and } \nabla^{2} V$.

Describe in words the surface whose equation is given. $\phi=3 \pi / 4$

Show that $\varepsilon \text { and } d \Phi _ { \mathrm { m } } / d t$ have the same units.

Consider the initial value problem x$^t$=Ax+g(t), x(0)=x$^0$.

By referring to the problem in the said section, show that

$$\mathbf{x}=\Phi(t) \mathbf{x}^0+\int_0^t \Phi(t-s) \mathbf{g}(s) d s .$$

Evaluate $\int_C \boldsymbol{\Phi}(\mathbf{r}) \cdot d \mathbf{r}$, where $\boldsymbol{\Phi}(x, y, z)=\sin z \mathbf{i}+$ $\cos \sqrt{y} \mathbf{j}+x^3 \mathbf{k}$ and $C$ is the line segment from $(1,0,0)$ to $(0,0,3)$.

Let $m=a^{n}-1$ where a and n are positive integers. Show that $\text { ord }_{m} a=n$, and conclude that $n | \phi(m).$

Show that if n is a positive integer with $n \neq 2$ and $n \neq 6,$ then $\phi(n) \geq \sqrt{n}$

For the following exercise, points P(1.5, 0) and $Q(\phi, y)$ are on the graph of the function $f(\phi)=\cos (\pi \phi)$. Complete the following table with the appropriate values: y-coordinate of Q, the point Q(x, y), and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.

$$\begin{matrix} \text{x} & \text{y} & {Q(\phi, y)} & {m_{sec}}\\ \text{1.4} & \text{a.} & \text{e.} & \text{i.}\\ \text{1.49} & \text{b.} & \text{f.} & \text{j.}\\ \text{1.499} & \text{c.} & \text{g.} & \text{k.}\\ \text{1.4999} & \text{d.} & \text{h.} & \text{l.}\\ \end{matrix}$$

Find all of the homomorphisms $\phi: \mathbb{Z} \rightarrow \mathbb{Z}$. What is $A u t(\mathbb{Z})$?

Consider the Volterra integral equation (see the previous problem)

$$\phi(t)+\int_0^t(t-\xi) \phi(\xi) d \xi=\sin 2 t .$$

Solve the initial value problem in part (b) and verify that the solution is the same as the one in part (a).