A particle of mass $m_{1}$ with an initial kinetic energy ${T}_{1}$ makes an elastic collision with a particle of mass $m_{2}$ initially at rest. $m_{1}$ is deflected from its original direction with a kinetic energy $T_{1}^{\prime}$ through an angle ${\phi}_{1}$. Letting $\alpha=m_{2} / m_{1} \text { and } \gamma=\cos \phi_{1}$, show that the fractional kinetic energy lost by $m_{1}, \Delta T_{1} / T_{1}=\left(T_{1}-T_{1}^{\prime}\right) / T_{1}$, is given by $\frac{\Delta T_{1}}{T_{1}}=\frac{2}{1+\alpha}-\frac{2 \gamma}{(1+\alpha)^{2}}(\gamma+\sqrt{\alpha^{2}+\gamma^{2}-1})$

(a) Suppose that $\frac{f(x)}{x}$ is integrable on every interval [a, b] for 0 < a < b, and that $\lim _{x \rightarrow 0} f(x)=A \text { and } \lim _{x \rightarrow \infty} f(x)=B$. Prove that for all $\alpha, \beta>0$ we have $\int_{0}^{\infty} \frac{f(a x)-f(\beta x)}{x} d x=(A-B) \log \frac{\beta}{\alpha}$. (b) Now suppose instead that $\int_{a}^{\infty} \frac{f(x)}{x} d x$ converges for all a > 0 and that $\lim _{x \rightarrow 0} f(x)=A$. Prove that $\int_{0}^{\infty} \frac{f(\alpha x)-f(\beta x)}{x} d x=A \log \frac{\beta}{\alpha}$. (c) Compute the following integrals: (i) $\int_{0}^{\infty} \frac{e^{-\alpha x}-e^{-\beta x}}{x} d x$. (ii) $\int_{0}^{\infty} \frac{\cos (\alpha \cdot x)-\cos (\beta x)}{x} d x$.

Underline the verb in parentheses that best completes the sentence. The rhythms of the percussion section (was, were) the highlight of the concert.

Find the sum of each infinite geometric series. $1-\frac{1}{2}+\frac{1}{4}-\ldots$

Complete each sentence by writing the correct present-tense form of the verb indicated. Most of the neighbors _____ spending time together. (enjoy)

Check the ISBN-13 codes of a selection of recently published books to see whether the check digit was computed correctly.

Show that every odd composite integer is a pseudoprime to both the base 1 and the base -1.

Solve each equation. Check your solution. $\frac{x}{x+1}-\frac{x}{x-3}=9$

Show that the Fourier series for a periodic square wave is $f(t)=\frac{4}{\pi}\left[\sin (\omega t)+\frac{1}{3} \sin (3 \omega t)+\frac{1}{5} \sin (5 \omega t)+\cdots\right]$ where f(t) = +1 $\text { for } 0<\omega t<\pi, 2 \pi<\omega t<3 \pi, \text { and so on }$ f(t) = -1 $\text { for } \pi<\omega t<2 \pi, 3 \pi<\omega t<4 \pi, \text { and so on }$

Show that if f and g are completely multiplicative functions, then fg is also completely multiplicative.

Identify the part of speech (POS) and the use in the sentence (USE) of the italicized word or words. Be very specific. $\mathit{To \ improve \ my \ typing \ skills}$ is one of my goals for this vacation. POS: _______ USE: _______

Factor the perfect square trinomial. $x^{2}-4 x+4$

Draw the graph of $f(x)=a x^{2}+b x+c$.

Find $f^{\prime}$ in terms of $g^{\prime}$ if (i) $f(x)=g(x+g(a))$. (ii) $f(x)=g(x \cdot g(a))$. (iii) $f(x)=g(x+g(x))$. (iv) $f(x)=g(x)(x-a)$. (v) $f(x)=g(a)(x-a)$. (vi) $f(x+3)=g\left(x^{2}\right)$.