Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. $y=x^{3}-x^{8}+1$ and y=1; about the y-axis

Solve each inequality, if possible. If the inequality has no solution, write no solution. If the solutions are all real numbers, write all real numbers. $2(t+2)-3 t \geq-1$

Give a $\text {big-} \Theta$ estimate of each function. Use the estimation targets when possible. $\text {(a)} 8 n \log _{2} n+(n-1)^{2}+2^{n}$ $\text {(b)}\left(1+n+n^{2}+n^{3}+n^{4}+n^{5}\right)\left(10+n^{2}\right)$

Find $f^{\prime}(0)$ if

$$f(x)=\left\{\begin{array}{ll}{\frac{g(x)}{x},} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.$$

and $g(0)=g^{\prime}(0)=0$ and $g^{\prime \prime}(0)=17$.

A clockwise rotation of $90^{\circ}$ is equivalent to a counterclockwise rotation of how many degrees?

Solve each equation. Check your answer. $1.2=2.4-0.6 x$

(a) Prove that $\pi$ is the maximum possible length of an interval on which sin is one-one, and that such an interval must be of the form $[2 k \pi-\pi / 2.2 k \pi+\pi / 2]$ or $[2 k \pi+\pi / 2,2(k+1) \pi-\pi / 2]$. (b) Suppose we let $g(x)=\sin x$ for x in $(2 k \pi-\pi / 2,2 k \pi+\pi / 2)$. What is $\left(g^{-1}\right)^{\prime}$?

The reading on a mercury manometer at $70\left(^{\circ} \mathrm{F}\right)$ (open to the atmosphere at one end) is 25.62(in). The local acceleration of gravity is $32.243(\mathrm{ft}) \cdot(\mathrm{s})^{-2}.$ Atmospheric pressure is 29.86(in Hg). What is the absolute pressure in (psia) being measured? The density of mercury at $70\left(^{\circ} \mathrm{F}\right)$ is $13.543 \mathrm{g} \cdot \mathrm{cm}^{-3}.$

Underline the word in parentheses that best completes each sentence. The warm weather will (affect, effect) the candy we are planning to make.

Write and draw an ad for a clothing store. Include the information in the box.

1. El nombre de la tienda
2. Prendas y colores
3. Productos en oferta
4. Modos de pago

Graph the direct variation that includes the given point. Write an equation of the line. (-2, 5)

A full journal bearing has a shaft diameter of 3.000 in with a unilateral tolerance of - 0.0004 in. The l/d ratio is unity. The bushing has a bore diameter of 3 .003 in with a unilateral tolerance of 0.0012 in. The SAE 40 oil supply is in an axial-groove sump with a steady-state temperature of $140^{\circ} \mathrm{F}$. The radial load is 675 lbf. Estimate the average film temperature, the minimum film thickness, the heat loss rate, and the lubricant side-flow rate for the minimum clearance assembly, if the journal speed is 10 rev/s.

Underline each personal pronoun and its antecedent. Hawaii, with its warm climate, fascinating culture, and gorgeous scenery, is a favorite tourist destination.

Let f and g be continuous functions such that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty$. We say that f grows faster than $g(f \gg g)$ if $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\infty$, and we say that f and g grow at the same rate $(f \sim g)$ if $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}$ exists and is $\neq 0. \infty$. For example, for any polynomial function P with $\lim _{x \rightarrow \infty} P(x)=\infty$ (i.e., P is non-constant and has positive leading coefficient) we have $\exp \gg P$ and $P \gg \log ^{n}$ for any positive integer n. (a) Given / and g, with $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty$, is it necessarily true that one of the three conditions $f \gg g$ or $g \gg f$ or $f \sim g$ holds? (b) If $f \gg g$, then $f+g \sim f$. (c) If $\frac{\log f}{\log g} \geq c>1$ for sufficiently large x, then $f \gg g$. (d) If $f \gg g$ and $F(x)=\int_{0}^{x} f, G(x)=\int_{0}^{x} g$, does it necessarily follow that $F \gg G$? (e) Arrange each of the following sets of functions in increasing order of growth (for convenience, we indicate each function simply by giving its value at x): (i) $x^{3}, e^{x}, x^{3}+\log \left(x^{3}\right), \log 4 x,(\log x)^{x}, x^{x}, x+e^{-5 x}, x^{3} \log x$. (ii) $x \log ^{2} x, e^{5 x}, \log \left(x^{x}\right), e^{x^{2}}, x^{x}, x^{\log x},(\log x)^{x}$. (iii) $e^{x}, x^{e}, x^{x}, e^{x^{2}}, 2^{x}, e^{x / 2},(\log x)^{2 x}$.