Find the centroid of the region lying between the graphs of the functions over the given interval.

$$y = e ^ { x },$$

y = 1, [0, 1]

Find the centroid of the region lying underneath the graph of the function over the given interval.

$$f ( x ) = e ^ { - x },$$

[0, 4]

Let $\theta$ and $\phi$ be fixed angles, and consider the following two curves: (a) $x=\sin \phi \cos t$,

$$\begin{aligned} & y=\sin \phi \sin t, \quad 0 \leqslant t \leqslant 2 \pi \\ & z=\cos \phi ; \end{aligned}$$

(b)

$$\begin{aligned} x & =\sin t \cos \theta, \\ y & =\sin t \sin \theta, \quad 0 \leqslant t \leqslant 2 \pi \\ z & =\cos t . \end{aligned}$$

Show that each curve is a circle lying on the sphere of radius 1 centered at the origin. Find the center and radius of each circle. Sketch the curves for $\phi=45^{\circ}$ and for $\theta=45^{\circ}$.

Consider the point (x, y) lying on the graph of $y=\sqrt{x-3}.$ Let L be the distance between the points (x, y) and (4, 0). Write L as a function of y.

(a) An electron is moving east in a uniform electric field of 1.50 N/C directed to the west. At point A, the velocity of the electron is $4.50 \times 10 ^ { 5 } \mathrm { m } / \mathrm { s }$ toward the east. What is the speed of the electron when it reaches point B, 0.375 m east of point A? (b) A proton is moving in the uniform electric field of part (a). At point A, the velocity of the proton is $1.90 \times 10 ^ { 4 } \mathrm { m } / \mathrm { s }$, east. What is the speed of the proton at point B?

Draw the vector field plot of the differential equation. Then, using the given initial conditions, sketch the solutions (i.e., draw a graph showing the dependent variable as a function of the independent variable).
$\frac{dN}{dt}$ = (N − 1)(N + 1)(N − 4)
(a) N(0) = 0,
(b) N(0) = 2,
(c) N(0) = 6,
(d) N(0) = −2.

Consider the point (x, y) lying on the graph of the line 2x+4 y=5. Let L be the distance from the point (x, y) to the origin (0,0). Write L as a function of $x$.

Without using a calculator, find two consecutive integers, one lying above and the other lying below the logarithm of the number.

$$0.99 \cdot 10^5$$

Read each question. Then write the letter of the correct answer on your paper. Find all the roots of $2 x^{4}+x^{3}-8 x^{2}-4 x=0.$ A. $x=-2, x=-0.5, x=0, x=2$ B. $x=-2, x=-0.5, x=2$ C. $x=-2, x=0.5, x=0, x=2$ D. $x=-2, x=0.5, x=2$

Write the equilibrium expression for each of the following reactions: (a) $2 \mathrm{~N}_2 \mathrm{O}(g) \rightleftarrows 2 \mathrm{~N}_2(g)+\mathrm{O}_2(g)$ (d) $\mathrm{CO}(g)+\mathrm{H}_2 \mathrm{O}(g) \rightleftarrows \mathrm{CO}_2(g)+\mathrm{H}_2(g)$ (b) $2 \mathrm{NOBr}(g) \rightleftarrows 2 \mathrm{NO}(g)+\mathrm{Br}_2(g)$ (e) $\mathrm{CH}_4(g)+2 \mathrm{H}_2 \mathrm{~S}(g) \rightleftarrows \mathrm{CS}_2(g)+4 \mathrm{H}_2(g)$ (c) $\mathrm{HF}(a q) \rightleftarrows \mathrm{H}^{+}(a q)+\mathrm{F}^{-}(a q)$ (f) $\mathrm{H}_2 \mathrm{C}_2 \mathrm{O}_4(a q) \rightleftarrows 2 \mathrm{H}^{+}(a q)+\mathrm{C}_2 \mathrm{O}_4^{2-}(a q)$

Tienes que crear una mascota y un cartel para una celebración deportiva internacional en tu comunidad. ¿Qué pones en el cartel? ¿Cómo es la mascota?

Find the area of the region lying to the right of

$$x = y ^ { 2 } + 4 y - 22$$

and to the left of x = 3y + 8.

The distance $d$, in meters, that an object falls after $t$ seconds is given by the formula $d=4.9 t^2$. Graph this equation for $0 \leq t \leq 6$, where $t$ is the independent variable.

Simplify $(x^3+x)+n(x^2+2x)$ by distributing the n. Show that this is equivalent for n = 2 to $(x^2+x)+(x^2+2x)+(x^2+2x)$.