Assuming that

$$n _ { 1 } = n _ { 2 }$$

find the sample sizes needed to estimate

$$\left( p _ { 1 } - p _ { 2 } \right)$$

for each of the following situations: SE = .01 with 99% confidence. Assume that

$$p _ { 1 } \approx .4 \ and \ p _ { 2 } \approx .7.$$

In "Daughter of Invention," how does the narrator's father react to his daughter's speech?

Evaluate the integral.

$$\int \frac { x ^ { 3 } d x } { x ^ { 2 } - 2 x + 1 }$$

Two vectors $u$ and $v$ are given. Compute the quantities $||u||$, $||v||$, and $||u+v||$. Use your results to illustrate the triangle inequality.

$$\begin{equation*} u=\begin{bmatrix}3\\2\end{bmatrix}\text{ and } v=\begin{bmatrix}-6\\-4\end{bmatrix} \end{equation*}$$

You are dealt one card from a standard 52-card deck. Find the probability of being dealt. a card greater than 3 and less than 7.

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse , or a hyperbola. (b) Use a rotation of axes to eliminate the xy -term . (c) Sketch the graph.

$$2 \sqrt { 3 } x ^ { 2 } - 6 x y + \sqrt { 3 } x + 3 y = 0$$

Simplify.

$$2 z ^ { - 8 }$$

What is the difference between the graph of an even function and the graph of an odd function?

Recall that the velocity of the free-falling bungee jumper can be computed analytically as:

$$v(t)=\sqrt{\frac{g m}{c_{d}}} \tanh (\sqrt{\frac{g c_{d}}{m}} t)$$

where v(t) = velocity (m/s), t = time (s), g = 9.81 m/s$^{2}$, m = mass (kg), $c_{d}$ = drag coefficient (kg/m). (a) Use Romberg integration to compute how far the jumper travels during the first 8 seconds of free fall given m = 80kg and $c_{d}$ = 0.2kg/m.Compute the answer to $\varepsilon_{s}=1 \%$. (b) Perform the same computation with quad.

Use the formula to determine the value of the indicated variable for the values given. Use a calculator when one is needed. When necessary, use the π key on your calculator and round answers to the nearest hundredth.

$$A=\dfrac{1}{2}h(b_1+b_2)$$

; determine h when A=36,

$$b_1=4$$

, and

$$b_2=8$$

(geometry).

Find a value for w so that the equation has (a) two solutions and (b) three solutions. Explain your reasoning.

$$5 x ^ { 3 } + w x ^ { 2 } + 80 x = 0$$

Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write

$$\lim _ { x \rightarrow a } f ( x ) \neq L$$

if for some

$$\varepsilon > 0$$

, there is no value of

$$\delta > 0$$

satisfying the condition |f(x)=L|<

$$\boldsymbol { \varepsilon }$$

whenever 0<|x-a|<

$$\delta$$

. Prove that

$$\lim _ { x \rightarrow 0 } \frac { | x | } { x }$$

does not exist.

Find the sum or difference.

$$\left( 5 x ^ { 2 } - 2 \right) + \left( 8 x ^ { 3 } + 2 x ^ { 2 } - x + 9 \right)$$

$$w - 3 \frac { 2 } { 3 } = 1 \frac { 1 } { 2 }$$