Simplify each expression.

$$( 6 + 2 \cdot 3 ) \div ( 9 - 7 ) ^ { 2 }$$

The length of a curve given by a parametric equation x(t), y(t) is given by:

$$\int _ { a } ^ { b } \sqrt { \left[ x ^ { \prime } ( t ) \right] ^ { 2 } + \left[ y ^ { \prime } ( t ) \right] ^ { 2 } } d t$$

The cardioid curve is given

$$x = 2 b \cos t - b \cos 2 t , \text{ and } y = 2 b \sin t - b \sin 2 t$$

with $0\leq t\leq 2\pi$ . Plot the cardioid with b = 5 and determine the length of a the curve. by:

find the center, vertices, foci, and eccentricity of the ellipse. Then sketch the ellipse. 16x^2 + 25y^2 - 32x + 50y + 16 = 0

Find the compositions $f \circ g$ and $g \circ f$, and identify their respective domains. $f(x)=\frac{1}{x^{2}-1}$, $g(x)=x^{2}-2$

If two lines are intersected by a third line, is the third line necessarily a transversal? Justify your answer with a diagram.

Find the critical value(s) and rejection region(s) for the type of z-test with level of significance

$$\alpha$$

. Include a graph with your answer. Right-tailed test,

$$\alpha = 0.08$$

Find an equation for the function f that has the given derivative and whose graph passes through the given point. Derivative: f’(x) = sec²(2x) Point: (π/2, 2)

Jiffy Lube has franchises in two different parts of a city. The number of oil changes made daily, for 25 days , is given below. East store

$$\left. \begin{array} { r l l l l } { 33 } & { 59 } & { 27 } & { 30 } & { 42 } \\ { 19 } & { 42 } & { 25 } & { 22 } & { 32 } \\ { 43 } & { 27 } & { 57 } & { 37 } & { 52 } \\ { 40 } & { 67 } & { 38 } & { 44 } & { 43 } \\ { 15 } & { 31 } & { 49 } & { 41 } & { 35 } \end{array} \right.$$

West Store

$$\left. \begin{array} { l l l l l } { 38 } & { 46 } & { 38 } & { 38 } & { 30 } \\ { 38 } & { 38 } & { 37 } & { 39 } & { 31 } \\ { 39 } & { 36 } & { 40 } & { 37 } & { 47 } \\ { 30 } & { 34 } & { 42 } & { 45 } & { 29 } \\ { 31 } & { 46 } & { 28 } & { 45 } & { 48 } \end{array} \right.$$

(a) Construct a frequency distribution for each store with a first class of 15-20. (b) Draw a histogram indicating the number of oil changes at each store. (c) Using the histograms, determine which store appears to have a greater mean , or do the means appear about the same? Explain. (d) Using the histogram, determine which store appears to have the greater standard deviation. Explain. (e) Calculate the mean for each store and determine whether your answer in part (c) was correct. Calculate the standard deviation for each store and determine whether your answer in part (d) was correct.

Factor the polynomial, if possible. If the polynomial cannot be factored using integers, write prime.

$$-6x^2 - 23x - 20$$

Simplify each radical expression.

$$\frac { \sqrt { 5 } } { \sqrt { 3 } }$$

Draw a correctly labeled graph of the market for taxicab rides. On the graph, draw and label a vertical line showing the level of an effective quota. Label the demand price, the supply price, and the quota rent.

A homogeneous second-order linear differential equation, two functions $y_1$ and $y_2$, and a pair of initial conditions are given. First verify that $y_1$ and $y_2$ are solutions of the differential equation. Then find a particular solution of the form $y = c _ { 1 } y _ { 1 } + c _ { 2 } y _ { 2 }$ that satisfies the given initial conditions. Primes denote derivatives with respect to $x$.

$$y ^ { \prime \prime } - y = 0 ; y _ { 1 } = e ^ { x } , y _ { 2 } = e ^ { - x } ; y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 5$$

Evaluate:

$$\sqrt { 81 }.$$

A boy swims across a river in the shortest time he can, by always heading straight for the opposite bank. He can swim at a speed of 0.500 m/s relative to the water. The river is 25.0 m wide and the boy ends up at 50.0 m downstream from his starting point. (a) How fast is the current flowing in the river? (b) What is the speed of the boy relative to a friend standing on the riverbank?