A couple plans to have two children. (a) Determine the number of sample points in the sample space of the possible arrangements of boys and girls. (b) Construct a tree diagram and list the sample...

You decide to take a science course, an English course, and a mathematics course during the next term. The available courses that you can take are listed below. If the courses are selected at random,...

Mr. and Mrs. Frank just moved into a new home and need to do some landscaping. They are going to purchase one tree, one shrub, and one lilac bush at random from the list below. (a) Determine the...

Consider the following wheel. If the wheel is spun and each section is equally likely to stop under the pointer, determine the probability that the pointer lands on. An even number, given that the...

Consider the circles shown. Assume that one circle is selected at random and each circle is equally likely to be selected. Determine the probability of selecting. A 3, given that the circle is...

Consider the circles shown. Assume that one circle is selected at random and each circle is equally likely to be selected. Determine the probability of selecting. A red number, given that the circle...

Two fair dice are rolled one after the other. Construct a sample space and determine the probability that the sum of the dots on the dice total. 3 if the first die is a 1.

Two fair dice are rolled one after the other. Construct a sample space and determine the probability that the sum of the dots on the dice total. 3 if the first die is a 3.

Two fair dice are rolled one after the other. Construct a sample space and determine the probability that the sum of the dots on the dice total. A number greater than 7 if the second die is a 5.

Assume that a hat contains four bills: a $1 bill, a $5 bill, a $10 bill, and a $20 bill. Each bill is equally likely to be selected. Two bills are to be selected at random with replacement. Construct...

What percent of head circumferences were greater than 35.75 cm? Assume that anthropologists have determined that the akidolestes, a small primitive mammal believed to have lived with the dinosaurs,...

What percent of head circumferences were less than 50 cm? Assume that anthropologists have determined that the akidolestes, a small primitive mammal believed to have lived with the dinosaurs, had a...

Use the following data on weekly salaries at Donovans Construction Company. What salary represents one standard deviation above the mean? First quartile Third quartile $825 79th percentile $832 $770...

Determine the equation of the line of best fit from the data indicated in the exercise. Round both m and b to the nearest hundredth. Exercise 19. Data from exercise 19: x . . . . . . . . y 4 . . . ....

Determine the equation of the line of best fit from the data indicated in the exercise. Round both m and b to the nearest hundredth. Exercise 20. Data from exercise 20: x . . . . . . . . y 8 . . . ....

A mean average of 80 or greater for five exams is needed for a final grade of B in a course. Jorges first four exam grades are 73, 69, 85, and 80. What grade does Jorge need on the fifth exam to get...

The following table gives the distribution of annual salaries for employees at Kulzers Home Improvement. Annual Salary . . . . . . . . . . . . . . . . Number Receiving Salary $100,000 . . . . . . . ....

Consider the following two normal curves. (a) Do these distributions have the same mean? If so, what is the mean? (b) One of these curves corresponds to a normal distribution with a standard...

The drawing below shows the floor plan of the first floor of the Oleander model home offered by Nisely Builders of Albuquerque, New Mexico. Place vertex O near the bottom of the graph. Use a graph to...

Use Table 12.8 to determine the percent of data specified. Table 12.8: Less than z = 0.79 Table of Areas to the Left of z When z Is Negative 00 .02 .04 .05 .01 .03 .06 .07 .08 .09 -3.4 .0003 .0003...

Evaluate the following limits. 4 sint x3 100 lim ( 5 + x x-

The following table gives the position s(t) of an object moving along a line at time t. Determine the average velocities over the time intervals [1, 1.01], [1, 1.001], and [1, 1.0001]. Then make a...

Given the function f(x) = 1 - cos x and the points A(/2, f(/2)), B(/2 + 0.05, f(/2 + 0.05)), C(/2 + 0.5, f(/2 + 0.5)), and D(p, f(p)) (see figure), find the slopes of the secant lines through A and...

Analyze the following limits and find the vertical asymptotes of a. b. c. F(x) x 25 lim f(x) x5*

Determine the following limits. -6 lim x

Determine the following limits. -11 lim x

Determine and for the following rational functions. Then give the horizontal asymptote of f (if any). lim f(x) X' lim f(x) x-00

Determine and for the following rational functions. Then give the horizontal asymptote of f (if any). lim f(x) X' lim f(x) x-00

The amount of an antibiotic (in mg) in the blood t hours after an intravenous line is opened is given by m(t) = 100(e -0.1t - e -0.3t ). a. Use the Intermediate Value Theorem to show the amount of...

Find the derivative of the following functions. y = tan t / 1 + sec t

The graph of f (t) = e -kt sin t with k > 0 is called a damped sine wave; it is used in a variety of applications, such as modeling the vibrations of a shock absorber. a. Use a graphing utility to...

The economic advisor of a large tire store proposes the demand function D(p) = 1800 / p - 40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p. a....

Determine and for the following rational functions. Then give the horizontal asymptote of f (if any). lim f(x) X' lim f(x) x-00

Determine and for the following rational functions. Then give the horizontal asymptote of f (if any). lim f(x) X' lim f(x) x-00

For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. 2 %3 2

The table lists average annual cost (in dollars) of tuition and fees at private four-year colleges for selected years. (a) Determine a linear function (x) = ax + b that models the data, where x = 0...

(a) Draw a bar graph that appears to show a small increase in the median age at first marriage for females. (b) Draw a bar graph that appears to show a large increase in the median age at first...

Evaluate the expression. -4 - 4

Fill in the blanks with an appropriate word, phrase, or symbol(s). With the zero exponent rule, the expression 7 0 can be simplified to ________.

Twelve red chips and 8 blue chips are in a bag. If four chips from the bag are to be selected at random, determine the probability of selecting four red chips. Set up the problem as if it were to be...

Each of the digits 09 is written on a slip of paper, and the slips are placed in a hat. If three slips of paper are selected at random, determine the probability that the three numbers selected are...

The number of hours of daylight at any point on Earth fluctuates throughout the year. In the northern hemisphere, the shortest day is on the winter solstice and the longest day is on the summer...

The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 10 1.5M . (This equation can be solved for M to define the magnitude of a given earthquake; it is...

Carry out the following steps. a. Use implicit differentiation to find dy/dx. b. Find the slope of the curve at the given point. y 2 + 3x = 8; (1, 5)

Use logarithmic differentiation to evaluate f'(x). (x + 1)10 (2x 4) f(x)

A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its...

From the graph of g, state the intervals on which g is continuous. y. + -3 -2 1 2.

If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y = 10t - 1.86t 2 . (a) Find the average velocity over the given time...

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin t 1 3 cos t, where t is measured in seconds. (a) Find the...

If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H = 10t - 1.86t 2 . (a) Find the velocity of the rock after one second. (b)...

Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square based box with the...

a. Multiply the numerator and denominator of sec x by sec x + tan x; then use a change of variables to show that sec x dx = ln |sec x + tan x| + C. b. Use a change of variables to show that csc x...

Find the area of the shaded region enclosed in a semicircle of diameter 10 inches. The length of the chord PQ is 8 inches. R. 10

Solve equation. Give a general formula for all the solutions. List six solutions. cos = -3/2

Solve each linear programming problem. Maximize z = 2x + 4y subject to x 0, y 0, 2x + y 4, x + y 9

How many different arrangements are there of the letters in the word ROSE?

A square plate 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then...

A cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed, then the...

A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P. 60 N 70 ) 10 P

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 2 i + 2t k, v(0) = 3 i - j, r(0) = j + k

A ball is thrown eastward into the air from the origin (in the direction of the positive x-axis). The initial velocity is 50 i + 80 k, with speed measured in feet per second. The spin of the ball...

Use a table of numerical values of f (x, y) for (x, y) near the origin to make a conjecture about the value of the limit of f (x, y) as (x, y) (0, 0). Then explain why your guess is correct. f (x,...

Find the limit, if it exists, or show that the limit does not exist. e tan(xz) lim (x, y, z)- (7, 0, 1/3)

Find the limit, if it exists, or show that the limit does not exist. xy + yz (r, y. 2(0,0,0) x? + y? + z? lim

Show that the function f given by f (x) = |x| is continuous on R n . Consider |x - a| 2 = (x - a) (x - a).]

Determine the signs of the partial derivatives for the function f whose graph is shown. (a) f x x (-1, 2) (b) f yy (-1, 2) ZA .

Determine the signs of the partial derivatives for the function f whose graph is shown. (a) f xy (1, 2) (b) f xy (-1, 2) ZA .

Find the first partial derivatives of the function. f (x, y, z) = xy 2 e -xz

Use implicit differentiation to find z/x and z/y. x 2 + 2y 2 + 3z 2 = 1

Find the indicated partial derivative(s). w = x/y + 2z; 3 W/zyx, 3 W/x 2 y

Use the Chain Rule to find dz/dt or dw/dt. w = lnx 2 + y 2 + z 2 , x = sin t, y = cos t, z = tan t

Find all points at which the direction of fastest change of the function f (x, y) = x 2 + y 2 - 2x - 4y is i + j.

Suppose you are climbing a hill whose shape is given by the equation z = 1000 - 0.005x 2 - 0.01y 2 , where x, y, and z are measured in meters, and you are standing at a point with coordinates (60,...

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the...

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the...

Calculate the double integral. R=[0, 1] [0, 1] -dA, 1+

Evaluate the given integral by changing to polar coordinates. where D is the top half of the disk with center the origin and radius 5 SS,r*y dA,

Evaluate the given integral by changing to polar coordinates. where R is the region in the first quadrant between the circles with center the origin and radii 1 and 3 SS, sin(x? + y) dA,

Evaluate the iterated integral by converting to polar coordinates. 1/2 -y xv dx dy /1-y2 /3y

Evaluate the iterated integral by converting to polar coordinates. V2x-x2 Vx2 + y2 dy dx 0.

Find the center of mass of the lamina in Exercise 11 if the density at any point is proportional to the square of its distance from the origin.

Evaluate the iterated integral. V1-z2 z sin x dy dz dx Jo Jo

Write five other iterated integrals that are equal to the given iterated integral. f(x, y, z) dx dz dy Jy

Change from rectangular to cylindrical coordinates. (a) (-1, 1, 1) (b) (-2, 23 , 3)

Sketch the solid whose volume is given by the integral and evaluate the integral.

At time t = 1, a particle is located at position (1, 3). If it moves in a velocity field F(x, y) = (xy - 2, y 2 - 10) find its approximate location at time t = 1.05.

(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle x 2 + y 2 = 1. (b) Is this also true for a force field F(x) = kx, where k is a constant...

Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the...

(a) Find a function f such that F = f and (b) use part (a) to evaluate C F dr along the given curve C. F(x, y, z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1, 0, -2) to (4, 6, 3)

Use Greens Theorem to evaluate C F dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y - cos y, x sin y), C is the circle (x - 3) 2 + (y + 4) 2 = 4 oriented clockwise

Use Greens Theorem to find the work done by the force F(x, y) = x(x + y) i + xy 2 j in moving a particle from the origin along the x-axis to (1, 0), then along the line segment to (0, 1), and then...

Use Exercise 22 to find the centroid of the triangle with vertices (0, 0), (a, 0), and (a, b), where a > 0 and b > 0.

Find (a) the curl and (b) the divergence of the vector field. F(x, y, z) = xye z i + yze x k

Use Greens Theorem to prove the change of variables formula for a double integral (Formula 15.9.9) for the case where f (x, y) = 1: Here R is the region in the xy-plane that corresponds to the region...

At t = 0, a train approaching a station begins decelerating from a speed of 80 mi/hr according to the acceleration function a(t) = -1280(1 + 8t) -3 , where t 0 is measured in hours. How far does the...

At rush hours, substantial traffic congestion is encountered at the traffic intersections shown in the figure. (All streets are one-way.) The city wishes to improve the signals at these corners to...

Give a geometric description of the set of points (x, y, z) that lie on the intersection of the sphere x 2 + y 2 + z 2 = 5 and the plane z = 1.

Show that the first five nonzero coefficients of the Taylor series (binomial series) for f(x) = 1 + 4x about 0 are integers. (In fact, all the coefficients are integers.)

For what values of p does the series converge (initial index is 10)? For what values of p does it diverge? kP k=10

Find the area of the region enclosed by one loop of the curve. r 2 = 4 cos 2