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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 Seawater has density 1025 kg/m 3 and flows in a velocity field v = y i + x j, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward... Let F(x, y, z) = z tan-1(y 2 ) i + z 3 ln(x 2 + 1) j + z k. Find the flux of F across the part of the paraboloid x 2 + y 2 + z = 2 that lies above the plane z = 1 and is oriented upward. Evaluate the line integral. C xy dx + y 2 dy + yz dz, C is the line segment from (1, 0, -1), to (3, 4, 2) Use power series to solve the differential equation y'' - xy' - 2y = 0 If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. (b) P(0, 0, -3), Q(4, 2, 0), R(3, 3, 1) Find a vector function that represents the curve of intersection of the cylinder x 2 + y 2 = 16 and the plane x + z = 5. For the curve given by r(t) = (sin 3 t, cos 3 t, sin 2 t), 0 < t < y 2 , find (a) The unit tangent vector, (b) The unit normal vector, (c) The unit binormal vector, and (d) The curvature. Suppose that over a certain region of space the electrical potential V is given by V(x, y, z) = 5x 2 - 3xy + x/z. (a) Find the rate of change of the potential at P(3, 4, 5) in the direction of the... The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the... Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y) = x 2 y; x 2 + y 2 = 1 Select a random sample of three without replacement from the following (very small) population of firms: IBM, GM, Ford, Shell, HP, Boeing, and ITT. Use the following sequence of random digits:... Without using a calculator, evaluate (a) 10 (2) (b) (1) (3) (c) (8) 2 (d) (5) (5) (e) 24 (2) (f) (10) (5) (g) 20/-4 (h) -27/-9 (i) (6) 5 (1) ( j) 2 x (-6) x 3 (-9) On graph paper draw axes with values of x and y between 3 and 10, and plot the following points: P(4, 0), Q(2, 9), R(5, 8), S(1, 2) Hence find the coordinates of the point of intersection of the line... An airline charges $300 for a flight of 2000 km and $700 for a flight of 4000 km. (a) Plot these points on graph paper with distance on the horizontal axis and cost on the vertical axis. (b) Assuming... Amazon.com is an e-commerce firm that has shown considerable growth since its founding in 1995, and its quarterly net sales are shown in Table 14.4.4. Their 2014 annual report includes a section... The number of people, N, employed in a chain of cafes is related to the number of cafes, n, by the equation: N = 10n + 120 (a) Illustrate this relation by plotting a graph of N against n for 0 n ... Write down a formula for each situation: (a) A plumber has a fixed call-out charge of $80 and has an hourly rate of $60. Work out the total charge, C, for a job that takes L hours in which the cost... The demand, Q, for a certain good depends on its own price, P, and the price of an alternative good, PA, according to Q = 30 3P + PA (a) Find Q if P = 4 and PA = 5. (b) Is the alternative good... The total cost, TC, of producing 100 units of a good is 600 and the total cost of producing 150 units is 850. Assuming that the total cost function is linear, find an expression for TC in terms of Q,... Table 3.12 gives the annual rate of inflation during a five-year period. If a nominal house price at the end of Year 1 was $10.8 million, find the real house price adjusted to prices prevailing at... A bank offers a return of 7% interest compounded annually. Find the future value of a principal of $4500 after six years. What is the overall percentage rise over this period? A department store has its own credit card facilities, for which it charges interest at a rate of 2% each month. Explain briefly why this is not the same as an annual rate of 24%. What is the annual... An individual saves $5000 in a bank account at the beginning of each year for 10 years. No further savings or withdrawals are made from the account. Determine the total amount saved if the annual... A builder is offered one of two methods of payment: Option 1: A single sum of $73 000 to be paid now. Option 2: Five equal payments of $15 000 to be paid quarterly with the first instalment to be... Differentiate the following functions, giving your answer in a similar form, without negative or fractional indices: (d) y = xVx (b) fx) = Jx (c) f(x) = (a) f(x) fx) = Differentiate (a) y = 5x 2 (b) y = 3/x (c) y = 2x + 3 (d) y = x 2 + x + 1 (e) y = x 2 3x + 2 (f) y = 3x (g) y = 2x 3 6x 2 + 49x 54 (h) y = ax + b (i) y = ax 2 + bx + c ( j) y = 4x - 3/x + 7/x 2 (a) Find the elasticity of demand in terms of Q for the demand function, P = 20 0.05Q. (b) For what value of Q is demand unit elastic? (c) Find an expression for MR and verify that MR = 0 when... If the supply equation is Q = 7 + 0.1P + 0.004P 2 find the price elasticity of supply if the current price is 80. (a) Is supply elastic, inelastic or unit elastic at this price? (b) Estimate the... Given the demand function Q = 200 2P P A + 0.1Y 2 where P = 10, P A = 15 and Y = 100, find (a) the price elasticity of demand; (b) the cross-price elasticity of demand; (c) the income elasticity of... Use Lagrange multipliers to find the maximum value of z = x + 2xy subject to the constraint x + 2y = 5 Find the producers surplus at Q = 9 for the following supply functions: (a) P = 12 + 2Q (b) P = 20Q + 15 A publisher decides to use one section of its plant to produce two textbooks called Microeconomics and Macroeconomics. The profit made on each copy is $12 for Microeconomics and $18 for... National income Y varies over time t according to the following model: dY/dt = 0.6(C + I - Y) C = 0.8Y + 600 I = 800 where C is consumption and I is planned investment. Initially, Y = 2000. (a) Find... Use Greens Theorem in the form of Equation 13 to prove Greens first identity: where D and C satisfy the hypotheses of Greens Theorem and the appropriate partial derivatives of f and g exist and are... Determine whether you would take a census or use a sampling. If you would use a sampling, determine which sampling technique you would use. Explain. (a) The most popular type of investment among... Determine whether each number describes a population parameter or a sample statistic. Explain. (a) A survey of 1003 U.S. adults ages 18 years and older found that 72% own a smartphone. (b) In a... Determine whether the data are qualitative or quantitative, and determine the level of measurement of the data set. Explain your reasoning. (a) The numbers of employees at fast-food restaurants in a... Data at the ratio level cannot be put in order. Whether the statement is true or false. If it is false, rewrite it as a true statement. What is an inherent zero? Describe three examples of data sets that have inherent zeros and three that do not. In a survey of 1033 U.S. adults, 51% said U.S. presidents should release all medical information that might affect their ability to serve. Determine whether the study is an observational study or an... Researchers demonstrated that adults using an intensive program to lower systolic blood pressure to less than 120 millimeters of mercury reduce the risk of death from all causes by 27%. Determine... Every tenth person entering a mall is asked to name his or her favorite store. Identify the sampling technique used, and discuss potential sources of bias (if any). Explain. Why is a sample used more often than a population? What is the difference between a parameter and a statistic? It is impossible to obtain all the census data about the U.S. population. Determine whether the statement is true or false. If it is false, rewrite it as a true statement. The amount of energy collected from every solar panel on a photovoltaic power plant. Determine whether the data set is a population or a sample. Explain your reasoning.
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