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mathematics
precalculus

Questions and Answers of Precalculus

Solve the differential equation.(ey - 1) y' = 2 + cos x
Solve the differential equation.y' + xey = 0
Solve the differential equation.xyy' = x2 + 1
Solve the differential equation.dy/dx = x√y
Solve the differential equation.dy/dx = 3x2y2
(a) Use Euler’s method with step size 0.2 to estimate y(0.6), where y(x) is the solution of the initial-value problem y' = cos(x + y), y(0) = 0.(b) Repeat part (a) with step size 0.1.
Use Euler’s method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y' = x2 y - 1/2 y2, y(0) = 1.
Use Euler’s method with step size 0.5 to compute the approximate y-values y1, y2, y3, and y4 of the solution of the initial-value problem y' = y - 2x, y(1) = 0.
Use a computer algebra system to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through (0, 1). Then use the CAS to draw
Use a computer algebra system to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through (0, 1). Then use the CAS to draw
Sketch the direction field of the differential equation.Then use it to sketch a solution curve that passes through the given point.y' = x + y2, (0, 0)
Sketch the direction field of the differential equation.Then use it to sketch a solution curve that passes through the given point.y' = xy - x2, (0, 1)
Sketch a direction field for the differential equation. Then use it to sketch three solution curves.y' = x - y + 1
Sketch a direction field for the differential equation. Then use it to sketch three solution curves.y' = 1/2y
Use the direction field labeled III (above) to sketch the graphs of the solutions that satisfy the given initial conditions.(a) y(0) = 1(b) y(0) = 2.5(c) y(0) = 3.5
Use the direction field labeled I (above) to sketch the graphs of the solutions that satisfy the given initial conditions.(a) y(0) = 1 (b) y(0) = 2.5 (c) y(0) = 3.5
Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer.y' = x + y - 1 II I IV -- -- -- II -- -- -- -2 1. /// ///! ///1|I\ ///||\\ ///||\ ////
Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer.y' = x(2 - y) II I IV -- -- -- II -- -- -- -2 1. /// ///! ///1|I\ ///||\\ ///||\ ////
Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer.y' = 2 - y II I IV -- -- -- II -- -- -- -2 1. /// ///! ///1|I\ ///||\\ ///||\ ////
Von Bertalanffy’s equation states that the rate of growth in length of an individual fish is proportional to the difference between the current length L and the asymptotic length L∞ (in
Match the differential equations with the solution graphs labeled I–IV. Give reasons for your choices. (a) y' = 1 + x? + y? (b) у' — хе-т*-у (c) y' (d) y' = sin(xy) cos(ry) y 1+ er*+. II y.
The Fitzhugh-Nagumo model for the electrical impulse in a neuron states that, in the absence of relaxation effects, the electrical potential in a neuron v(t) obeys the differential equationwhere a is
Verify that y = -t cos t - t is a solution of the initialvalue problemt dy/dt = y + t2 sin ty(π) = 0
Find the centroid of the region bounded by the given curves.y = 1/2x, y = √x
Find the centroid of the region shown. y х -8
Find the centroid of the region shown. ул (4, 2) х
Find the length of the curve y= Vi - 1 dt 1
Use Simpson’s Rule with n = 10 to estimate the area of the surface obtained by rotating the sine curve in Exercise 7 about the x-axis.
Let C be the arc of the curve y = 2/(x + 1) from the point (0, 2) to (3, 1/2). Use a calculator or other device to find the value of each of the following, correct to four decimal places.(a) The
Find the length of the curve.12x = 4y3 + 3y-1, 1 < y < 3
Find the length of the curve.y = 2 In(sin 1/2x), π/3 < x < π
Show that the probability density function for a normally distributed random variable has inflection points at x = µ ± σ
REM sleep is the phase of sleep when most active dreaming occurs. In a study, the amount of REM sleep during the first four hours of sleep was described by a random variable T with probability
An online retailer has determined that the average time for credit card transactions to be electronically approved is 1.6 seconds.(a) Use an exponential density function to find the probability that
Let f (x) = c/(1 + x2).(a) For what value of c is f a probability density function?(b) For that value of c, find P (-1 < X < 1).
The density function f(x) = e3-x/(1 + e3-x)2 is an example of a logistic distribution.(a) Verify that f is a probability density function.(b) Find P(3 < X < 4).(c) Graph f. What does the mean
After a 5.5-mg injection of dye, the readings of dye concentration, in mg/L, at two-second intervals are as shown in the table. Use Simpson’s Rule to estimate the cardiac output. c(t) c(t) 0.0 10
Pareto’s Law of Income states that the number of people with incomes between x = a and x = b is where A and k are constants with A > 0 and k > 1. The average income of these people
The present value of an income stream is the amount that would need to be invested now to match the future value as described in Exercise 15 and is given by
If revenue flows into a company at a rate of f(t) = 9000√1 + 2t, where t is measured in years and f(t) is measured in dollars per year, find the total revenue obtained in the first four years.
A movie theater has been charging $10.00 per person and selling about 500 tickets on a typical weeknight. After surveying their customers, the theater management estimates that for every 50 cents
A camera company estimates that the demand function forits new digital camera is p(x) = 312e-0.14x and the supply function is estimated to be pS(x) = 26e0.2x, where x is measured in thousands.
If a supply curve is modeled by the equation p = 125 + 0.002x2, find the producer surplus when the selling price is $625.
The demand function for a particular vacation package is p(x) = 2000 - 46√x . Find the consumer surplus when the sales level for the packages is 400. Illustrate by drawing the demand curve and
A company estimates that the marginal revenue (in dollars per unit) realized by selling x units of a product is 48 - 0.0012x. Assuming the estimate is accurate, find the increase in revenue if sales
Use the Second Theorem of Pappus described in Exercise 48 to find the surface area of the torus in Example 7.
Use the Theorem of Pappus to find the volume of the given solid.A sphere of radius r (Use Example 4.)
A rectangle 5 with sides a and b is divided into two parts R1 and R2 by an arc of a parabola that has its vertex at one corner of R and passes through the opposite corner. Find the centroids of both
Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments. y A 1+ 2 х 2.
Use a graph to find approximate x-coordinates of the points of intersection of the curves y = ex and y = 2 - x2. Then find (approximately) the centroid of the region bounded by these curves.
Use Simpson’s Rule to estimate the centroid of the region shown. Ул 4 4
Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape.P = 6 ул 4 х -2
Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape.P = 4 ум У 3 2 х
Find the centroid of the region bounded by the given curves.x + y = 2, x = y2
Find the centroid of the region bounded by the given curves.y = x3, x + y = 2, y = 0
Find the centroid of the region bounded by the given curves.y = 2 - x2, y = x
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.y = sin x, y = 0, 0 < x < π
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.y = √x, y = 0, x = 4
The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. т — 3, т, тz — 4, тз P:(-4, 2), P2(0, 5), P3(3, 2), Pa(1, –2) 5, — 6;
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.y = 2x. y = 0, x = 1
Point-masses mi are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass x. m, = 12 + m2=15 2 + M3= 20 -3 х -3
A milk truck carries milk with density 64.6 lb/ft3 in a horizontal cylindrical tank with diameter 6 ft.(a) Find the force exerted by the milk on one end of the tank when the tank is full.(b) What if
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then
A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 820 kg/m3 to a depth of 1.5 m. Find(a) The hydrostatic pressure on the bottom of the tank,(b) The hydrostatic force on the
Show that if we rotate the curve y = ex/2 + e-x/2 about the x-axis, the area of the resulting surface is the same value as the enclosed volume for any interval a < x < b.
Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator.y
Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator.y
Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator.y
The given curve is rotated about the y-axis. Find the area of the resulting surface.y = 1/4x2 - 1/2, 1 < x < 2
The given curve is rotated about the y-axis. Find the area of the resulting surface.x3/2, + y3/2, = 1, 0 < x < 1
The given curve is rotated about the y-axis. Find the area of the resulting surface.y = 1/3x3/2, 0 < x < 12
Find the exact area of the surface obtained by rotating the curve about the x-axis.x = 1 + 2y2, 1 < y < 2
Find the exact area of the surface obtained by rotating the curve about the x-axis.x = 1/3(y2 +2)3/2, 1 < y < 2
Find the exact area of the surface obtained by rotating the curve about the x-axis.y = x3/6 + 1/2x, 1/2 < x < 1
Find the exact area of the surface obtained by rotating the curve about the x-axis.y = cos(1/2 x), 0 < x < π
Find the exact area of the surface obtained by rotating the curve about the x-axis.y = √1 + ex, 0 < x < 1
Find the exact area of the surface obtained by rotating the curve about the x-axis.y2 = x + 1, 0 < x < 3
Find the exact area of the surface obtained by rotating the curve about the x-axis.y = √5 - x, 3 < x < 5
Find the exact area of the surface obtained by rotating the curve about the x-axis.y = x3, 0 < x < 2
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.(b) Use the numerical integration capability of a calculator to evaluate the
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.(b) Use the numerical integration capability of a calculator to evaluate the
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.(b) Use the numerical integration capability of a calculator to evaluate the
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.(b) Use the numerical integration capability of a calculator to evaluate the
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.(b) Use the numerical integration capability of a calculator to evaluate the
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.(b) Use the numerical integration capability of a calculator to evaluate the
Find the length of the curve y= 1< x< 4 Vt3 – 1 dt
For the function f (x) = 1/4 ex + e-x, prove that the arc length on any interval has the same value as the area under the curve.
Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve y = ex that lies between the points (0, 1) and (2, e2).
Repeat Exercise 29 for the curvey = x + sin x 0 < x < 2πExercise 29 (a) Graph the curve y = x 3√4 - x , 0 < x < 4.(b) Compute the lengths of inscribed polygons with n = 1, 2, and
(a) Graph the curve y = x 3√4 - x , 0 < x < 4.(b) Compute the lengths of inscribed polygons with n = 1, 2, and 4 sides. (Divide the interval into equal sub-intervals.) Illustrate by sketching
Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator.y = e-x2, 0 < x < 2
Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator.y = In (1 + x3), 0 < x < 5

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