A demand curve is given by p = 450/ (x + 8). Find the consumer surplus when the selling price is $10.
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. tan x y = Jo
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = 4 - x², y = 0
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. %3D %3D P.(1, - 2), Р:(3, 4), Р:(-3, —7), Р.(6, — 1)
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. т — 6, т — 5, ms — 10; P.(1,5), Р.(3, —2), Р.{(-2, —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,= 25 ++ -2 m3= 10 ++++ 7 m2 = 20 ++ 3
> A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpson’s Rule to estimate the force of the water against the plate. Depth (m) 2.0 2.5 3.0 3.
> A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate. 2 m water level 12 m 4 m
> Find the volume of the described solid S. The base of S is an elliptical region with boundary curve 9x2 + 4y2 = 36. Cross-sections perpendicular to the -axis are isosceles right triangles with hypotenuse in the base.
> A large tank is designed with ends in the shape of the region between the curves y = 1/2x2 and y = 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline’s density is
> A trough is filled with a liquid of density 840 kg/m3. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.
> Sketch the region in the xy-plane defined by the inequalities x – 2y2 > 0, 1 – x – |y| > 0 and find its area.
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) = " VT+ r³ dr
> Use cylindrical shells to find the volume of the solid. The solid torus of Exercise 45 in Section 6.2. Exercise 45 in Section 6.2: (a). Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii r and R.
> Find the volume of the described solid S. A pyramid with height and rectangular base with dimensions a and 2b.
> Let T be the triangular region with vertices (0, 0), (1, 0), (1, 2), and let V be the volume of the solid generated when T is rotated about the line x = a, where a > 1. Express a in terms of V.
> 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 express the force as an integral and evaluate it. -2 m
> 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 express the force as an integral and evaluate it. 5 m
> Show that the total length of the ellipse x = a sin θ, y = b cos θ, a > b > 0, is Where e is the eccentricity of the ellipse (e = c/a, where c = √a2 - b2). (T/2 L = 4a Jo VT - e² sin²0 do
> 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 bottom, and (c) the hydrostatic force on one end of th
> An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) the hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.
> A steady wind blows a kite due west. The kite’s height above ground from horizontal position x = 0 to x = 80 ft is given by y = 150 – 1/40 (x – 50)2. Find the distance traveled by the kite.
> (a). Newton’s Law of Gravitation states that two bodies with masses m1 and m2 attract each other with a force where is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) = [" arctan t dt 12
> Use Newton’s method to find the coordinates of the inflection point of the curve y = ecosx, 0 < x < π, correct to six decimal places.
> In a steam engine the pressure P and volume V of steam satisfy the equation PV1.4 = k, where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Use Exe
> Use either a CAS or a table of integrals to find the exact length of the curve. y = In(cos x), 0 sxS T/4
> Use either a CAS or a table of integrals to find the exact length of the curve. y? = 4x, 0 < y< 2 %3D
> Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = sin?x, y = 0, 0 <xS T; about y = -1
> If fave [a, b] denotes the average value of f on the interval [a, b] and a b — с с — а fave[a, c] + b - a fwe[c, b] fave [a, b] - b - a
> Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.3) to the function F (x) = fxaf (t) dt.
> The velocity of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is Where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood (see Example 7 in
> The graph of the concentration function c (t) is shown after a 7-mg injection of dye into a heart. Use Simpson’s Rule to estimate the cardiac output. (mg/L) 6. 4 2 4 6 8 10 1 i (seconds) 14 2.
> The standard deviation for a random variable with probability density function f and mean µ is defined by Find the standard deviation for an exponential density function with mean µ. 11/2 (x-씨)?f(x) dx
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. G(x) = [' cos /F dt
> For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.
> Repeat Exercise 21 for the integral f1-1 √4 – x3, dx. Exercise 21: The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for |f (4)(x) | by hand. But computer algebra systems
> Show that the probability density function for a normally distributed random variable has inflection points at x = µ ± σ.
> Use Poiseuille’s Law to calculate the rate of flow in a small human artery where we can take η = 0.027, R = 0.008 cm, t = 2 cm, and P = 4000 dynes/cm2.
> A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitos is increasing at an estimated rate of 2200 + 10e0.8t per week (where is measured in weeks). By how much does the mosquito population increase betw
> Pareto’s Law of Income states that the number of people with incomes between x = a and x = b is N = fba Ax-k dx, where A and k are constants with A > 0 and k > 1. The average income of these people is Calculate x. ī=÷"Ax!-*
> According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a). What is the probability that an adult male chosen at random is between 65 inch
> The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 minutes. (a). Find the probability that a customer has to wait more than 4 minutes. (b). Find the probability that a customer is served wit
> (a). A type of lightbulb is labeled as having an average lifetime of 1000 hours. It’s reasonable to model the probability of failure of these bulbs by an exponential density function with mean µ = 100. Use this model to find the probability that a bulb (
> Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. F(x) = [" VI + sec t dt Hint:"V1 + sec t dt = - V1 + sec t dt
> For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve p = 20 + 1/10x, and the supply curve p
> If a supply curve is modeled by the equation p = 200 + 0.2x3/2, find the producer surplus when the selling price is $400.
> The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for |f (4)(x) | by hand. But computer algebra systems have no problem computing and graphing it, so we can easily find a v
> The supply function px (x) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so ps is an increasing function of x
> Let f (x) = xe-x if x > 0 and f (x) = 0 for if x < 0. (a). Verify that is a probability density function. (b). Find P (1 < X < 2).
> Let f (x) = 3/64 x √16 – x2, for 0 < x < 4 and f (x) = 0 for all other values of x. (a). Verify that f is a probability density function. (b). Find P (X < 2).
> Let f (t) be the probability density function for the time it takes you to drive to school in the morning, where t is measured in minutes. Express the following probabilities as integrals. (a). The probability that you drive to school in less than 15 min
> Let f (x) be the probability density function for the lifetime of a manufacturer’s highest quality car tire, where is measured in miles. Explain the meaning of each integral. *40,000 (а) 30,000 f(x) dx (b) m f(x) dx 25,000
> A curve is defined by the parametric equations Find the length of the arc of the curve from the origin to the nearest point where there is a vertical tangent line. cos u - du sin u -du y = %3D
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(r) = V? + 4 dx
> A circular disk of radius is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a heig
> Evaluate limx→o = 1/x fx0(1 – tan 2t)1/t, dt.
> Find the point on the parabola y = 1 – x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.
> Estimate the area of the region enclosed by the loop of the curve x = t3 -12t, y = 3t2 + 2t + 5.
> If f is a differentiable function such that f (x) is never 0 and fx0 f (t) dt = [f (x)]2 for all x, find f.
> If f (x) = fx0x2 sin (t2) dt, find f'(x).
> If x sin πx = fx20f (t) dt, where f is a continuous function, find f (4).
> If f40e(x-2)4 dx = k, find the value of f40xe(x-2)4 dx.
> Show that |sin x – cos x | < √2 for all x.
> A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tange
> Evaluate the integral. f-23 (x2 – 3) dx
> Find d2/dx2 fx0(fsint1 √1 + u4, du) dt.
> One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceil
> Evaluate limx→∞ (1/√n √n + 1, + 1/√n √n + 2, + . . . + 1/√n √n + n.
> The figure shows a horizontal line y = c intersecting the curve y = 8x – 27x3. Find the number c such that the areas of the shaded regions are equal. y y = 8x – 27x y=c
> (a). Let An be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2π/n, show that (b). Show that limn→∞ An =
> Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. (a). If the target weight is 500 g, what is the probability that the machine produces a box with l
> Evaluate f10(3√1 – x7 - 7√1 – x3) dx.
> Suppose f is continuous, f (0) = 0, f (1) = 1, f'(x) > 0, and f10f (x) dx = 1/3. Find the value of the integral f10f-1 (y) dy.
> Evaluate f∞-1(x4/1 + x6) dx.
> Determine the values of the number for which the function f has no critical number: f(x) = (a? + a – 6) cos 2x + (a – 2)x + cos 1 %3D
> Evaluate the integral. f12 (2x – ex) dx
> Suppose the graph of a cubic polynomial intersects the parabola when y = x2, x = 0, x = a and x = b, where 0 < a < b. If the two regions between the curves have the same area, how is b related to a?
> A triangle with area is cut from a corner of a square with side 10 cm, as shown in the figure. If the centroid of the remaining region is 4 cm from the right side of the square, how far is it from the bottom of the square? 10 cm
> Find the highest and lowest points on the curve x2 + xy + y2 = 12.
> For what value of a is the following equation true? + a = e lim x - a х —
> A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius r and center O and the initial position of is P (r, 0), and
> If f (x) = fg(x)01/√1 + t3, dt, where g (x) = fcosx0[1 + sin (t2)], dt, find f'(π/2).
> Find the approximations Ln, Rn, Tn and Mn for n = 5, 10 and 20. Then compute the corresponding errors EL, ER and ET. and EM (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations ca
> A solid is generated by rotating about the -axis the region under the curve y = f (x), where f is a positive function and x > 0. The volume generated by the part of the curve from x = 0 to x = b is b2 for all b > 0. Find the function f.
> In lies on ∆ABC, D, lies on AB, |CD| = 5cm, |AD| = 4 cm, |BD = 4cm, and CD ⊥ AB. Where should a point p be chosen on CD so that the sum |PA| + |PB| + |PC| is a minimum? What if |CD| = 2 cm?
> Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.
> (a). Use an improper integral and information from Exercise 27 to find the work needed to propel a 1000-kg satellite out of the earth’s gravitational field. Exercise 27: (a). Newton’s Law of Gravitation states that t
> Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.
> Find the point on the hyperbola xy = 8 that is closest to the point (3, 0).
> A water skier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp? -15 ft
> A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?
> A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3/s, how fast is the water level rising when the water is 5 cm deep?
> The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is π/6?
> Evaluate the limit. lim x→(π/2)- (tan x) cos x
> The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate f20 f (x) dx, where f is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each
> Evaluate the limit. lim x→1+ (x / 1 – x – 1/ ln x)
> Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines. (a). The x-axis (b). The y-axis (c). y = 2
> Evaluate the integral. f-55 e dx
> A 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?
