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Condition |
free/or 0.5$ |
| m52792 | (a) Suppose that ∑an and ∑bn are series with positive terms and ∑bn is divergent. Prove that if
Then ∑an is also divergent.
(b) Use part (a) to show that the series diverges.
(i)
(ii) |
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| m52795 | A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate the hydro static force on
(a) The shallow end,
(b) The deep end,
(c) One of the sides,
(d) The bottom of the pool. |
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| m52796 | A table of values for f, g, f , and g is given.
(a) If h(x) = f(g (x)), find h (1).
(b) If H(x) = g(f(x)). Find H (1). |
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| m52797 | A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for |
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| m52800 | A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank
(a) After minutes
(b) After 20 minutes? |
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| m52801 | A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis.
(a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank.
b) After 4000 ft-lb of work has been done, what is the depth of the water remaining in the tank? |
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| m52802 | A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes.
(a) If P is the point (15, 250) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t = 5, 10, 20, 25, and 30.
(b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines.
(c) Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.) |
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| m52803 | A tank is full of water. Find the work required to pump the water out of the spout. the fact that water weighs 62.5 lb/ft3.
(a)
(b) |
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| m52807 | A telephone line hangs between two poles 14 m apart in the shape of the catenary y = 20 cosh(x/20) - 15, where x and y are measured in meters.
(a) Find the slope of this curve where it meets the right pole.
(b) Find the angle between the line and the pole. |
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| m52808 | A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let s assume the rocket rises vertically and its speed is 600 ft/s when it has risen 3000 ft.
(a) How fast is the distance from the television camera to the rocket changing at that moment?
(b) If the television camera is always kept aimed at the rocket, how fast is the camera s angle of elevation changing at that same moment? |
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| m52810 | (a) The curve with equation y2 = 5x4 - x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2).
(b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.) |
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| m52811 | (a) The curve y = 1/(1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point (- 1, 1/2).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. |
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| m52812 | (a) The ellipse x2/a2 + y2 /b2 = 1 a > b is rotated about the -axis to form a surface called an ellipsoid, or prolate spheroid. Find the surface area of this ellipsoid.
(b) If the ellipse in part (a) is rotated about its minor axis (the -axis), the resulting ellipsoid is called an oblate spheroid. Find the surface area of this ellipsoid. |
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| m52816 | a. The terms of a series are defined recursively by the equations
a1 = 2
Determine whether (an converge or diverges. |
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| m52817 | (a) The van der Waals equation for moles of a gas is
where P is the pressure, V is the volume, and is the temperature of the gas. The constant R is the universal gas constant and and are positive constants that are characteristic of a particular gas. If remains constant, use implicit differentiation to find dv/dP.
(b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of V = 10 L and a pressure P = 2.5 atm, Use a = 3.592 L2-atm/mole2 and b = 0.04267 L/mole. |
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| m52818 | A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path back, arriving at the monastery at 7:00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days. |
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| m52821 | A torus is generated by rotating the circle x2 + (y - R)2 = r2 about the x-axis. Find the volume enclosed by the torus. |
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| m52827 | 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. |
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| m52828 | (a) Try to find an appropriate viewing rectangle for
F(x) = (x -10)3 2-x.
(b) Do you need more than one window? Why? |
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| m52829 | A uniform disk with radius 1 m is to be cut by a line so that the center of mass of the smaller piece lies halfway along a radius. How close to the center of the disk should the cut be made? (Express your answer correct to two decimal places.) |
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