| № |
Condition |
free/or 0.5$ |
| m52991 | An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation.
(a)
(b) |
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| m52997 | An object with mass is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s(t) and the acceleration is a = v (t). If g is the acceleration due to gravity, then the downward force on the object is mg - cv, where c is a positive constant, and Newton s Second Law gives
m dv / dt = mg - c v
(a) Solve this as a linear equation to show that
v = mg / c (1 - e-ct / m)
(b) What is the limiting velocity?
(c) Find the distance the object has fallen after t seconds. |
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| m53000 | An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t) = 100e-0.01t liters per minute. How much oil leaks out during the first hour? |
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| m53028 | Approximate the sum of the series correct to four decimal places.
a.
b. |
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| m53064 | Assume that all of the functions are twice differentiable and the second derivatives are never 0.
(a) If f and are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I.
(b) Show that part (a) remains true if f and are both decreasing.
(c) Suppose f is increasing and is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesn t the argument in parts (a) and (b) work in this case? |
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| m53067 | Assume that is a one-to-one function.
(a) If f(6) = 17, what is f-1 (17)?
(b) If f-1 (3), what is f(2)? |
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| m53096 | At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM? |
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| m53102 | At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 Ib/in2. Below the surface, the water pressure increases by 4.34 Ib/in2 for every 10 ft of descent.
(a) Express the water pressure as a function of the depth below the ocean surface.
(b) At what depth is the pressure 100 Ib/in2? |
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| m53103 | At what point of the curve y = cosh x does the tangent have slope 1? |
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| m53104 | At what point on the curve y = [In (x + 4)]2 is the tangent horizontal? |
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| m53105 | At what points does the curve
x = 2a cos t - a cos 2t y = 2a sin t - a sin 2t
Here vertical or horizontal tangents? Use this information to help sketch the curve. |
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| m53106 | At what points on the curve x = 2t3, y = 1 + 4t - t2, does the tangent line have slope 1? |
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| m53107 | At what points on the curve y = sin x + cos x, 0 ≤ x ≤ 2π, is the tangent line horizontal? |
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| m53110 | (b) Deduce that lim n→( xn/n! = 0 for all x. |
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| m53111 | (b) Find a value of n so that sn is within 0.00005 of the sum. Use this value of n to approximate the sum of the series. |
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| m53112 | (b) Graph f(x) = (sin x)/x. How many times does the graph cross the asymptote? |
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| m53113 | (b) Use part (a) to find the Maclaurin series for sin- 1 x. |
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| m53143 | Between 0°C and30°C, the volume (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula
V = 999.87 - 0.06426T + 0.0085043T2 - 0.0000679T3
Find the temperature at which water has its maximum density. |
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| m53145 | Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70oF and 173 chirps per minute at 80oF.
(a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. |
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| m53148 | Boyle s Law states that when a sample of gas is compressed at a constant pressure, the pressure P of the gas is inversely proportional to the volume V of the gas.
(a) Suppose that the pressure of a sample of air that occupies 0.106 m3 at 25°C is 50kPa. Write V as a function of P.
(b) Calculate dV/dP when p = 50 kPa. What is the meaning of the derivative? What are its units? |
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