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free/or 0.5$ |
| m52854 | (a) Use the table of integrals to evaluate F(x) = ∫ f(x) dx, where
f(x) = 1/(x√(1 - x2))
What is the domain of f and F?
(b) Use a CAS to evaluate F(x). What is the domain of the function that the CAS produces? Is there a discrepancy between this domain and the domain of the function F that you found in part (a)? |
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| m52857 | (a) Use trigonometric substitution to show that
∫ dx/√(x2+a2) = ln(x+√(x2+a2) + C
(b) Use the hyperbolic substitution to show that
∫ dx/√(x2 + a2) = sinh-1(x/a) + C
These formulas are connected by formula 3.11.3. |
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| m52858 | (a) Use trigonometric substitution to verify that
(b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a). |
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| m52859 | A variable force of 5x-2 pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from x= 1 ft to x = 10 ft. |
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| m52862 | A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of 5 gal / min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour? |
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| m52864 | 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.
a.
b. |
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| m52866 | A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour? |
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| m52867 | A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m3/min, how fast is the water level rising when the water is 30 cm deep? |
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| m52870 | (a) What are the domain and range of f?
(b) What is the -intercept of the graph of f?
(c) Sketch the graph of f.
F(x) = ln x + 2 |
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| m52872 | (a) What can you say about a solution of the equation y = - y2 just by looking at the differential equation?
(b) Verify that all members of the family y = 1/(x + C) are solutions of the equation in part (a).
(c) Can you think of a solution of the differential equation y = - y2 that is not a member of the family in part (b)?
(d) Find a solution of the initial-value problem y = - y2 y(0) = 0.5 |
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| m52875 | (a) What does it mean for f to be continuous at a?
(b) What does it mean for f to be continuous on the interval (-∞, ∞)? What can you say about the graph of such a function? |
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| m52876 | (a) What does it mean for f to be differentiable at a?
(b) Sketch the graph of a function that is continuous but not differentiable at a = 2. |
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| m52877 | (a) What is a differential equation?
(b) What is the order of a differential equation?
(c) What is an initial condition? |
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| m52878 | (a) What is a function? What are its domain and range?
(b) What is the graph of a function?
(c) How can you tell whether a given curve is the graph of a function? |
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| m52879 | (a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph?
(b) If f is a one-to-one function, how is its inverse function f-1 defined? How do you obtain the graph of f-1 from the graph of f? |
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| m52880 | (a) What is a sequence?
(b) What does it mean to say that limn ( 8 an?
(c) What does it mean to say that limn ( ( an? |
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| m52881 | (a) What is an alternating series?
(b) Under what conditions does an alternating series converge?
(c) If these conditions are satisfied, what can you say about the remainder after terms? |
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| m52882 | (a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function.
(b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function. |
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| m52883 | (a) What is parametric curve?
(b) How do you sketch a parametric curve? |
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| m52884 | (a) What is the average value of a function f on an interval [a, b]?
(b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation? |
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