© 2005 by Karl Hahn
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Archived Forum Discussion for May/June 2005
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Some bacteria reproduce themselves, others do not. If 80 percent of a certain culture alwasy reproductive, and there are no deaths, show that the original population is eventually multiplied by 5.
thats all that is given what is the series? or what is done? gabe el paso, tx USA - Friday, June 24, 2005 at 03:33:52 (EDT) |
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I would like to hear from people and hear their comments about my question.
"Where and how is calculus used in people's lives, either for a profession or for everyday use?"
Thanks Sarah Alexandria, VA USA - Monday, June 13, 2005 at 13:07:29 (EDT) |
Reply to Aaron:
Remember that
So integrate the last expression. Use the substitution,
u = cos(x) and du = -sin(x)dx. Break the integral
you get in u into partial fractions and integrate them.
Then back-substitute and put the pieces together using log identities. |
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Reply to Crystal:
Presumably you are allowed to use measuring equipment like rulers and calipers.
And presumably the wine glass is radially symmetric about its axis.
Have a look at this
and see if you can't see the parallel on how you would estimate the volume
of a wine glass by using calipers to measure the diameter of the glass at
different heights above its base. To find the amount of glass, measure the
thickness of the glass at each height and combine that with the diameter
measurement at the respective heights to find the amount of glass in each
"hoop" of glass. Karl <Click to Send Email to Karl> USA - Thursday, May 26, 2005 at 22:57:06 (EDT) |
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How can I integrate the cosecant function? Aaron Bronx, NY USA - Wednesday, May 25, 2005 at 18:56:24 (EDT) |
Reply to Martin:
I have included a plot of such a path to the right.
For the general problem of a particle moving in polar coordinates as a function
of time, t, we use the unit vectors, r for radial
unit vector, and θ for the tangential unit vector. Unlike
unit vectors in Cartesian coordinates, these unit vectors are not constant
with respect to each other. You have, for their derivatives:
drand dθIf you have a function like r = eaθ, and θ(t) = bt, then your function really is
f(t) = eabtr
where r is now a function of time (because the direction
in which it points is constantly changing with θ). So by use of the
chain rule, you find:
dr dθwhere dθYou can use the equations so far to find the second derivative of f with respect to time also to give you acceleration. If you apply everything correctly you will find that you get terms corresponding to "centrifugal" and "Coriolis". In the case you give, the Coriolis force will be nonzero because r is changing with time. With your equation I find for acceleration: f" = [(ab)2 - b2]eabtr + 2ab2eabtθThe (ab)2 term is the radial acceleration due strictly to the second derivative of the spiral. The -b2 term is the centripedal acceleration (which is responsible for the sensation of "centrifugal" force). The remaining term is the Coriolis term. I hope this is the kind of treatment you were looking for. Karl <Click to Send Email to Karl> USA - Wednesday, May 25, 2005 at 18:30:41 (EDT) |
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Obejective: Twofold
1. Determine, by using methods of calculus, the volume of liquid the wine glass wil hold. Assume it will be filled to the lip of the wine glass.
2. Determine, by using methods of estimation, curve fitting, and calculus, the volume of glass contained in your wine glass.
Approaches to the problem:
You cannot use a graduated measuring device to directly measure the volume that the wine glass will hold.
You cannot use volume displacement to measure the volume of glass in the wine glass
You cannot use specific gravity of the glass to measure the volume of the glass.
You cannot destroy the wine glass in any way shape or form.
You must develop, with the aid of a graphing calculator, equations that can help you determine the volume of the wine glass and the volume of the glass in the goblet. You must use methods of calculus, using the aforementioned equations to determine these volumes. Crystal Richland, WA USA - Wednesday, May 25, 2005 at 01:38:37 (EDT) |
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Hello!
I think this might be a bit trickier, but here it goes:
If a particle is moving (I am not sure if this is relevant, but- at a constant angular speed) along a trajectory line that is a logarithmic (aka equiangular) spiral, does the tangential vector's always being at a constant angle to the radial vector at any given point have a physical interpretation? (I know the tangential vector itself represents velocity, but how about the angle it forms with the radial vector?) Martin Bulgaria - Monday, May 23, 2005 at 14:35:45 (EDT) |
Reply to Rai:
Write the coeffiecients inside the trig functions as fractions reduced to
lowest terms. So the first one would be
Y = sin((2/1)x) - sin((3/4)x)Now find the least common denominator. In this case its 4. Then multiply that common denominator by 2π to get the period. Karl <Click to Send Email to Karl> USA - Saturday, May 14, 2005 at 19:05:16 (EDT) |
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I'm in a precalculus class, and we were given a challenge I'm having trouble finding. I was hoping someone out there might now. Here's the exact text:
Algebraic sums, differences, products, and quotients of periodic functions produce periodic functions. Determine the period of each component function and the period of the combined function. based on this information determine an algorithm for finding the period of the combined function a. Y= sin2x-sin.75x b. Y= (cos3x)/(cos.25x) c. Y= 2sin4x+3sinx My main trouble is the algorithm. Any engineers able to lend me a hand? =)Rai USA - Thursday, May 12, 2005 at 13:38:35 (EDT) |
Reply to Ralph Swanson:
Look at the page for the area of a circle. The method Archimedes used to approximate π
is shown there. I am including a diagram from that page here (shown to the
right). Observe that each of the lengths identified in the diagram is
a multiple of the circle's radius, r. So when you take ratios
of those lengths, as you would when you approximate π, the
r's always cancel. That means that the approximation is independent
of the circles radius.
As for the question of proving that all circles are similar, look at the
way you define a circle. It is the locus of points on a plane that are
equidistant from some specified point. To prove similarity between two circles,
is sufficient to show that you can translate and scale them so that they
coincide. Every circle can be translated so that its center is at the origin.
So translate both circles in question to the origin. Now scale each one by
the reciprocal of its radius, and they will then coincide. All that remains
for you to show is how the definition given above implies this. |
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I'm looking for a proof that pi is a constant. Everyone knows it is and all discussions start with assuming that without question but I'm interested in seeing an actual proof. There is one of course but I can't seem to find it. Any help or suggestions in finding a source would be very much appreciated. I'm interested in seeing an actual proof and not in hints or suggestions on how such a proof could be constructed. I believe the key starting point is proving all circles are similar. Ralph Swanson Laurel, MT USA - Monday, May 09, 2005 at 20:32:04 (EDT) |
Reply to Jeff Bechtel:
If r is the rate or growth (r = $500,000/year per year
note the units carefully -- sales are $ per year, so growth must be
$ per year per year) and
x0 is the current sales rate (x0 = $2,000,000/year), then sales rate at time t years after
this year is:
x(t) = x0 + rtSo in part 1 of the problem, t = 1/2. Apply the above formula to that. For part 2, remember that f'(5) approximates f(5.5) - f(5)So equate that with f'(5) (whose value is given in the problem), then use algebra to solve for f(5.5) (note that the value of f(5) is also given in the problem). For part 3, you have q(p) = 40,000,000 - 3000p2. Put in $40 and $45 for p and find each consequent value of q. Take the second minus the first and divide that by the difference between the two p's. For part 4, putting x+h into the function gives
4(x+h)2 - 2(x+h) = 4x2 + 8xh + 16h2 - 2x - 2h. Use algebra to subtract from that the expression for f(x),
which is 4x2 - 2x. You will get some cancellations. |
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I'm currently deployed to Germany for the Air Force and am trying to finish a class that I started
1. A company's sales after 5 years of existence are $2,000,000 per year and growing at $500,000 per year.
What will the rate of sales be in 6 months?
2. Suppose that f(5) = 2 and f '(5) = .5.
What is the best estimate for f(5.5)
3. Suppose that the demand (q: in barrels) for oil is a function of the price (p: in dollars) and is given by the formula q = 40,000,000 - 3,000p^2. What is the average rate of change of demand if the price goes from $40 to $45? Give the units of the answer.
4. Let f(x) = 4x^2 - 2x. Compute and simplify: f(x+h) - f(x) Jeff Bechtel Amherst, Ohio USA - Sunday, May 08, 2005 at 10:52:48 (EDT) |
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