Here are the first twenty H problems.
Notes: (1) >= is the equivalent of "greater than or equal to."
(2) "X^Y" is read as "X to the power of Y."
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Show, for n=2,3,4,..., that
1 + 1/2 + 1/3 + ... + 1/(n-1) > ln(n) > 1/2 + 1/3 + 1/4 + ... + 1/n.
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If f maps the reals to the reals, is differentiable, and f'=f, then show that f
is of the form f(x) = C(e^x).
Hint: Consider (d/dx)[(f(x))/(e^x)].
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Suppose b is a member of the reals and x>0. Show that:
(d/dx)[x^b] = b(x^(b-1)).
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Differentiate the following:
(x^x)(1+ln(x)).
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Suppose f maps the reals to the reals and is differentiable, and
suppose the limit as x goes to infinity of f'(x) = 3.
Does this imply that f has an asymptote which is a line of slope 3?
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(A) Which Trig Identity is obtained by flipping the unit circle over the 45
degree line?
(B) Which Trig Identity is obtained by flipping the unit circle over the 60
degree line?
(C) Which Trig Identity is obtained by rotating the unit circle 45 degrees
in the positive direction?
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In "The Wizard of Oz", what error(s) did the scarecrow make in stating the
Pythagorian theorem?
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The Unicyle Problem.
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Note: The "_" character denotes subscript.
Let I_n(x) := Indefinite Integral of (1/((x^2 + 1)^n)) dx.
Using integration by parts, derive a formula of the form
I_{n+1} (x) = f(x) + [g(n) times I_n(x)].
<<>>
Referring to Question A1 (h) on Take Home Exam A,
(A) Calculate f composed with itself one million times, and
(B) Calculate g composed with itself one million times,
for all integer values of x from 1 to 12, inclusive.
<<>>
Two solid cylinders, both of radius R, intersect so that their axes of symmetry
meet at right angles. Let S denote the resulting solid of intersection.
Determine the Volume of S.
<<>>
Determine the speed of a bicylist in parsecs per fortnight if she is traveling
at a speed of 20 miles per hour.
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(A) Given an umbrella of length L feet and a uniform density of P lb/ft.
Using the center of mass, how much energy is required for the unbrella to
be lifted from theta radians (with respect to the ground) to pi/2 radians.
(B) Umbrella problem, pt 2.
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A bucket o' cement weighing 200 pounds is hoisted by means of a windless of
negligable mass and radius from the ground to the tenth story of an office
building, 80 feet above the ground.
Assume the chain has a density of one pound per foot, and that the chain
hangs from the other side of the windlass as the backet o' cement is
hoisted.
Find the work, W, required to make the lift.
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(A) For hemidisc,H, of radius rho whose origin is at (0,0), determine y-bar
(the y-coordinate at the centroid of H; x-bar = 0).
(B) Determine y-bar for the the BOUNDARY (perimeter) of the above
mentioned hemidisc.
<<>>
"Burried Treasure Problem"
Sitting by the beach on the coast of New Zealand, a message in a bottle
floats up to you, and being the curious person you are, you immediately
open it. The message turns out to be directions to buried treasure, which
reads as follows:
The treasure, m'lad, lies midway between two spikes that ye put
into the ground.
To place the first spike, stand at the gallows and walk to the
rock, countin' yer steps. Now turn right (by 90 degrees, matey) and
walk an equal distance. Place the spike in the ground.
Return to the gallows. Now walk to the tree, countin' yer steps.
Turn left (by 90 degrees) and walk an equal distance. Place the
second spike in the ground.
You immediately find the rock and the tree; unfortunately, the gallows have
rotted beyond recognition over the past 250 years. Using complex numbers,
show that it is not necessary to know the location of the gallows to find
the buried treasure.
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Note: X^n := "X to the power of n"
Compute the summation of [3^(-n)] from n=14 to infinity
<<>>
Given: A domino weighs 1 ounce, and is 2 inches long and one quarter
inch high.
(A) How many dominos do you need to equal the weight of the Statue of
Liberty?
(B) After discarding the number of dominos calculated in part (A), how
many more dominos are needed to balence the statue of liberty five miles out
over the Grand Canyon? How high in the air is she? Is she higher than
the moon?
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Using the telescoping series idea, calculate the summation from n=1 to
infinity of (1/(N(N+2))).
<<>>
Note: "x_n" is read as "x sub n".
Suppose you have a decreasing series, (y_n)_1^infinity,
of positive numbers, which decreases TO ZERO.
Prove that
the sum from n=1 to infinity of
[y_n - y_(n+1)]/[y_n + y_(n+1)]
equals infinity.
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