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Derive the Euler-Lagrange Differential Equation (rigorously). Explain any theorems or results that you use in the derivation (e.g. don't just say "by the fundamental theorem of the calculus of variations" without telling what the fundamental theorem of the calculus of variations is and what it means). Also explain in words and/or pictures why the Euler-Lagrange Differential Equation is believable (intuitively).
The Orthogonal group On is the set of all nxn matrices with entries in R such that AA^t = 1. The special orthogonal group SOn is the subset of all matrices in On with det(A) = 1.
The number of staff personnel needed to handle patients at a medical center is determined by the mean time that patients must wait before being attended to by a physician. For a random sample of 100 previously recorded emergencies, the sample mean..
A ski chalet roof is isosceles triangle and has a vertical angle of 36 degrees. Width of base is 14 m. What is the slant height of roof?
You can select 5 values for each of the two data sets and practically calculate the standard deviation and coefficient of variation for each data set to clarify your explanation.
An element e of a monoid M is called an idempotent if e^2 = e. If M is finite, show that some positive power of every element is an idempotent.
Discuss differences between neutral geometry and Euclidean geometry. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.
Assuming the amount of fill is normally distributed, what is the probability that the next randomly checked bottle contains more than 32.02 ounces.
The cost of renting tuxes for the Choral Society's formal is $20 down, plus $88 per tux.
Derive an identity for nCk via inclusion-exclusion by counting the k-multisets of [n] in which each element of [n] appears at most once. use p_i = "element i appears more than once in the multiset" as the i-th property, for 1
At Shaw's Market apples cost $10 per case and bananas cost $6 per case. If an order comes in for a total of 300 cases for $2,000, what was the specific number of cases of apples? (Hint, Let A = cases of apples)
Probability that Lounging Larry wins the horse race. The odds against Lounging Larry winning the horse race are 5:12.
Among U.S. cities with a population of more than 250,000 the mean one-way commute to work is 24.3 minutes. The longest one-way travel time is New York City, where the mean time is 37.9 minutes.
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