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Suppose instead of working on the Euclidean plane we study geometry on a sphere in (Euclidean) three space. We interpret point to mean any point on the sphere and we interpret line to mean any great circle on the sphere (that is any circumference of the sphere).
a. Is Euclids parallel postulate true in this setting?
b. How would you define the angle which two great circles make at a point where they intersect on the sphere?
c. How would you define a triangle on the sphere? Give some examples.
d. Is the sum of the angles in a triangle greater than, less than, or equal to 180 degrees in this setting?
e. Does the angle sum depend on the area of the triangle? How?
f. How does part d relate to showing that the parallel postulate implies the triangle postulate?
Suppose we want to determine the (binomial) probability (p) of getting 5 heads in 15 flips of a 2-sided coin. Using the Binomial table in the appendix of the text, what values of n, x, and p would we use to look up this probability, and what would..
Use Gram-Schmidt algorithm to the sequence {x_1,x_2, x_3} to find an orthonormal basis of S. Use the result above to find the QR factorization of the matrix A=(x_1l x_2l x_3).
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Suppose f and g are functions analytic in a domain D. If z_n is a bounded sequence of distinct points in D and if f(z_n) = g(z_n) for all n, show that f(z) = g(z) for all z in D.
Let A be the real 2x2 matrix with bc greater than or equal to 0. Prove there exists a real 2x2 invertible matrix S so that S^-1 A S is either diagonal or of the form
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Find the probability of 14 or more widowed people in a random sample of 290 bowling league members?
If the current levels of input are x = 10 and y = 20, use calculus to estimate the change in input y that should be made to offset an increase of 0.5 in input x so that output will be maintained at its current level.
Matrice operations
Translate the following argument in symbolic form and determine whether it's logically correct by constructing a truth table. If affirmative action policies are adopted, then minorities will be hired. If minorities get hired, then discrimination wi..
Find the derivatives of the following functions (Do not simplify): f(x) = 20x^(11/5) - 13x^(5/4) - 4x(-7/2)
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