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1. (‡) Prove asymptotic bounds for the following recursion relations. Tighter bounds will receive more marks. You may use the Master Theorem if it applies.
1. C(n) = 3C(n/2) + n
2. G(n) = G(n - 1) + 1/n
3. I(n) = I(n/2) + n/ lg(n)
2. Define a (p,q)-tree as a rooted tree where every internal node has between p and q (inclusive) children. Use the Master Theorem to give asymptotic bounds for the height of the tree. You can assume both p and q are constants with 2 ≤ p ≤ q.
3. (‡) Dominos
A 2 × 10 rectangle filled with ten dominos, and a 2 × 2 × 10 box filled with ten slabs.
1. A domino is a 2×1 or 1×2 rectangle. How many different ways are there to completely fill a 2 × n rectangle with n dominos?
2. A slab is a three-dimensional box with dimensions 1 × 2 × 2, 2 × 1 × 2, or 2 × 2 × 1. How many different ways are there to fill a 2 × 2 × n box with n slabs? Set up a recurrence relation and give reasonable exponential upper and lower bounds.
Consider a class of 55 students. The student names are placed in a hat and 3 names are randomly drawn without replacement. a) If the first person drawn was named the class presi
From a fixed point directly in front of the center of a bull's eye, Kim aims two arrows at the bull's eye. The first arrow nicks one point on the edge of the bull's eye; the second
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For the initial value problem y' + 2y = 2 - e -4t , y(0) = 1 By using Euler's Method along with a step size of h = 0.1 to get approximate values of the solution at t = 0.1, 0
(1 0 3 21 -1 1 -1 1) find A-1
test is tomorrow, don''t know anything lol, please help
This topic is specified its own section for a couple of purposes. Firstly, understanding direction fields and what they tell us regarding a differential equation as well as its sol
All differential equations will doesn't have solutions thus it's useful to identify ahead of time if there is a solution or not. Why waste our time trying to get something that doe
An inground pool is pooring with water. The shallow end is 3 ft deep and gradually slopes to the deepest end, which is 10 ft deep. The width of the pool is 30 ft and the length is
The area of a square is 64 cm 2 . What is the length of one side of the square? To find out the area of a square, you multiply the length of a side through itself, because all
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