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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.
Evaluate the given definite integral. Solution Let's begin looking at the first way of dealing along with the evaluation step. We'll have to be c
round 200 to nearest hundreds
Describe the Properties of Inequalities ? Postulate In comparing two quantities, say a and b, there are exactly three possibilities. (1) a is less than b. (a b)
1. a) Given a digraph G = (V,E), prove that if we add a constant k to the length of every arc coming out from the root node r, the shortest path tree remains the same. Do this by
Proof of the Properties of vector arithmetic Proof of a(v → + w → ) = av → + aw → We will begin with the two vectors, v → = (v 1 , v 2 ,..., v n )and w? = w
Recognizes the absolute extrema & relative extrema for the given function. f ( x ) = x 3 on [-2, 2] Solution :
Explain the Graphical Technique of Linear Equations by using this figure.
explanation
a) Determine the distance traveled among t = 0 and t =∏/2 by a particle P(x, y) whose position at time t is given by Also check your result geometrically. (5) b) D
Six times as many people voted in the 2012 election as in the 2008 election.If 162 people voted in 2008,how many people voted in both elections?
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