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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.
Solve the subsequent proportion: Example: Solve the subsequent proportion for x. Solution: 5:x = 4:15 The product of the extremes is (5)(15) = 75. The produ
Horizontal asymptotes : Such as we can have vertical asymptotes defined in terms of limits we can also have horizontal asymptotes explained in terms of limits. Definition
Carmen bought 3 pounds of bananas for $1.08. June paid for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy?
The following graph shows the growth of the median home value in a particular region of the United States starting in 1996. The graphs starts in 1996 and shows the trend through t
what is the meaning of volume of plane figures?
Compute the value of the following limit. Solution: Notice as well that I did say estimate the value of the limit. Again, we will not directly compute limits in this sec
Method to determine solution is absolute minimum/maximum value Let's spend a little time discussing some methods for determining if our solution is in fact the absolute minimum
On dividing p(X)=5x^(4)-4x^(3)+3x^(2)-2x+1 by g(x)=x^(2)+2 if q(x)=ax^(2)+bx+c, find a,b and c.
six men and Eight boys can finish a piece of work in 14 days while eight men and twelve boys can do it in 10 days. Find the time taken by 1man alone and that by 1boy alone to fin
What is Pythagorean Triples? A set of three numbers a, b, and c that can satisfy the equation A 2 +b 2 = c 2 , is called a Pythagorean triple. The following is a list of
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