Notes
... One way to do this is pick the team first, which can be done in m ways, k since order is not important. Then we can choose any of the k members of the m team to be the captain. Thus the answer is k k . But this isn’t necessarily the way you’d actually do it in practice: you might choose the captai ...
... One way to do this is pick the team first, which can be done in m ways, k since order is not important. Then we can choose any of the k members of the m team to be the captain. Thus the answer is k k . But this isn’t necessarily the way you’d actually do it in practice: you might choose the captai ...
Programming Contest Practice Problems
... smallest numbers of coins necessary to obtain 15, assuming that we can use all 4 coins), we have to compute the whole array. However, an optimization is possible, because we do not need to have access to each row of the array at the same time. Show how to compute the last row using only two vectors: ...
... smallest numbers of coins necessary to obtain 15, assuming that we can use all 4 coins), we have to compute the whole array. However, an optimization is possible, because we do not need to have access to each row of the array at the same time. Show how to compute the last row using only two vectors: ...
Review of Real Numbers
... 1. Can a number be both a rational and an irrational number? 2. Are there any integers that are not rational numbers? Are there any rational numbers that are not integers? 3. What are the real numbers? 4. Is there a smallest positive rational number? Is there a largest positive rational number? 5. G ...
... 1. Can a number be both a rational and an irrational number? 2. Are there any integers that are not rational numbers? Are there any rational numbers that are not integers? 3. What are the real numbers? 4. Is there a smallest positive rational number? Is there a largest positive rational number? 5. G ...
Advanced Calculus
... The issue of convergence must not be ignored or casually assumed. The following example illustrates this: Consider the sequence ( xn ) defined by x1 1, xn 1 2 xn 1. Assuming the ‘convergence’ (actually wrong! The sequence is not convergent) with lim( xn ) x, we would obtain x 2x 1, so t ...
... The issue of convergence must not be ignored or casually assumed. The following example illustrates this: Consider the sequence ( xn ) defined by x1 1, xn 1 2 xn 1. Assuming the ‘convergence’ (actually wrong! The sequence is not convergent) with lim( xn ) x, we would obtain x 2x 1, so t ...
Section 4.1
... 1. There are an infinite number of primes. 2. Every natural number can be factored into a product of primes (Fundamental Theorem of Arithmetic). Determining the Primality of Larger Positive Integers Because of its use in cryptology and other applications, mathematical techniques for determining whet ...
... 1. There are an infinite number of primes. 2. Every natural number can be factored into a product of primes (Fundamental Theorem of Arithmetic). Determining the Primality of Larger Positive Integers Because of its use in cryptology and other applications, mathematical techniques for determining whet ...
Section 4.3 - math-clix
... Division of Polynomials When dividing a polynomial P(x) by a divisor d(x), a polynomial Q(x) is the quotient and a polynomial R(x) is the remainder. The quotient must have degree less than that of the dividend, P(x). The remainder must be either 0 or have degree less than that of the divisor. P(x) ...
... Division of Polynomials When dividing a polynomial P(x) by a divisor d(x), a polynomial Q(x) is the quotient and a polynomial R(x) is the remainder. The quotient must have degree less than that of the dividend, P(x). The remainder must be either 0 or have degree less than that of the divisor. P(x) ...
fn (x) = f(x). n2x if 0 ≤ x if 1 n ≤ x 0 if 2 n ≤ x ≤1
... The first example fn(x) = xn is not uniformly convergent. Although, if you picked any number smaller than 1 to “end” the interval, it would be. Informally, the reason it is not uniformly convergent is because the closer you get to 1, the bigger N you will need to get within epsilon of zero. The way ...
... The first example fn(x) = xn is not uniformly convergent. Although, if you picked any number smaller than 1 to “end” the interval, it would be. Informally, the reason it is not uniformly convergent is because the closer you get to 1, the bigger N you will need to get within epsilon of zero. The way ...