you must memorize these!
... 2x + y = 15 4x + 3y = 37 . By Substitution: Step 1: 2x + y = 15 y = 15 – 2x Step 2: (substitution) 4x + 3(15 – 2x) = 37 Step 3: (distribute) 4x – 6x + 45 = 37 Step 4: (subtract 45 from both sides) -2x = -8 Step 5: (multiply both sides by –1) 2x = 8 Step 6: (divide both sides by 2) x = 4 Step 7: subs ...
... 2x + y = 15 4x + 3y = 37 . By Substitution: Step 1: 2x + y = 15 y = 15 – 2x Step 2: (substitution) 4x + 3(15 – 2x) = 37 Step 3: (distribute) 4x – 6x + 45 = 37 Step 4: (subtract 45 from both sides) -2x = -8 Step 5: (multiply both sides by –1) 2x = 8 Step 6: (divide both sides by 2) x = 4 Step 7: subs ...
![[2014 question paper]](http://s1.studyres.com/store/data/008844916_1-6bcf832b30821138ae4b8d8f7d9aa434-300x300.png)
[2014 question paper]
... 7. Let f, g be real valued continuous functions on [0, 1]. Which of the following statements is/are always true? (a) f is uniformly continuous. (b) g is bounded. (c) If f (r) = 1 − g(r) for all rational numbers r ∈ [0, 1] then f (x) = 1 − g(x) for all x ∈ [0, 1]. (d) The function v defined by v(x) = ...
... 7. Let f, g be real valued continuous functions on [0, 1]. Which of the following statements is/are always true? (a) f is uniformly continuous. (b) g is bounded. (c) If f (r) = 1 − g(r) for all rational numbers r ∈ [0, 1] then f (x) = 1 − g(x) for all x ∈ [0, 1]. (d) The function v defined by v(x) = ...
CS342 Data Structures - William Paterson University
... • Similar to big-O situation, there can be infinite number solutions for c and N pair. For practical purposes, only the closest s are the interest, which represents the largest lower bound. ...
... • Similar to big-O situation, there can be infinite number solutions for c and N pair. For practical purposes, only the closest s are the interest, which represents the largest lower bound. ...
PDF
... Theorem. Given a Mersenne number m = 2n − 1 (with n a nonnegative integer), the Collatz sequence starting with m reaches 3n − 1 in precisely 2n steps. Also, the parity of such a sequence consistenly alternates parity until 3n − 1 is reached. For example, given 22 − 1 = 3 gives the Collatz sequence 3 ...
... Theorem. Given a Mersenne number m = 2n − 1 (with n a nonnegative integer), the Collatz sequence starting with m reaches 3n − 1 in precisely 2n steps. Also, the parity of such a sequence consistenly alternates parity until 3n − 1 is reached. For example, given 22 − 1 = 3 gives the Collatz sequence 3 ...
Kevin McGown: Computing Bernoulli Numbers Quickly
... order to compute (4), it is useful to first compute all primes p ≤ N ; this may be done quickly using the Sieve of Eratosthenes. One may also compute the product in (2) via a sieving process. Finally, for the value of N we may choose any integer greater than or equal to the one specified in (3), so ...
... order to compute (4), it is useful to first compute all primes p ≤ N ; this may be done quickly using the Sieve of Eratosthenes. One may also compute the product in (2) via a sieving process. Finally, for the value of N we may choose any integer greater than or equal to the one specified in (3), so ...
Sequences, Functions and Graphs
... Using your knowledge of the meanings of the numbers in the function, draw the graph of y=7-3x Which of these graphs are parallel: y=2x+2, y=x+2, y=2-2x, y=½x+2, y=2x-½, y=½+2x ? Which of these graphs have the same y-intercept: y=2x+2, y=x+2, y=2-2x, y=½x+2, y=2x-½, y=½+2x ? Plot the graph of ...
... Using your knowledge of the meanings of the numbers in the function, draw the graph of y=7-3x Which of these graphs are parallel: y=2x+2, y=x+2, y=2-2x, y=½x+2, y=2x-½, y=½+2x ? Which of these graphs have the same y-intercept: y=2x+2, y=x+2, y=2-2x, y=½x+2, y=2x-½, y=½+2x ? Plot the graph of ...
The Basel Problem - David Louis Levine
... • Euler also proved a profound formula that equates a sum of powers of all the natural numbers with a product of powers of all the prime numbers ...
... • Euler also proved a profound formula that equates a sum of powers of all the natural numbers with a product of powers of all the prime numbers ...
Example 1: Determine the possible number of positive and negative
... • The number of positive real zeros of a polynomial function, P(x), with real coefficients, is equal to the number of variations in sign of the terms of P(x) or is less than this number by a multiple of 2. • The number of negative real zeros is equal to the number of variations in sign of the terms ...
... • The number of positive real zeros of a polynomial function, P(x), with real coefficients, is equal to the number of variations in sign of the terms of P(x) or is less than this number by a multiple of 2. • The number of negative real zeros is equal to the number of variations in sign of the terms ...
