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... The vertical line test can be used to determine whether or not an equation in two variables specifies a function. The functions specified by equations of the form y = mx + b, where m is not equal to 0, are called linear functions. Functions specified by equations of the form y = b are called con ...
... The vertical line test can be used to determine whether or not an equation in two variables specifies a function. The functions specified by equations of the form y = mx + b, where m is not equal to 0, are called linear functions. Functions specified by equations of the form y = b are called con ...
Generating Functions and the Fibonacci Sequence
... the Fibonacci sequence is Fn+1 = Fn + Fn−1 for all n ≥ 2 with F0 = 0 and F1 = 1. So, the sequence would be 1, 1, 2, 3, 5, 8, 13... What if we wanted to find the 10th term of the Fibonacci sequence? This would not be a very difficult task, right? We could walk through the sequence and find the 10th t ...
... the Fibonacci sequence is Fn+1 = Fn + Fn−1 for all n ≥ 2 with F0 = 0 and F1 = 1. So, the sequence would be 1, 1, 2, 3, 5, 8, 13... What if we wanted to find the 10th term of the Fibonacci sequence? This would not be a very difficult task, right? We could walk through the sequence and find the 10th t ...
Floating-Point Representation and Approximation Errors
... k=1 appears to converge if the terms are added in the given (descending) order. Determine the least value N0 for which ...
... k=1 appears to converge if the terms are added in the given (descending) order. Determine the least value N0 for which ...
Full text
... natural number (i.e., k = 4, 5,8, 9,12, 13, etc.). Schinzel provided a general form for generating an infinite number of solutions to (2) when k = 4. He provided one case for k = 5 (viz., 1,1, 1,-1, - 1 ) , but failed to demonstrate any solutions at all for (2) when the values of k are of the form 4 ...
... natural number (i.e., k = 4, 5,8, 9,12, 13, etc.). Schinzel provided a general form for generating an infinite number of solutions to (2) when k = 4. He provided one case for k = 5 (viz., 1,1, 1,-1, - 1 ) , but failed to demonstrate any solutions at all for (2) when the values of k are of the form 4 ...
Solution
... (x + 5)(x + 2) x+5 x+2 There is no indicator of what the numerators should be, so there is work to be done to find them. If we let the numerators be variables, we can use algebra to solve. That is, we want to find constants A and B that make equation (1) below true for all x 6= −5, −2, which are the ...
... (x + 5)(x + 2) x+5 x+2 There is no indicator of what the numerators should be, so there is work to be done to find them. If we let the numerators be variables, we can use algebra to solve. That is, we want to find constants A and B that make equation (1) below true for all x 6= −5, −2, which are the ...
What Shape is the Universe?
... but I have the superstitious feeling that I should try to win a little favor with their ghosts before proceeding with my talk. In a spirit of humility, therefore, I offer a few apt lines from Boris Pasternak. I beg you, and any ghosts who may be listening, to excuse ...
... but I have the superstitious feeling that I should try to win a little favor with their ghosts before proceeding with my talk. In a spirit of humility, therefore, I offer a few apt lines from Boris Pasternak. I beg you, and any ghosts who may be listening, to excuse ...
Arithmetic Sequences
... Some sequences can be described using an algebraic model. An explicit rule allows you to find the n-th term of the function. Ex 3 – Find the first 4 terms of a sequence that follows the explicit rule f(n) = 3n + 2, assuming the domain starts with n = 1. What do you notice about the terms and how is ...
... Some sequences can be described using an algebraic model. An explicit rule allows you to find the n-th term of the function. Ex 3 – Find the first 4 terms of a sequence that follows the explicit rule f(n) = 3n + 2, assuming the domain starts with n = 1. What do you notice about the terms and how is ...
Non-standard Simplex Problems
... 1. If neccessary, rewrite the problem as a max. Minimizing C is equivalent as maximizing −C 2. If necessary, rewrite all constraints using ≤ signs 3. Introduce slack variables and set up initial simplex table 4. Scan the column of constants for negative numbers (a) If there are no negatives, complet ...
... 1. If neccessary, rewrite the problem as a max. Minimizing C is equivalent as maximizing −C 2. If necessary, rewrite all constraints using ≤ signs 3. Introduce slack variables and set up initial simplex table 4. Scan the column of constants for negative numbers (a) If there are no negatives, complet ...
Digital Subsequences
... Disregarding squares of the form NOO ... 0, where N is also a perfect square, how many numbers belong to this subsequence? b) Given the sequence of perfect cubes, 0,1,8,27,64,125, ... only those terms whose digits are all perfect cubes {O, 1, 8} are chosen. The first few terms are 0,1, 8, 1000, 8000 ...
... Disregarding squares of the form NOO ... 0, where N is also a perfect square, how many numbers belong to this subsequence? b) Given the sequence of perfect cubes, 0,1,8,27,64,125, ... only those terms whose digits are all perfect cubes {O, 1, 8} are chosen. The first few terms are 0,1, 8, 1000, 8000 ...