Proofs

== == Cos2x - 1 = [1 - 2sin2 x] - 1 = - 2sin2 x; so [Cos2x - 1] / x = -2 [sinx] [sinx / x] As x approaches 0, sinx / x app 1 while 2 sinx app 0; hence the limit is 0.

data is a set of words like a phrase

Yes.

There is an injective function from rational numbers to positive rational numbers*.

Every positive rational number can be written in lowest terms as a/b, so there is an injective function from positive rationals to pairs of positive integers.

The function f(a,b) = a^2 + 2ab + b^2 + a + 3b maps maps every pair of positive integers (a,b) to a unique integer.

So there is an injective function from rationals to integers.

Since every integer is rational, the identity function is an injective function from integers to rationals.

Then By the Cantor-Schroder-Bernstein theorem, there is a bijective function from rationals to integers, so the rationals are countably infinite.

*This is left as an exercise for the reader.

Link under Related Links above this answer.

a lot of other solution manuals of engineering. for free...

5 dived 328 is i don't know

cube

Yes, here's the proof.

Let's start out with the basic inequality 9 < 13 < 16.

Now, we'll take the square root of this inequality:

3 < √13 < 4.

If you subtract all numbers by 3, you get:

0 < √13 - 3 < 1.

If √13 is rational, then it can be expressed as a fraction of two integers, m/n. This next part is the only remotely tricky part of this proof, so pay attention. We're going to assume that m/n is in its most reduced form; i.e., that the value for n is the smallest it can be and still be able to represent √13. Therefore, √13n must be an integer, and n must be the smallest multiple of √13 to make this true. If you don't understand this part, read it again, because this is the heart of the proof.

Now, we're going to multiply √13n by (√13 - 3). This gives 13n - 3√13n. Well, 13n is an integer, and, as we explained above, √13n is also an integer, so 3√13n is an integer too; therefore, 13n - 3√13n is an integer as well. We're going to rearrange this expression to (√13n - 3n)√13 and then set the term (√13n - 3n) equal to p, for simplicity. This gives us the expression √13p, which is equal to 13n - 3√13n, and is an integer.

Remember, from above, that 0 < √13 - 3 < 1.

If we multiply this inequality by n, we get 0 < √13n - 3n < n, or, from what we defined above, 0 < p < n. This means that p < n and thus √13p < √13n. We've already determined that both √13p and √13n are integers, but recall that we said n was the smallest multiple of √13 to yield an integer value. Thus, √13p < √13n is a contradiction; therefore √13 can't be rational and so must be irrational.

Q.E.D.

It is not, so the question is misinformed.

AAS (apex)

Not necessarily. Suppose f(x) is non monotone. Consider g(x) = x - f(x) which will also be monotone if f'(x) <1 for some x and > 1 for some x. Then f(x) + g(x) = x which is strictly monotone. As a simple example, think of f(x) = x^2

odd feet going west. always.

It was in 1875 when a man of Oliver Winston got it on his whole body claiming his life within 2 days. It was in 1875 when a man of Oliver Winston got it on his whole body claiming his life within 2 days.

the same way you would use them if you were not a baker

10/21

First, observe that for the sum of the three balls to be odd, either one is odd and two are even, or all three are odd. Since the sum of two odd numbers is even, the sum of two even number is even, and the sum of an even number and an odd number is odd.

P(exactly one is odd)=(5/9)*(4/8)*(3/7)*3=180/504=5/14

P(all three are odd)=(5/9)*(4/8)*(3/7)=60/504=5/42

P(sum is odd)=5/14+5/42=15/42+5/42=20/42=10/21, which is approximately .48

The starter unit is a complete standalone section of gondola that contains both end uprights. The add-on unit only contains 1 additional upright.

I think the following piecewise function satisfies the two criteria: when x is rational: f(x)=x

when x is irrational: f(x)=x*, where x* is the largest rational number smaller than x.

I think not. When x is irrational, there is no largest rational number less than x. No matter what rational number you pick, there is a larger one that is less than x. For example, between 3.1415926 and pi, there is 3.14159265.

The usual answer is the one given by Weierstrass, which is the sum of an infinite series of functions. The first term in the series is a periodic sawtooth (piecewise linear) function, which is = x from x=0 to x=1, and then descends back to 0 between x=1 and x=2 (i.e., it is = -x+2 in that interval). It repeats that pattern between x=2 and x=4, and so on. The second term is just like it, but with 1/10 the frequency and 1/10 the amplitude, and so on. The first function is continuous everywhere and differentiable except at x= an integer. The sum of the first 2 is continuous everywhere and differentiable except for the multiples of 1/10, and so on. It turns out that the series converges to a function that is continuous everywhere and differentiable nowhere.

By the way, if you can take the derivative of a function at a given point, it is said to be differentiable, not derivable at that point.

Yes. Every perfect square has two roots: one positive and one negative.

There are four requirements that need to be satisfied:

A. Closure: For any two elements of the group, a and b, the operation a*b is also a member of the group.

B. Associativity: For any three members of the group, a*(b*c) = (a*b)*c

C. Identity: There exists an element in the group, called the identity and denoted by i, such that a*i = i*a for all a in the group. For real numbers with multiplication, this element is 1.

D. Inverse: For any member of the group, a, there exists a member of the group, b, such that a*b = b*a = 1 (the identity). b is called the inverse of a and denoted by a-1.

.1 = one tenth

.01 = one hundredth

.001 = one thousandth

So

* .001 is the smallest (one thousandth) * .012 is next (twelve thousandth) * 2 is next * 12 is the biggest.

240

anyway this is our homework in school the importance is to help us to compute Student's Grades and to our computation in business...

No such word. Loom can be a weaving machine or threatening actions coming into light. Do you mean Lupus. That is an auto immune disease that affects humans, especially women..... A german-welsh last name

It wouldn't be what it is unless if it didn't have to be that way

Original data compiled and studied for a specific purpose. For example, a structured survey might be conducted for the purpose of discovering current attitudes on a particular topic; raw survey responses would be primary data.

3x - 7 + 4x = 28

7x - 7 = 28 I added 4x and 3x to get 7x

7x = 35 I subtracted 7 from both sides

x = 5 I divided by 7 on both sides