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Steven wrote me a private message asking if I objected to his proposed method of solving. The answer is "No." Perhaps others will find his question as interesting as I do so I am making my more extensive answer public. Being super-pedantic, I do not think his approach is perfect because it...

You clearly read my post #4 because you thanked me for it, and you're welcome. But you did not use it. Substitution is NEVER necessary in differential calculus, but it is frequently very useful. \text {Let } p = (3x - 2)^2 \text { and } q = (2x - 6)^2 \implies\\ p'= 2(3x - 2)(3) = 6p \text {...

You cannot find V as a NUMBER by normal algebraic methods. Why? You have four unknowns, A, G, V, and W but only three equations. It is a sensible first step to go 10G + 5V + 9W = A\\ 4G + 10V + 12W = A\\ 2G + 27W = A. Notice that you did not quite do that first step correctly. But, if you...

First, the problem formulation itself gives you a hint. You are to use the chain rule. That suggests a substirution. Of course it does not sugesst which substitution. Selecting USEFUL substitutions is an art that you develop with experience. Second, I shall give a hint on a helpful...

Let's see if this will help. There is no rule that says a cubic with rational coefficients must have a rational root. An example is f(x) = x^3 - 2. It has a REAL root at x = \sqrt{2}. But that root is an irrational number, and that function's slope is non-negative everywhere. Consequently...

I think that we need to recognize that the vocabulary around numbers and numerals is a product of history and so is sometimes confusing. To prevent confusion, I think what we should say when we NEED to be exact is: A numeral system is a set of symbols, called digits, and rules for combining...

We keep forgetting that the person who submitted this is a HOBBYIST, NOT a student. For those who are not math students, help must take a different form. It is true that the problem as presented does not formally make any mathematical sense. f(x) = \int 2x - 733 \ dx defines a family of...

I am far from sure I understand what the real question is because it is obvious that you can never answer a question about an infinitie number of pairs of numbers by testing every pair. However, the question seems to be to determine some sort of description of or limits on n(b) where n(b) is the...

Odd problem. It is soluble by algebra and arithmetic, but it is quite ugly (I used the rational root theorem to solve the implied quartic). The solution set of a and b is four distinct real pairs. I wonder if the original problem specified integer solutions. I am not sure what the problem is...

First off, let's get the nomenclature straight. y = 4x^2 -12x + 3 is an example of a quadratic function, a rule that relates values of x to a value of y. There are a number of properties that are true of every quadratic function so recognizing that something is a quadratic function gives you...

The reddit thread is wrong. One of the fundamental laws of probability is: \text {P(A or B) = P(A) + P(B) - P(A and B).} So, the desired probability is \text {P(one ace and one 2) + P(one king and one queen) } \\ - \text { P(one ace, one king, one queen, and one 2)}

Almost always, one of the first things I do after seeing one or more fractions is to consider clearing fractions. 2 \ln(x) - 3 = \dfrac{2 \ln(x) + 3}{\ln(x)} \implies 2 \ln(x) * \ln(x) - 3 \ln (x) = 2 \ln(x) + 3 And when I see an ugly-looking expression multiplied by itself, I think...

Start by NAMING things that may be relevant. Let (u, v) identify the the point of tangency for x^2/8 \ - 2. Let w = the slope at (u, v). Let (p, q) identify the the point of tangency for x^2/2 \ - 8. Let r = the slope at (p, q). We are given that u > 0 and p > 0. Let the line coincident...

The initial question from the great khan of khans really needs to be answered. If the domain is integers, the best solution is via a computer table.

If by the set N, we mean the set of all natural numbers, it is infinite, meaning without an upper bound. First, If by n you mean a natural number and by S a set of n consecutive natural numbers, it simply is not true that n is necessarily in S. For example, if n = 3 and S = {6, 7, 8}, S has n...

A couple of thoughts. Basis step n = 0 \implies \sum_{i=0}^n (2i + 1)^2 = (2 * 0 + 1)^2 = 1^2 = 1.\\ \text {And } n = 0 \implies \dfrac{(n + 1)(2 * n + 1)(2n + 3)}{3} =\\ \dfrac{(0 + 1)(2 * 0 + 1) * (2 * 0 + 3)}{3} = \dfrac{1 * 1 * 3}{3} = 1.\\ \therefore \ n = 0 \implies \sum_{i=0}^n (2i +...

It is impossible. Let's take a simpler example. A has a weight of 60% and B has a weight of of 40%. We want numbers that will make the weighted average come out to 48. One possible answer is the number of A is 40 and the number of B is 60. Another possible answer is B is zero and A is 80. To...

I am not sure what you mean by that question, but I would answer that by saying that we can "zoom in" on real and rational numbers in a way that we cannot for natural numbers. There is no such thing as a closest neighbor to a rational or real number. With the natural numbers, you cannot get more...

Here is the most intuitive description as far as I am concerned. Imagine a very small number, so small it is no longer finite and is indistinguishable from zero but is not equal to zero. The difference is too small to measure. Differential calculus is about dividing two such infinitessimals...

A slightly different "angle." The practical use of the trigonometric functions goes back at least as far as the Greek-speaking mathematicians, who used it for practical things like architecture and astronomy. All sorts of angles may arise in the course of a practical application. In general...