Dear TRS: Reality sets in

That's right folks, it's time for another exciting edition of ask the RS. Incidentally, all of these are real questions, sent in by real people. Scary, I know. Here we go:

Dear Sir:

Can you think of an English rhyme to aid baalei tshuva in understanding the talmudic and practical concept of "der oilam's ah goilem?"
sincerely,

Shliach to the poets-in-residence
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Dear Shliach to the poets-in-residence,

Are you trying to suggest that the masses are asses? And not even the asses of Rebbi Pinchas Ben Yair?
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Dear Sir:
Is the prohibition of music in effect on hallel days that occur during sfira? If so does yom ha'atzmaut (known to certain chagas kreizin as yoim huhhatzomos) count?

What about stuff that is technically produced on instruments but is not music by any sane definition? (Insert singer/group you dislike)
sincerely,

Metal Fan
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Dear Metal Fan,

The answer to your first question is that if you're willing to suspend rational thought and hold that you can't listen to whatever it is you're not listening to then the same would apply on hallel days as well, obviously excepting the 5th of Iyar, when it's a mitzvah to play Hatikvah on loudspeakers from a Mitzvah Tank in Williamsburg.

Any music which is not music by any sane definition shouldn't be listened to anyway, so what's the question? You mean if you're of a snaggish disposition and have masochistic tendencies?
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Dear Sir:

What is the fundamental question of integral calculus and what does it have to do with differential calculus? And oh yeah, what is the point of that little carrot shaped button on my blue calculator?

Sincerely,

Mathematically Mystified
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Dear Mathematically Mystified,

I.

Let f be a continuous real-valued function defined on a closed interval a,">b. If F is the function defined for x in a,">b by

$F(x)\; =\; \backslash int\_a^x\; f(t)\backslash ,\; dt$

then

$F\text{'}(x)\; =\; f(x)\backslash ,$

for every x in a,">b.

II.

Let F be a continuous real-valued function defined on a closed interval a,">b. If f is the function such that

$f(x)\; =\; F\text{'}(x)\backslash ,$ for all x in a,">b

then

$\backslash int\_a^b\; f(x)\; dx\; =\; F(b)\; -\; F(a)$.

Corollary

Let F be a real-valued continuous function defined on a closed interval a,">b. If f is the function defined by

$f(x)\; =\; F\text{'}(x)\backslash ,$ for all x in a,">b

then

$F(x)\; =\; \backslash int\_a^x\; f(t)\; dt\; +\; F(a)$

and

$f(x)\; =\; \backslash frac\{d\}\{dx\}\; \backslash int\_a^x\; f(x)\; dx$.

Proof

Part I

It is given that

$F(x)\; =\; \backslash int\_\{a\}^\{x\}\; f(t)\; dt$

Let there be two numbers x₁ and x₁ + Δx in a,">b. So we have

$F(x\_1)\; =\; \backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt$

and

$F(x\_1\; +\; \backslash Delta\; x)\; =\; \backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt$.

Subtracting the two equations gives

$F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\; =\; \backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; -\; \backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt\; \backslash qquad\; (1)$.

It can be shown that

$\backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt\; +\; \backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; =\; \backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt$.

(The sum of the areas of two adjacent regions is equal to the area of both regions combined.)

Manipulating this equation gives

$\backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; -\; \backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt\; =\; \backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt$.

Substituting the above into (1) results in

$F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\; =\; \backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; \backslash qquad\; (2)$.

According to the mean value theorem for integration, there exists a c in x₁,">x₁ + Δx such that

$\backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; =\; f(c)\; \backslash Delta\; x$.

Substituting the above into (2) we get

$F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\; =\; f(c)\; \backslash Delta\; x\; \backslash ,$.

Dividing both sides by Δx gives

$\backslash frac\{F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\}\{\backslash Delta\; x\}\; =\; f(c)$.

Notice that the expression on the left side of the equation is Newton's difference quotient for F at x₁.

Take the limit as Δx → 0 on both sides of the equation.

$\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\}\{\backslash Delta\; x\}\; =\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; f(c)$

The expression on the left side of the equation is the definition of the derivative of F at x₁.

$F\text{'}(x\_1)\; =\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; f(c)\; \backslash qquad\; (3)$.

To find the other limit, we will use the squeeze theorem. The number c is in the interval x₁,">x₁ + Δx, so x₁ ≤ c ≤ x₁ + Δx. Also, $\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; x\_1\; =\; x\_1$ and $\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; x\_1\; +\; \backslash Delta\; x\; =\; x\_1$. Therefore, according to the squeeze theorem,

$\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; c\; =\; x\_1$.

Substituting into (3), we get

$F\text{'}(x\_1)\; =\; \backslash lim\_\{c\; \backslash to\; x\_1\}\; f(c)$.

The function f is continuous at c, so the limit can be taken inside the function. Therefore, we get

$F\text{'}(x\_1)\; =\; f(x\_1)\; \backslash ,$.

which completes the proof. (Leithold et al, 1996)

Part II

This is a limit proof by Riemann Sums. Let f be continuous on the interval a,">b, and let F be an antiderivative of f. Begin with the quantity

$F(b)\; -\; F(a)$.

Let there be numbers x₁ thru x_n such that

$F(b)\; -\; F(a)\; =\; F(x\_n)\; -\; F(x\_0)\; \backslash ,$.

Now, we add each F(x_i) along with its additive inverse, so that the resulting quantity is equal:

& = & F(x_n)\,-\,F(x_{n-1})\,+\,F(x_{n-1})\,+\,\ldots\,-\,F(x_1)\,+\,F(x_1)\,-\,F(x_0) \, \end{matrix} The above quantity can be written as the following sum:

$F(b)\; -\; F(a)\; =\; \backslash sum\_\{i=1\}^n-\; F(x\_\{i-1\})\backslash qquad\; (1)$

Here we employ the Mean Value Theorem. In brief, it is as follows:

Let f be continuous on the closed interval a,">b and differentiable on the open interval (a, b). Then there exists some c in (a, b) such that

$f\text{'}(c)\; =\; \backslash frac\{f(b)\; -\; f(a)\}\{b\; -\; a\}$.

It follows that

$f\text{'}(c)(b\; -\; a)\; =\; f(b)\; -\; f(a)\; \backslash ,$.

The function F is differentiable on the interval a,">b; therefore, it is also differentiable and continuous on each interval x_i-1. Therefore, according to the Mean Value Theorem (above),

$F(x\_i)\; -\; F(x\_\{i-1\})\; =\; F\text{'}(c\_i)(x\_i\; -\; x\_\{i-1\})\; \backslash ,$.

Substituting the above into (1), we get

$F(b)\; -\; F(a)\; =\; \backslash sum\_\{i=1\}^n-\; x\_\{i-1\})$.

As the derivative of the antiderivative is the original function, $F\text{'}(c\_i)\; =\; f(c\_i)$. Also, $x\_i\; -\; x\_\{i-1\}$ can be expressed as $\backslash Delta\; x$ of partition $i$.

$F(b)\; -\; F(a)\; =\; \backslash sum\_\{i=1\}^nx\_i)\backslash qquad\; (2)$

Notice that we are describing the area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the Mean Value Theorem, describes an approximation of the curve section it is drawn over. Also notice that $\backslash Delta\; x\_i$ does not need to be the same for any value of $i$, or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with $n$ rectangles. Now, as the size of the partitions get smaller and n increases, resulting in more partitions to cover the space, we will get closer and closer to the actual area of the curve. By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity. So, we take the limit on both sides of (3). This gives us

$\backslash lim\_\{\backslash |\; \backslash Delta\; \backslash |\; \backslash to\; 0\}\; F(b)\; -\; F(a)\; =\; \backslash lim\_\{\backslash |\; \backslash Delta\; \backslash |\; \backslash to\; 0\}\; \backslash sum\_\{i=1\}^nx\_i)\backslash ,dx$

The expressions F(b) and F(a) are not dependent on ||Δ||, so the limit on the left side remains F(b) - F(a).

$F(b)\; -\; F(a)\; =\; \backslash lim\_\{\backslash |\; \backslash Delta\; \backslash |\; \backslash to\; 0\}\; \backslash sum\_\{i=1\}^nx\_i)$

The expression on the right side of the equation defines an integral over f from a to b. Therefore, we obtain

$F(b)\; -\; F(a)\; =\; \backslash int\_\{a\}^\{b\}\; f(x)\backslash ,dx$

which completes the proof.

Happy?

In answer to your second question, "What's up Doc?"
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Dear Sir:

My recently engaged friend was appointed to oversee all dorm-food preparation and thus is at her wits end. How might I go about telling her that raw eggs mixed with mushy bananas doesn't create ice cream, without losing my head?

Sincerely,

A'feared of Wrath

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Dear A'feared of Wrath

Just give her this simple recipe, and mention that TRS is now accepting donations:

Blend 6 very ripe bananas until smooth. Add 2 egg yolks and 3/4 cup of ground nuts, filberts, or almonds. Blend for one minute. Beat 4 egg whites until stiff. Mix slowly into banana mixture with wooden spoon. Pour into a glass dish and spread remaining 1/4 cup nuts on top. Freeze at least 4 hours. When serving poor 1 teaspoon wine on each scoop or portion.

Serves 12
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Dear Sir:

My brother insists on not cutting his afro and I fear that by the end of sefira all hope, not to mention sight, will be lost. Do you think he's mind if I trim it during the night? Then again, it would make for a lovely sheitel...

Sincerely,

Distressed
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Dear Distressed,

Do I think he'll mind if you sin against all that is holiness and sacred? Of course, locks for love is a wonderful organization.
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Dear Sir:

What's a Chol Hamoed trip that will be enjoyable for a family aged 1.5 through 55?

Sincerely,

Even more desperate
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Dear Even more desperate,

What exactly do 1.5 year olds enjoy doing anyway? From what I can tell, they're the same happy whether here or there.
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Dear Sir:

Why is borscht in America made of beets and borscht in Russia made with cabbage?

Sincerely,

Ari
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Dear Ari,

Perhaps because Americans prefer sour cream nine times out of ten?
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And that's all folks!