User:Mathfantasia

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Divisibility rules: for 7

Rule 3: "Tail-End divisibility." Note. This only tells you if it is divisible and NOT the remainder. Take a number say 12345. Look at the last digit and add or subtract a multiple of 7 to make it zero. In this case we get 12370 or 12310. Lop off the ending 0's and repeat. 1237 ==> 123 ==> 130 ==> 13 NOPE. Works in general with numbers that are relatively prime to the base (and works GREAT in binary).

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Estimating Compound Interest

Look up and understand the "Rule of 72." My short synopsis here says that certain small percentages, say n%, will double in "72 divided by n" years.

I will now estimate $2^x$ between x=0 and x=1 by forming segments between the previous x values and x=.5. So for x<.5 we use 1 + .8x and otherwise 1.2x + .8.

I will now give an example of this method!

6% compounded yearly for 4 years with principal 20000. We see that 6% interest will make the principal double every 12 years (by the law of 72). But we only need 4 years, so with no more estimation we would need to know $2^{1/3}$ . We estimate this with the above linear equations. So we want to multiply the principal, 20000, by ( $1 + .8/3$ ). We end up getting 25336. And there ya go! Actual answer: 25250 (less than 1% off ... actually .3%)

LOL needed a place to latex on the go. Putting up some calculus notes.

We wish to find $\int^2_1 x^2dx$ . Recall that the formula for integration is $\int^b_a f(x)dx = \sum_{i=1}^n (height)(width) = \sum_{i=1}^n f(x_i)((b-a)/2)$ . So $x_i= a +\frac{(b-a)i}{n}$ = 1 + \frac{i}{n}. Width = $\frac{i}{n}$ . $f(x_i) = (1 + \frac{i}{n})^2$

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User:Mathfantasia

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