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See also the proof of e u du = e u PROOF 2 You need not memorize this theorem Derive it each time you use it Consider this example if you have the integral 2 x dx There is no need to memorize the formula We will get this integral into the easier form, e u du Recall that e ln(2) = 2 2 x dx = ( e ln (2)) x dx = e ln (2) x dx set u = ln(2) x. DYaåchgr cV gXdä \^cr ^ g`V\h X ^ibac^^ «U chdh madX`, `V`^b Wqa fVcrn U cVim^agå kdZ^hr X @ik, shd edacdghrä ^ cVXgYZV ^bc^ad bdä \^cr!» $ 9 E8 GHED 6 7 М ое мД ух гл б ина др ст , чтобы Духом Моим ты мог приобретать знание, говорит Дух Благодати. # F « ª e O \ K % 1 !.

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For positive m and nonnegative kThe theorem is Every nonnegative integer is the sum of m2 polygonal numbers of order m2 When m=0, u 0 (k)=kAnd with nonnegative k we trivially have f(NxN)=N when f(x,y)=xyEven simpler is f(N)=N when f(x)=x But the open question is for inputs both positive and negative. Â 5 Ê ³ % é W ÿ ó ê ° ¥ v 4 ´ k g 0 › é 6 W < § * ‘ z € m P Ý « G Ä š › ä < A ` Ñ v ˜ ç “ Ê ¡ Q !. Oh, so what they want you to do is as follows a For f(g(x) you take the f(x) equation, except, instead of the "x"s, you put in the "g(x)" equation.

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Solution Any diagonal n n matrix looks like 0 BBB BBB BBB B@ a1 an 1 CCC CCC CCC CA = a1E11 anEnn where Eii is the matrix with entries all 0 except a 1 at the i’th diagonal entry This tells us that (E11;. SOLUTION 13 Begin with x 2 xy y 2 = 1 Differentiate both sides of the equation, getting D ( x 2 xy y 2) = D ( 1 ) , 2x ( xy' (1)y) 2 y y' = 0 , so that (Now solve for y' ) xy' 2 y y' = 2x y, (Factor out y' ) y' x 2y = 2 x y, and the first derivative as a function of x and y is (Equation 1). ( 6 ) u ~tjfs q f¶6 g f ;.

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Ae Aeœ E µe A Ae C Aºœae Ae Aº E Vicjuan S A Aººa A Cs A C

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Ae Aeœ E µe A Ae C Aºœae Ae Aº E Vicjuan S A Aººa A Cs A C

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