By Harley Flanders

Moment path in Calculus

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Extra resources for A Second Course in Calculus

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This function is certainly continuous for (x, y) ^ (0, 0). Now ,,n' f(x,0) fx(0, 0) = rhm z-»>0 -/(0,0) = rhm n0 =• n0. % x->0 But for x 9^ 0, fx(x, x) = i, so fx is discontinuous at (0, 0). A similar discus­ sion holds for fy. 6. Let fi(a + x,b + y) = /,-(a, b) + J^x + k{y + et-(s, y) for i = 1, 2. /). But [ei(z, 2/) + e2(x, y)]/|x| = ei(z, y)/|x| + e2(>, 2/)/|x| >0 + 0 = O a s x > 0. 8. Let g(a, b) = c, so g(x, y) = c + h{x - a) + fc(y - 6) + e(s, y), where e(z, 2/)/|x — (a, b)\ identity >0 as x > (a, 6).

Paraboloid of revolution *~y CHAPTER 8 Section J, page 287 2. h = / x ( l , 2) = — | , fc = / y ( l , 2) = — J. To compute the error efficiently, start with the one variable expansions 1 = 1 - ( s - 1) +r(x), 1 ( y - 2 ) +«(y). y By a little algebra, r(a:) = (x-1)2 s(y) = (y - 2) 2 42/ Multiply: — xy =2J - 2o (* ~ X) " 47 (2/ "" 2) + e(*' y)> where each term in the numerator of e(x, y) involves (x — l ) 2 or (y — 2) 2 , or (a: — 1) (y — 2), and the denominator is xy. Since (x — l ) 2 / l x ~" (1?

If / = / ( p ) , then g r a d / = /'(p) (p*, py, pz) = p" 1 /'(p)x. By the same formula, applied to p - 1 /'(p), and by Ex. 19, d i v [ g r a d / ( P ) ] = divCp-Tx] = grad(p" 1 /0 x + p" 1 /' divx = [p- 1 (p- 1 /0 , x]-x + 3p-T = p(p" 1 / , ) / + 3 p - r = p-Kpf" - / ' ) + 3P"1/' = / " + 2p~y = p-i(pf)" (or p - W ) ' ) . Section 5, page 306 2. 6, 2, - 6 4. -12, - 4 , -12 6. Vs 8. 6V2 Section 6, page 309 2. F = grad(#32/22), 1 4. grad(cos 0) = grad[x/(x 2 + y2)112], hence ( - s i n 0) (grad 0) = ( - s i n 0) (-y, x)/(x2 + y2).

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