Question

Derive the propagated errors SG or SGul for the following formulas (x, y, z denote variables and a, b constants) a • G(x) = ax b • G(x) = 1/2 c • G(x) = 1/(x + a) d • G(x) = a log(bx) e • (x) = ax! • G(x) = a exp(-bx) g • G(x, y, z) = xyz h • G(x, y) = xy

Answer 1

Propagated Error for a general function G = 6(x,y,z) can be expressed as * 86 @* (8x)2 + ( )*(su) 39 Joz) ....O Sx, dy, dz are uncertainties in X, Y, Z respectively, @ For G(x) = ax ; We have 26 = a. Hence applying ☺ where G= G(X) we have the propagated error as – 8G - Jea)* (8x)? Va com a (6x) 22 ISG = a (sx) © For Gce) - ; we have, ali os Aplying ® - sa - )?. x)? te exa

© For Gw) cation, we have seen We have he ones 22 Hence, by applying ③ we have → (x+2)4 ² (xera)² ox SG = (x+a)² @ For 6(x) = a log (6x) we have, 26 s by applying ☺ ~ 66 02-82) :: S = a robes ab 2. november (972 - 24 (54) .

© For 6() = axb; vee abxb-- abz 6 (x) Applying ☺ we have → 86 - Cw164)*8w£462) = 60(8) : 36 = benu) where (6 = azº) for 6(x) = aebe 26 = -abe bx = -bG; Aplying → S J 8x- JebG)*(8x) – → 86 - VBG+ (62) 66 (6x) • S = 6G (Sx) | (where 6 = aebe). 2x bt where Gae

for G(xyz) = xyz We have, a teyz , rannya azanary 2. Applying ③ we have → SG • **•*+ 384)*+ ) 62) = N yaz(8x) + 2z89)*7 de zy* (62) • xyz Jellyt + ( - G Net ( S6 = 6 (3x ( ) + ( 32 )² (where 6 = xyz). XN

-1 n = nx' 0 0 ЭЧ for 6(x,y) = x we have, as = y241 - 4 x = nx2-17 24 = xenx = (lnx) 6 : a* (ne)] Applying © → S JC)*8x)* tes yeus = net 2 (Box)* + (8n z)(Sy) a • G (8x++ (na(By)? Hence, saz ve bu) lente)Buja where, (G=x4)