z=(5a^3 - (2cm)b^2)/C a= 2.0cm ±1%, b+3.0 cm ±1%, C= 11.0cm±2% using df= v{[(df/

Question

z=(5a^3 - (2cm)b^2)/C

a= 2.0cm ±1%, b+3.0 cm ±1%, C= 11.0cm±2% using df= v{[(df/dx)(dx)]^2 + [(df/dy)(dy)]^2 + (df/dz)(dz)]^2} the we are using only partial derivitives for df/dx, df/dy etc I have the most trouble with these types of problems with the partial derivitives. Could you explain the derivitives part more thoroughly than the rest of the problem? Thanks so much
a= 2.0cm ±1%, b+3.0 cm ±1%, C= 11.0cm±2% using df= v{[(df/dx)(dx)]^2 + [(df/dy)(dy)]^2 + (df/dz)(dz)]^2} the we are using only partial derivitives for df/dx, df/dy etc df= v{[(df/dx)(dx)]^2 + [(df/dy)(dy)]^2 + (df/dz)(dz)]^2} the we are using only partial derivitives for df/dx, df/dy etc (df/dz)(dz)]^2} I have the most trouble with these types of problems with the partial derivitives. Could you explain the derivitives part more thoroughly than the rest of the problem? Thanks so much

Explanation / Answer

z = (5a^3 - 2b^2)/c a = 2.0 cm ± 1%, b = 3.0 cm ± 1%, c = 11.0 cm ± 2% so da = 2.0 * 1% = 0.02 cm, db = 3.0 * 1% = 0.03 cm, dc = 11.0 * 2% = 0.22 cm To find the partial derivative dz/da, assume b and c are constants, so dz/da = 15a^2/c, similarly, dz/db = -4b/c, dz/dc = -(5a^3 - 2b^2)/c^2 use a = 2.0 cm, b = 3.0 cm, c = 11.0 cm dz= sqrt{[(dz/da)(da)]^2 + [(dz/db)(db)]^2 + [(dz/dc)(dc)]^2} = sqrt{[15a^2/c * 0.02]^2 + [-4b/c * 0.03]^2 + [(5a^3 - 2b^2)/c^2 * 0.22]^2} = 0.1207 z = (5a^3 - 2b^2)/c = 2.00 cm^2 dz/z = 0.1207/2.00 = 6.0% answer: 2.00 ± 6.0%

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