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Introduction to Calculus-based Error Calculations for GE226
Purpose of Error Calculations
To determine the maximum possible error in any calculated quantity based upon the estimated errors in the measured data used to calculate the quantity.
You may encounter more sophisticated statistical techniques of error calculations in senior courses.
Calculus Method
The Calculus method of Error Calculations uses Differentials of mathematical expressions which is similar but not identical to the Derivatives of the same mathematical expressions.
Basic Rules of Derivatives and Differentials
1a) Derivative Exponent Rule
1b) Differential Exponent Rule
1)( −= nn ncxcxdxd
dxncxcxd nn 1)( −=
Basic Rules of Derivatives and Differentials
2a) Derivative Addition/Subtraction Rule
2b) Differential Addition/Subtraction Rule
dxdw
dxdv
dxduwvu
dxd
−+=−+ )(
dwdvduwvud −+=−+ )(
Basic Rules of Derivatives and Differentials
3a) Derivative Product Rule
3b) Differential Product Rule
dxdwuv
dxdvuw
dxduvwuvw
dxd
++=)(
uvdwuwdvvwduuvwd ++=)(
Basic Rules of Derivatives and Differentials
4a) Derivative Quotient Rule
4b) Differential Quotient Rule
2)/()/()(
vdxdvudxduv
vu
dxd −
=
2)()()(
vdvuduv
vud −
=
Tip – Avoid Quotient Rule
For error calculation purposes, the quotient rule is clumsy.
It is better to rewrite original expression as a product and use product and exponent rules.
e.g.
dvuvvduuvdvud 211 )1()()()( −−− −+==
Basic Rules of Derivatives and Differentials
5a) Derivative Chain Rule
5b) Differential Chain Rule
dxd
dxd θθθ )(cos)(sin =
θθθ dd )(cos)(sin =
Error Calculation Procedure
Step 1a) Must ensure that Original Expression is in Simplest Form with each variable appearing as few times as possible
Simplify: Y=am2/m to Y = am Simplify: Z = ax-bx to Z = (a-b)x
Error Calculation Procedure
Step 1b) Ensure that each variable is independent of all other variables in original expression
Independent variables are measured separately
If y = ax and Z = yx, then y and x are NOT independent
Solution: Substitute y=ax into Z equation, Z = (ax)x = ax2
Error Calculation Procedure
Step 2) Using rules from calculus, calculate the Differential of the Original Expression. (See previous calculus summary.)
If K = (1/2)mv2, dK = (1/2)(dm)v2 + (1/2)m(2v)(dv) or to simplify, dK = (1/2)(dm)v2 + mv(dv)
Error Calculation Procedure
Step 3) To avoid having to repeatedly insert both positive and negative error values into error equations, take the absolute value of each term and always ADD terms (even if calculus rules tell you to subtract terms)
dK = |(1/2)(dm)v2| + |mv(dv)|
Error Calculation Procedure
Step 4) Replace each differential (dx) with the corresponding absolute error (δx)
δK = |(1/2)(δm)v2| + |mv(δv)|
Error Calculation Procedure
Step 5) When angles are used, express the absolute error in RADIANS
You should keep the Angle in Degrees. Example: If θ = 30° ± 1°, and
Y = sin θ, δY = |cos (θ) δθ |= |cos(30°) (π / 180)|
Tip for Constants
Treat Constants as variables and then set error in constant equal to 0
Error Formula Check
Error Formulae should ALWAYS have one and only one error factor (δx, δm, etc) in each term
Error Factors should ALWAYS appear to exponent 1 (Error Factors are NEVER in the denominator and NEVER raised to a power other than 1 and NEVER appear under a square root.)
Sample Problem
Assume, m = 30 kg ± 1 kg v = 10 m/s ± 1 m/s
Given: Momentum: P = mv
Find error in P
Sample Problem Solution
Assume, m = 30 kg ± 1 kg , v = 10 m/s ± 1 m/s
Find error in P = mv dP = (dm)v + m (dv) δP = |(δm)v| + |m (δv)| δP = |v(δm)| + |m (δv)| δP = |(10 m/s)(1 kg)| + |(30 kg) (1 m/s)| δP = 10 kg m/s + 30 kg m/s δP = 40 kg m/s
Check Units
Always check that each term of an error equation has the same units.
If not, your error formula is wrong.
Dummy Variable Technique for complicated formulae If you need to determine an error calculation
for a complicated formula, then use dummy variables to represent part of the equation.
Work out the differential for each part and then substitute everything back into the original to obtain the final error equation.
Example of Dummy Variable Technique
1) Find error equation for “complex” formula Y = A(B-C) where A, B and C are variables [This formula is complex because calculus has different rules for multiplication and subtraction.]
2) Let Dummy Variable D = B-C 3) Then: Y = AD 4) Find differential: dD = dB – dC 5) Error formula for D: δD = |δB| + |δC|
continued next slide
Example of Dummy Variable Technique
6) Find differential of Y: dY = (dA)D + A(dD) 7) Error Formula for Y: δY = |(δA)D| + |A(δD)| 8) Substitute: δY = |(δA)(B-C)| + |A(|δB| + |δC|)|
9) Expand so that each term has only 1 error factor: δY = |(δA)(B-C)| + |A(δB)| + |A(δC)|
Error Calculations in Lab Reports
You do NOT need to show the derivation of any error formulae. With practice, you can do that in your head.
You need to document one sample error calculation for each formula using following format.
Error Calculation Format
Descriptive Title of Error Calculation State Formula with variables State Formula with substitution of numerical
values and units State Calculation Result with units to
appropriate number of significant figures.
Sample Error Calculation Format
Sample Error Calculation of Initial Momentum δP = VδM + MδV δP = (3.2 m/s)(0.0001 kg) + (0.1234 kg)(0.2 m/s) δP = 3 X 10-2 kg m/s