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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