Maximum Percent Error Calculus
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Whole Number Place Value of Whole Numbers Rounding Whole Numbers Whole Numbers on a using differentials to estimate error Number Line Comparing Whole Numbers Adding Whole Numbers Subtracting use differentials to estimate the maximum error in the calculated volume Whole Numbers Multiplying Whole Numbers Multiplication Table Dividing Whole Numbers (Long Division) Division with use differentials to estimate the maximum error in the calculated surface area Remainder Integers > Negative Numbers What is Integer Number Rounding Integers Number Line with Integers Ordering and Comparing Integers Adding Integers Adding Integers relative error differentials on a Number Line Subtracting Integers Subtracting Integers on a Number Line Multiplying Integers Dividing Integers Exponents and Integers Factors and Multiples > Divisibility of Integers Even Numbers (Integers) Odd Numbers (Integers) Divisibility Rules What are Factors and Multiples Integer Factorization What is a Prime Number
Maximum Error Formula
Composite Numbers How do you do Prime Factorization Greatest Common Divisor (GCD) Least Common Multiple (LCM) Fractions > What is Fraction Proper Fractions Improper Fractions Mixed Numbers/Fractions Mixed Numbers on a Number Line Equivalent Fractions Reducing Fractions Adding Fractions with Like Denominators Subtracting Fractions with Like Denominators Adding Fractions with Unlike Denominators Subtracting Fractions with Unlike Denominators Converting Mixed Numbers to Improper Fractions Converting Improper Fractions to Mixed Numbers Adding Fractions with Whole Numbers Subtracting Fractions with Whole Numbers Adding Mixed Numbers Subtracting Mixed Numbers Comparing Fractions Multiplying Fractions Multiplying Mixed Numbers Dividing Fractions by Whole Number Dividing Fractions Dividing Mixed Numbers Reciprocals Negative Exponents Rational Numbers Decimals > What is Decimal Decimals Place Value Rounding Decimals Decimal Number Line Comparing Decimals Powers of 10 Scientific Notation Decimal Fractions Converting Decimals To Fractions Converting Fractions to Dec
with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations in algebraic
Use Differentials To Estimate The Maximum Error In The Calculated Area Of The Rectangle
context. At this mathematical level our presentation can be briefer. We how to calculate percent error in volume can dispense with the tedious explanations and elaborations of previous chapters. 6.2 THE CHAIN RULE AND DETERMINATE differentials calculus ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: [6-1] ∂R ∂R ∂R dR = http://www.emathhelp.net/notes/calculus-1/differentials/using-differentials-to-estimate-errors/ —— dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation to DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations. They are also called determinate error equations, because they are strictly valid for determinate errors (not indeterminate errors). [We'll get to indeterminate eSolutions 1. Approximations If a quantity x (eg, side of a square) is obtained by measurement and a quantity y (eg, area of the square) is calculated as a function of x, say y = f(x), then any error involved in http://www.phengkimving.com/calc_of_one_real_var/08_app_of_the_der_part_2/08_04_approx_of_err_in_measrmnt.htm the measurement of x produces an error in the calculated value of y as well. Recall from Section 4.3 Part 2 that the Section 8.3 Part 1, we have: That is, the error in x is dx and the corresponding approximate error in y is dy = f '(x) dx. Fig. 1.1 Fig. 1.2 – 1st and 2nd axes: if 1,000 = xa – 1 then xa = 1,001, – differentials to 1st and 3rd axes: if 1,000 = xa + 1 then xa = 999, therefore xa is somewhere in [999, 1,001]. Example 1.1 Solution Let s be the side and A the area of the square. Then A = s2. The error of the side is ds = 1 m. The approximate error of the calculated area is: dA = 2s ds = 2(1,000)(1) = 2,000 m2. EOS Note that we calculate dA from the differentials to estimate equation A = s2, since the values of s and ds are given. To find the differential of A we must have an equation relating A to s. So even if the measured value of the side is given we still define the variable s that takes on as a value the measured value. In general, when the measured value say V of a quantity and the error say E in the measurement are given, we define a variable say x for the quantity, so that x = V and dx = E, which will be used later on in the solution. When using the quantity, first use the variable x, not the value V, then use the value V when a value is to be obtained. Go To Problems & Solutions Return To Top Of Page 2. Types Of Errors A measurement of distance d1 yields d1 = 100 m with an error of 1 m. A measurement of distance d2 yields d2 = 1,000 m with an error of 1 m. Both measurements have the same absolute error of 1 m. However, intuitively we feel that measurement of d2 has a smaller error because it's 10 times larger and yet has the same absolute error. Clearly the effect of 1 m out of 1,000 m is smaller than that of 1 m out of 100 m. This leads us to consider an error relative to the size of the quantity being expressed. This r
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